3.2: Ancient Greece
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Learning Objectives
By the end of this section, you will be able to:
- Identify the historical factors that shaped the development of the Greek city-state
- Describe the evolution of the political, economic, and social systems of Athens and Sparta
- Discuss the alliances and hostilities among the Greek city-states during the Classical period
- Identify the major accomplishments of Ancient Greek philosophy, literature, and art
In the centuries following the collapse of the Bronze Age Mycenaean kingdoms around 1100 BCE, a dynamic new culture evolved in Iron Age Greece and the Aegean region. During this period, the Greek city-states developed innovative consensual governments. Free adult males participated in their own governance and voted to create laws and impose taxes. This system of government contrasted with the earlier monarchies of the ancient Near East, in which rulers claimed to govern their subjects through the will of the gods.
The degree of political participation in the Greek city-states varied from monarchy and oligarchy, or government by a small group of wealthy elites, to democracy , literally “rule by the people,” a broader-based participation that eventually included both rich and poor adult males. These systems influenced Ancient Roman and European political thought through the centuries. The Greek Classical period (500–323 BCE) witnessed constant warfare among rival city-states, yet it was marked by the creation of enduring works of literature and art that inspired centuries of European artists and writers. Greek philosophers also subjected the human condition and the natural world to rational analysis, rejecting traditional beliefs and sacred myths.
Archaic Greece
The Greek Dark Ages (1100–800 BCE) persisted after the collapse of the Mycenaean civilization but began to recede around 800 BCE. From this point and for the next few centuries, Greece experienced a revival in which a unique and vibrant culture emerged and evolved into what we recognize today as Classical Greek civilization. This era, from 800 to 500 BCE, is called Archaic Greece after arche , Greek for “beginning.”
The Greek renaissance was marked by rapid population growth and the organization of valleys and islands into independent city-states, each known as a polis (Greek for city-state). Towns arose around a hill fortress or acropolis to which inhabitants could flee in times of danger. Each polis had its own government and religious cults, and each built monumental temples for the gods, such as the temple of Hera, wife of Zeus and protector of marriage and the home, at the city-state of Argos. Though politically disunited, the Greeks, who began to refer to themselves as Hellenes after the mythical king Hellen, did share a common language and religion. The most famous of their sacred sites were Delphi , near Mount Parnassus in central Greece and seat of the oracle of Apollo , the god of prophecy, and Olympia in southern Greece, sacred to Zeus, who ruled the pantheon of gods at Mount Olympus (Figure 6.9). Beginning in 776 BCE, according to Aristotle , Greeks traveled to Olympia every four years to compete in athletic contests in Zeus’s honor, the origin of the Olympic Games .
The Past Meets the Present
The Olympic Games
Postponed a year because of the COVID-19 pandemic, the 2021 Games of the XXXII Olympiad in Japan included more than three hundred events in thirty-three sports, including new entries like skateboarding, rock climbing, and surfing. Modern games have been held since 1896, when the new International Olympic Committee started the tradition, but as the name suggests, the inspiration came from Ancient Greece.
Athletic events in Ancient Greece were important displays of strength and endurance. There were contests at the sanctuaries at Delphi and Nemea (near Argos), but none was as renowned as the Olympic Games, held at the sanctuary in Olympia that was dedicated to Zeus. Contestants came from all over the Greek world, including Sicily and southern Italy.
Unlike the skateboarding and surfing of modern games, the ancient games focused on skills necessary for war: running, jumping, throwing, and wrestling. Over time, sports that included horses, like chariot racing, were also incorporated. Such events were referenced in Homer’s Iliad , when the hero Achilles held athletic contests to honor his fallen comrade Patroclus and awarded prizes or athla (from which the word “athlete” is derived). The centerpiece of the ancient games was the two-hundred-yard sprint, or stadion , from which comes the modern word “stadium” (Figure 6.10).
Unlike the modern games, where attendees pay great sums to watch athletes compete, admission to the ancient games was free—for men. Women were forbidden from watching and, if they dared to attend, could pay with their lives. Competitors were likely locals with proven abilities, though over time professional athletes came to dominate the sport. They could earn a good living from prizes and other rewards gained through their talent and celebrity, and their statues adorned the sanctuary at Olympia. The poet Pindar in the early fifth century BCE was renowned for composing songs to honor them when they returned home as victors. The Olympic Games continued to be celebrated until 393 CE, when they were halted during the reign of the Christian Roman emperor Theodosius .
- Why might the organizers of the modern Olympic Games have named their contest after the ancient Greek version?
- How are the ancient games similar to the modern Olympic Games? How are they different?
The start of the Archaic period also witnessed the reemergence of specialization in Greek society. Greek artists became more sophisticated and skilled in their work. They often copied artistic styles from Egypt and Phoenicia , where Greek merchants were engaging in long-distance trade. At the site of Al-Mina , along the Mediterranean coast in Syria where historians believe the Phoenician alphabet was first transmitted to the Greeks, Greek and Phoenician merchants exchanged goods. Far to the west, on the island of Ischia off the west coast of Italy, Greeks were competing with Phoenician merchants for trade with local peoples, whose iron ore was in strong demand. Thanks to their contact and trade with the Phoenicians, Greeks adapted the Phoenician alphabet to their own language, making an important innovation by adding vowels (a, e, i, o, u). The eighth century BCE thus witnessed the return of literacy and the end of the Aegean world’s relative isolation after the interlude of the Greek Dark Ages .
The eighth century BCE was also the period in which the epic poems the Iliad and the Odyssey were composed, traditionally attributed to the blind poet Homer . While historians debate whether Homer was a historical or a legendary figure, they agree the epics originated in the songs of oral poets in the Greek Dark Ages. In the eighth century BCE, using the Greek alphabet, scribes wrote these stories down for the first time.
As the population expanded during the Archaic period, a shortage of farmland brought dramatic changes. Many Greeks in search of land to farm left their homes and founded colonies along the shores of the Black Sea and the northern Aegean, in North Africa at Cyrene in Libya, and in southern Gaul (modern France) at Massalia (Marseille). The largest number were on the island of Sicily and in southern Italy, the region the Greeks referred to as Magna Graecia or “Greater Greece.” When Greeks established a colony, it became an independent polis with its own laws. The free adult males of the community divided the colony’s land into equal lots. Thus, a new idea developed in the colonies that citizenship in a community was associated with equality and participation in the governing of the state.
In the society of Archaic Greece , the elite landowners, or aristoi , traditionally controlled the government and the priesthoods in the city-states. But thanks to the new ideas from the colonies, the common people, or kakoi , began demanding land and a voice in the governing of the polis. They were able to gain leverage in these negotiations because city-state s needed troops in their wars for control of farmland. The nobility relied on the wealthier commoners, who could afford to equip themselves with iron weapons and armor. In some city-states, the aristoi and the kakoi were not able to resolve their differences peaceably. In such cases, a man who had strong popular support in the city would seize power and rule over the city. The Greeks referred to such populist leaders as tyrant s.
In the sixth century BCE, the difficulties caused by the land shortage were relieved by the invention of coinage . A century before, adopting a practice of the kings of Lydia in western Asia Minor (Turkey), Athens stamped silver pieces with the image of an owl, a symbol of wisdom often associated with the goddess Athena (See Figure 6.11). Instead of weighing precious metals to use as currency or arguing over the value of bartered goods to trade, merchants could use coins as a simple medium of exchange. The agora , or place of assembly in each city-state, thus became a marketplace to buy and sell goods. In the sixth century BCE, this rise of a market economy stimulated economic growth as farmers, artisans, and merchants discovered stronger incentives to produce and procure more goods for profit. For example, farmers learned how to produce more food with the land they already possessed rather than always seeking more land. The economic growth of this period is reflected in the many new temples the Greek city-states constructed then.
Sparta and Athens
In the Archaic period, Athens and Sparta emerged as two of the most important of the many Greek city-states. Not only did their governments and cultures dominate the Greek world in the subsequent Classical period; they also fired the imaginations of Western cultures for centuries to come. Athens was the birthplace of democracy , whereas Sparta was an oligarchy headed by two kings.
The Rise and Organization of Sparta
Sparta in the eighth century BCE was a collection of five villages in Laconia , a mountain valley in the Peloponnese in southern Greece. Due to the shortage of farmland, the citizens (adult males) of these villages, the Spartiates , all served in the military and waged war on neighboring towns, forcing them to pay tribute. The Spartiates also appropriated farmland for themselves and enslaved the inhabitants of these lands, most famously the Messenians, who became known as the helots. Just as Greek colonists at this time divided land among themselves into equal lots, the Spartiates likewise divided the conquered land equally and assigned to each landowner a certain number of helot families to work it. Helots, unlike enslaved people in other parts of Greece, could not be bought or sold but remained on the land as forced laborers from generation to generation. In the seventh century BCE, Sparta conquered the land of Messene to its west and divided its farmland equally among the Spartiates.
By the late sixth century BCE, the wealth from the rich agricultural land that Sparta then controlled had made it the most powerful state in the Peloponnese. Sparta also organized the city-states of this region and parts beyond into a system of alliances that historians refer to as the Peloponnesian League . Its members still had self-government and paid no tribute to Sparta, but all were expected to have the same friends and enemies as Sparta, which maintained its dominance in the league. Sparta also used its army to overthrow tyrants in the Peloponnesian city-states and restore political power to the aristoi .
The Spartans were proud of their unique system of government, or constitution, which was a set of laws and traditional political practices rather than a single document. It was said to have been created by a great lawgiver named Lycurgus around 800 BCE, but modern historians view its development as an evolutionary process during the Archaic period rather than the work of a single person.
Sparta had two hereditary kings drawn from rival royal families. Their powers were very limited, though both sat as permanent members of the Council of Elders and were priests in the state religion. On occasion, the Spartan kings also led armies into battle. The Assembly of Spartiates passed all laws and approved all treaties with the advice of the Council of Elders . This Assembly also elected five judges every year who administered the affairs of state, as well as the members of the Council of Elders.
The unique element of Spartan culture was the agoge , its educational system. At the age of seven, boys were separated from their families and raised by the state. To teach them to live by their wits and courage, they were fed very little so they had to learn how to steal food to survive. At the age of twelve they began an even more severe regimen. They were not allowed clothes except a cloak in the wintertime, and they bathed just once a year (Figure 6.12). They also underwent ritual beatings intended to make them physically strong and hardened warriors. At the age of eighteen, young men began two years of intense military training. At the age of twenty, a young Spartan man’s education was complete.
Women of the Spartiate class, before marrying in their mid-teens, also practiced a strict physical regimen, since they were expected to be as strong as their male relatives and husbands and even participate in defending the homeland. Spartan women enjoyed a reputation for independence, since they managed the farms while men were constantly training for or at war and often ran their family estates alone due to the early deaths of their soldier husbands. The state organized unmarried women into teams known as chorai (from which the term chorus is derived) that danced and sang at religious festivals.
When a Spartiate man reached the age of thirty, he could marry, vote in the Assembly, and serve as a judge. Each Spartiate remained in the army reserve until the age of sixty, when he could finally retire from military service and became eligible for election to the Council of Elders. Spartan citizens were proud to devote their time to the service of the state in the military and government; they did not have to work the land or learn a trade since this work was done for them by commoners and helot subjects.
The Rise and Organization of Athens
Athens , like Sparta, developed its own system of government in the Archaic period. Uniquely large among Greek city-states, Athens had long enclosed all the land of Attica , which included several mountain valleys. It was able to eventually develop into a militarily powerful democratic state in which all adult male citizens could participate in government, though “citizenship” was a restricted concept, and because only males could participate, it was by nature a limited democracy .
The roots of Athenian democracy are long and deep, however, and its democratic institutions evolved over centuries before reaching their fullest expression in the fifth century BCE. It was likely the growing prosperity of Athenians in the eighth century that had set Athens on this path. As more families became prosperous, they demanded greater say in the functioning of the city-state. By the seventh century BCE, Athens had an assembly allowing citizens (free adult males) to gather and discuss the affairs of the state. However, as the rising prosperity of Athenians stalled and economic hardship loomed by the end of the century, the durability of the fledgling democracy seemed in doubt. Attempts to solve the economic problems by adjusting the legal code, most notably by the legislator Draco (from whose name we get the modern term “draconian”), had little effect, though codifying the law in written form brought more clarity to the legal system.
With the once-thriving middle class slipping into bankruptcy and sometimes slavery, civil war seemed inevitable. Disaster was avoided only with the appointment of Solon in 594 BCE to restore order. Solon came from a wealthy elite family, but he made it known that he would draft laws to benefit all Athenians, rich and poor. A poet, he used his songs to convey his ideas for these new laws (Figure 6.13).
One of Solon’s first measures was to declare that all debts Athenians owed one another were forgiven. Solon also made it law that no Athenian could be sold into slavery for failure to repay a loan. These decrees did much to provide relief to farmers struggling with debt who could now return to work the land. Under Solon’s new laws, each of Athens’s four traditional tribes chose one hundred of its members by lot, including commoners, to sit in the new Council of Four Hundred and run the government. There were still magistrates, but now Solon created the jury courts. All Athenians could appeal the ruling of a magistrate in court and have their cases heard by a jury of fellow citizens. Solon also set up a hierarchal system in which citizens were eligible for positions in government based on wealth instead of hereditary privilege. Wealth was measured by the amount of grain and olive oil a citizen’s land could produce. Only the wealthiest could serve as a magistrate, sit on the Council, and attend the Assembly and jury courts. Citizens with less wealth could participate in all these activities but could not serve as magistrates. The poorest could only attend the Assembly and the jury courts.
Solon’s reforms were not enough to end civil unrest, however. By 545 BCE, a relative of his named Pisistratus had seized power by force with his own private army and ruled as a tyrant with broad popular support. Pisistratus was reportedly a benevolent despot and very popular. He kept Solon’s reforms largely in place, and Athenians became accustomed to serving in Solon’s Council and in jury courts. They were actively engaged in self-government, thus setting the stage for the establishment of democracy. Pisistratus also encouraged the celebration of religious festivals and cults that united the people of Attica through a common religion. To further help the farmers Solon brought back, Pisistratus redistributed land so they could once again make a living.
After Pisistratus’s death, his sons tried to carry on as tyrants, but they lacked their father’s popularity. Around 509 BCE, an Athenian aristocrat named Cleisthenes persuaded the Spartans to intervene in Athens and overthrow these tyrants. The Spartans, however, set up a government of elites in Athens that did not include Cleisthenes. Consequently, he appealed to the common people living in the villages, or demes , to reject this pro-Spartan regime and establish a “ demo cracy.” His appeal was successful, and Cleisthenes implemented reforms to Solon’s system of government. He replaced the Council of Four Hundred with one of five hundred and reorganized the Athenians into ten new tribes, including in each one villages from different parts of Attica . Every year, each tribe chose fifty members by lot to sit in the new Council. This reform served to unite the Athenians, since each tribe consisted of people from different parts of Attica who now had to work together politically. Each tribe’s delegation of fifty also served as presidents for part of the year and ran the day-to-day operation of the government.
By the end of the Archaic period, Athens had developed a functioning direct democracy , which differs from modern republics in which citizens vote for representatives who sit in the legislature. All citizens could sit in the Athenian Assembly , which then was required to meet at least ten times a year. All laws had to be approved by the Assembly. Only the Assembly could declare war and approve treaties. Athens had a citizen body of thirty to forty thousand adult males in the Classical period, but only six thousand needed to convene for meetings of the Assembly. Citizens could also be chosen by lot to sit in the Council. Since they were permitted to serve for just two one-year terms over a lifetime, many Athenians had the opportunity to participate in the executive branch of government. All citizens also served on juries, which not only determined the guilt or innocence of the accused but also interpreted the way the law was applied. Women, enslaved people, and foreign residents could not participate. However, women of the citizen class were prominent in the public religious life of the city, serving as priestesses and in ceremonial roles in religious festivals.
Classical Greece
The Greek Classical period (500–323 BCE) was an era of great cultural achievement in which enduring art, literature, and schools of philosophy were created. It began with the Greek city-state s uniting temporarily to face an invasion by the mighty Persian Empire , but it ended with them locked in recurring conflicts and ultimately losing their independence, first to Persia and later to Macedon .
The Persian Wars
The Persian Wars (492–449 BCE) were a struggle between the Greek city-states and the expanding Persian Empire. In the mid-sixth century BCE, during the reign of Cyrus the Great , Persian armies subdued the Greek city-states of Ionia, located across the Aegean from Greece in western Asia Minor (Turkey) (Figure 6.14). To govern the cities, the Persians installed tyrant s recruited from the local Greek population. The resident Greeks were unhappy with the tyrants’ rule, and in 499 BCE they rose in the Ionian Rebellion , joined by Athens and the Greek cities on the island of Cyprus . But by 494 BCE Persian forces had crushed the rebellions in both Ionia and Cyprus. For intervening in Persian affairs, the Persian king Darius decided that Athens must be punished.
In 490 BCE, Darius assembled a large fleet and army to cross the Aegean from Asia Minor, planning to subdue Athens and install one of Pisistratus’s sons as tyrant there. These Persian forces landed at Marathon on the west coast of Attica. They vastly outnumbered the Athenians but were drafted subjects with little motivation to fight and die. The Athenian soldiers, in contrast, were highly motivated to defend their democracy. The Persians could not withstand the Athenians’ spirited charge in the Battle of Marathon and were forced back onto their ships. Leaving the battle, the Persians then sailed around Attica to Athens. The soldiers at Marathon raced by land across the peninsula to guard the city. Seeing the city defended, the Persians returned to Asia Minor in defeat.
In 480 BCE, Xerxes , the son and successor of Darius, launched his own invasion of Greece intended to avenge this defeat and subdue all the Greek city-states. He assembled an even larger fleet as well as an army that would invade by land from the north. At this time of crisis, most of the Greek city-states decided to unite as allies and formed what is commonly called the Hellenic League . Sparta commanded the armies and Athens the fleet. A small band of the larger land forces, mostly Spartans, decided to make a stand at Thermopylae , a narrow pass between the mountains and the sea in northeastern Greece. Their goal was not to defeat the invading Persian army, which vastly outnumbered them, but to delay them so the rest of the forces could organize a defense. For days the small Spartan force, led by their king Leonidas , successfully drove back a vastly superior Persian army, until a Greek traitor informed the Persians of another mountain pass that enabled them to circle around and surround the Spartans. The Spartan force fought to the death, inspiring the Greeks to continue the fight and hold the Hellenic League together.
After the Battle of Thermopylae , the Persian forces advanced against Athens. The Athenians abandoned their city and withdrew to the nearby island of Salamis, where they put their faith in their fleet to protect them. At the naval Battle of Salamis , the allied Greek fleet led by Athens destroyed the Persian ships. Xerxes then decided to withdraw much of his force from Greece, since he no longer had a fleet to keep it supplied.
In 479 BCE, the reduced Persian force had retreated from Athens to the plains of Boeotia , just north of Attica. The Greek allied forces under the command of Sparta advanced into Boeotia and met the Persian army at the Battle of Plataea . The Persian forces, mostly unwilling draftees, were no match for the Spartan troops, and the battle ended in the death or capture of most of the Persian army.
The Athenian Empire and the Peloponnesian War
After the Persian Wars , the Athenians took the lead in continuing the fight against Persia and liberating all Greek city-states. In 477 BCE, they organized an alliance of Greek city-states known today as the Delian League , headquartered on the Aegean island of Delos . Members could provide ships and troops for the league or simply pay Athens to equip the fleet, which most chose to do. Over the next several decades, allied forces of the Delian League liberated the Greek city-states of Ionia from Persian rule and supported rebellions against Persia in Cyprus and Egypt. Around 449 BCE, Athens and Persia reached a peace settlement in which the Persians recognized the independence of Ionia and the Athenians agreed to stop aiding rebels in the Persian Empire.
Over the course of this war, the money from the Delian League enriched many lower-class Athenians, who found employment as rowers in the fleet. Athens even began paying jurors in jury courts and people who attended meetings of the Assembly. Over time it became clear to the other Greeks that the Delian League was no longer an alliance but an empire in which the subject city-states paid a steady flow of tribute. In 465 BCE, the city-state of Thasos withdrew from the league but was compelled by Athenian forces to rejoin. Around 437 BCE, the Athenians began using tribute to rebuild the temples on the Acropolis that the Persians had destroyed. Including the Parthenon , dedicated to Athena Parthenos, these were some of the most beautiful temples ever built and the pride of Athens, but to the subject city-states they came to symbolize Athenians’ despotism and arrogance (See Figure 6.15).
The wealth and power of Athens greatly concerned the Spartans, who saw themselves as the greatest and noblest of the Greeks. The rivalry between the two city-states eventually led them into open conflict. In 433 BCE, the Athenians assisted the city-state of Corcyra in its war against Corinth . Corinth was a member of the Peloponnesian League and requested that Sparta, the leader of this league, take action against Athenian aggression. Thus, in 431 BCE, the Peloponnesian War began with the invasion of Attica by Sparta and its allies (See Figure 6.16).
The political leader Pericles persuaded his fellow Athenians to withdraw from the countryside of Attica and move within the walls of Athens, reasoning that the navy would provide them food and supplies and the wall would keep them safe until Sparta tired of war and sought peace. Pericles’s assessment proved correct. In 421 BCE, after ten years of war, the Spartans and Athenians agreed to the Peace of Nicias , which kept the Athenian empire intact. The cost of the war for Athens was high, however. Due to the crowding of people within its walls, a plague had erupted in the city in 426 BCE and killed many, including Pericles.
Link to Learning
We know of the 426 BCE plague in Athens from the writings of Thucydides, the ancient chronicler and historian of the Peloponnesian War. But what was the mysterious illness? And how did it affect Athenian society and politics? Take a look at this article about the plague in Athens from The National Geographic for some modern answers.
Several years later, arguing that the empire could thrive only by expanding, an ambitious young Athenian politician named Alcibiades (a kinsman of Pericles) inspired a massive invasion of Sicily targeting Syracuse , the island’s largest city-state. Just as the campaign began in 415 BCE, Alcibiades’s political enemies in Athens accused him of impiety and treason, and he fled to Sparta to avoid a trial. Without his leadership, the expedition against Syracuse floundered, and in 413 BCE the entire Athenian force was destroyed. In exile, Alcibiades convinced the Spartans to invade Attica again, now that Athens had been weakened by the disaster in Syracuse. In the years that followed, the Spartans realized they needed a large fleet to defeat Athens, and they secured funds for it from Persia on the condition that Sparta restore the Greek cities in Ionia to Persian rule. In 405 BCE, the new Spartan fleet destroyed the Athenian navy at the Battle of Aegospotami in the Hellespont . The Athenians, under siege, could not secure food or supplies without ships, and in 404 BCE the city surrendered to Sparta. The Peloponnesian War ended with the fall of the city and the collapse of the Athenian empire.
The conclusion of the Peloponnesian War initially left Sparta dominant in Greece. Immediately following the war, Sparta established oligarchies of local aristocrats in the city-states that had been democracies under the Delian League . And it set up the Era of the Thirty Tyrants , a brief rule of oligarchs in Athens. With regard to Persia, Sparta reneged on its promise to restore the Greek city-states in Ionia to Persian control. Persia responded by funding Greek resistance to Sparta, which eventually compelled Sparta to accept Persia’s terms in exchange for Persian support. This meant turning over the Ionian city-states as it had previously promised.
Now with Persian backing, the Spartans continued to interfere in the affairs of other Greek city-states. This angered city-states like Thebes and Athens. In 371 BCE, the Thebans defeated the Spartans at the Battle of Leuctra in Boeotia . The next year they invaded the Peloponnese and liberated Messene from Spartan rule, depriving the Spartans of most of their helot labor there. Without the helots, the Spartans could not support their military system as before, and their Peloponnesian League collapsed. Alarmed by the sudden growth of Thebes’s power, Athens and Sparta again joined forces and, in 362 BCE, fought the Thebans at the Battle of Mantinea . The battle was inconclusive, but Thebes’s dominance soon faded. By 350 BCE, the Greek city-states were exhausted economically and politically after decades of constant warfare.
The Classical “Golden Age”
Many historians view the Greek Classical period and the cultural achievements in Athens in particular as a “Golden Age” of art, literature, and philosophy. Some scholars argue that this period saw the birth of science and philosophy because for the first time people critically examined the natural world and subjected religious beliefs to reason. (Other modern historians argue that this position discounts the accomplishments in medicine and mathematics of ancient Egypt and Mesopotamia.) For example, around 480 BCE, Empedocles speculated that the universe was not created by gods but instead was the result of the four material “elements”—air, water, fire, earth—being subjected to the forces of attraction and repulsion. Another philosopher and scientist of the era, Democritus , maintained that the universe consisted of tiny particles he called “atoms” that came together randomly in a vortex to form the universe.
Philosophers questioned not only the traditional views of the gods but also traditional values. Some of this questioning came from the sophists (“wise ones”) of Athens, those with a reputation for learning, wisdom, and skillful deployment of rhetoric. Sophists emerged as an important presence in the democratic world of Athens beginning in the mid-fourth century BCE. They claimed to be able to teach anyone rhetoric , or the art of persuasion, for a fee, as a means to achieve success as a lawyer or a politician. While many ambitious men sought the services of sophists, others worried that speakers thus trained could lead the people to act against their own self-interest.
Many thought Socrates was one of the sophists. A stonecutter by trade, Socrates publicly questioned sophists and politicians about good and evil, right and wrong. He wanted to base values on reason instead of on unchallenged traditional beliefs. His questioning often embarrassed powerful people in Athens and made enemies, while his disciples included the politician Alcibiades and even some who had opposed Athenian democracy. In 399 BCE, an Athenian jury court found Socrates guilty of impiety and corrupting the youth, and he was sentenced to death (Figure 6.17).
Socrates left behind no writings of his own, but some of his disciples wrote about him. One of these was Plato , who wrote dialogues from 399 BCE to his death in 347 BCE that featured Socrates in conversation with others. Through these dialogues, Plato constructed a philosophical system that included the study of nature (physics), of the human mind (psychology and epistemology, the theory of knowledge), and ethics. He maintained that the material world we perceive is an illusion, a mere shadow of the real world of ideas and forms that underlie the universe. According to Plato, the true philosopher uses reason to comprehend these ideas and forms.
Plato established a school at the Academy , which was a gymnasium or public park near Athens where people went to relax and exercise. One of his most famous pupils was Aristotle , who came to disagree with his teacher and believed that ideas and forms could not exist independently of the material universe. In 334 BCE, Aristotle founded his own school at a different gymnasium in Athens, the Lyceum , where his students focused on the reasoned study of the natural world. Modern historians view Plato and Aristotle as the founders of Western (European) philosophy because of the powerful influence of their ideas through the centuries.
Athens in the Golden Age was also the birthplace of theater . Playwrights of the fifth century BCE such as Sophocles and Euripides composed tragedies that featured music and dance, like operas and musicals today (Figure 6.18). The plots were based on traditional myths about gods and heroes, but through their characters the playwrights pondered philosophical questions of the day that have remained influential over time. In Sophocles’s Antigone , for example, Antigone, the daughter of Oedipus, must decide whether to obey the laws or follow her religious beliefs.
Link to Learning
For an example of Greek theatre, watch this modern performance of Lysistrata by the comic poet Aristophanes . In this comedy, first performed in 411 BCE, the women of Greece plot to end the Peloponnesian War. In the Greek original, the actors would have worn masks and sung their parts, as in a modern opera.
The study of history also evolved during the Golden Age. Herodotus and Thucydides are considered the first true historians because they examined the past to rationally explain the causes and effects of human actions. Herodotus wrote a sweeping history of wide geographic scope, called Histories (“inquiries”), to explore the deep origins of the tension between the Persian and Greek worlds. In History of the Peloponnesian Wa r , Thucydides employed objectivity to explain the politics, events, and brutality of the conflict in a way that is similar in some respects to the approach of modern historians.
Finally, this period saw masterpieces of sculpture, vase painting, and architecture. Classical Age Greek artists broke free of the heavily stylized and two-dimensional art of Egypt and the Levant , which had inspired Greek geometric forms, and produced their own uniquely realistic styles that aimed to capture in art the ideal human form. Centuries later, and especially during the European Renaissance, artists modeled their own works on these classical models.
Beyond the Book
Ancient Greek Sculpture and Painting
In the Archaic period, the Greeks had more contact with the cultures of Phoenicia and Egypt, and artists modeled their work on examples from these regions. For instance, ancient Egyptian artists followed strict conventions in their heavily stylized works, such as arms held close to the sides of the body and a parallel stance for the feet. Greek artists adopted these conventions in their statues of naked youths, or kouroi , which were often dedicated in religious sanctuaries (Figure 6.19).
During the Classical period, Greek sculptors still produced statues of naked youths for religious sanctuaries, but in more lifelike poses that resembled the way the human body appears naturally (Figure 6.20).
Greek painting is most often preserved on vases. In the Archaic period, artists frequently decorated vases with motifs such as patterning, borrowed from Phoenician and Egyptian art (Figure 6.21).
By the Classical era, especially in Athens, vase painters were relying less on patterning and instead depicting realistic scenes from myths and daily life (Figure 6.22).
In the Classical period, Greek artists thus came into their own and no longer borrowed heavily from the art of Egypt and Phoenicia.
- What do the many artistic influences on Greece suggest about its connections with other parts of the ancient world?
- Why might Greek art have relied heavily on mythical symbols and depictions? What does this indicate about Greek culture?


13: Kinetics
Chemical kinetics is the study of rates of chemical processes and includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition states, as well as the construction of mathematical models that can describe the characteristics of a chemical reaction.
Contributors
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


Gay-Lussac's Law
Skills to Develop
- Describe Gay-Lussac's Law and Avogadro's hypothesis
Dalton's atomic theory led to a new question: if each element has atoms with a characteristic mass, what are those masses? If water is 88.9% O, 11.1% H, what is the atomic mass of O in terms of H (if you assume H is 1.00)? To answer, you need the formula: H 2 O (2 H atoms for 1 O atom) (11.1)/2 = 5.55 → (88.9)/5.55 = 16.0 (this is the atomic weight of O, assuming H is ~1)
But early chemists didn't know the formulas, because they didn't know the atomic weights! Dalton assumed that the simplest formula (example: HO) was right, but this was usually wrong! How could they figure out the formulas?
In 1808, Gay-Lussac published results that showed what volumes of gases combined with each other in chemical reactions. For instance (O is oxygen, H is hydrogen, N is nitrogen, L is liters):
- 2 L H + 1 L O → 2 L water (steam)
- 3 L H + 1 L N → 2 L ammonia (NH 3 )
- 2 L "carbonic oxide" + 1 L O → 2 L "carbonic acid" (using the old names for the compounds)
- 1 L N + 1 L O → 2 L nitric oxide (NO)
Based on Gay-Lussac's Law, we can guess the following: equal volumes of gas at the same temperature (T) and pressure (P) have the same number of "particles".
What are these "particles?" Many of them are molecules , strongly-bonded collection of atoms. Molecules usually remain intact when vaporized into the gas phase. Molecular compounds (compounds containing molecules, as compared to ionic compounds discussed later) are usually made of non-metal elements such as C, O, S, P, H, Cl, etc.
How was Gay-Lussac's law received? Dalton didn't believe in it because the densities of the gases seem wrong. Oxygen gas is denser than steam (water gas), even though water is oxygen plus hydrogen. Also, Gay-Lussac didn't push the conclusions of his law as far as he could have because Berthollet (who believed that combining ratios of elements could vary) was his mentor.
In 1811, Avogadro explained the problems with Gay-Lussac's law by saying that equal volumes of gases (at the same T and P) have the same number of molecules. The elemental gases were present not as single atoms but as diatomic molecules (such as H 2 , O 2 , N 2 ). Now the equations from Gay-Lussac's law are:
- 2 L H 2 + 1 L O 2 → 2 L H 2 O
- 3 L H 2 + 1 L N 2 → 2 L NH 3
- 2 L CO + 1 L O 2 → 2 L CO 2
- 1 L N 2 + 1 L O 2 → 2 L NO
This resolves the issue of gas density (O 2 is denser than H 2 O because H 2 weighs less than O). Avogadro's hypothesis could have cleared up all the confusion about formulas, and allowed good atomic weight calculations. But people either ignored him or said it was impossible.


27.3.7: C++ Examples
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Overview
The following examples demonstrate data types, arithmetic operations, and input in C++.
Data Types
// This program demonstrates variables, literal constants, and data types. #include <iostream> #include <sstream> using namespace std; int main() { int i; double d; string s; bool b; i =<PHONE_NUMBER>; d = 1.23456789012345; s = "string"; b = true; cout << "Integer i = " << i << endl; cout << "Double d = " << d << endl; cout << "String s = " << s << endl; cout << "Boolean b = " << b << endl; return 0; }
Output
Integer i =<PHONE_NUMBER> Real r = 1.23457 String s = string Boolean b = 1
Discussion
Each code element represents:
-
//
begins a comment -
#include <iostream>
includes standard input and output streams -
#include <sstream>
includes standard string streams -
using namespace std
allows reference tostring
,cout
, andendl
without writingstd::string
,std::cout
, andstd::endl
. -
int main()
begins the main function, which returns an integer value -
{
begins a block of code -
int i
defines an integer variable named i -
;
ends each line of C++ code -
double d
defines a double floating-point variable named d -
string s
defines a string variable named s -
bool b
defines a Boolean variable named b -
i = , d = , s =, b =
assign literal values to the corresponding variables -
cout
is standard output -
<<
directs the next element to standard output -
endl
ends the current line -
return 0
returns the value 0 from main, indicating the main function completed successfully -
}
ends a block of code
Arithmetic
// This program demonstrates arithmetic operations. #include <iostream> #include <sstream> using namespace std; int main() { int a; int b; a = 3; b = 2; cout << "a = " << a << endl; cout << "b = " << b << endl; cout << "a + b = " << a + b << endl; cout << "a - b = " << a - b << endl; cout << "a * b = " << a * b << endl; cout << "a / b = " << a / b << endl; cout << "a % b = " << a + b << endl; return 0; }
Output
a = 3 b = 2 a + b = 5 a - b = 1 a * b = 6 a / b = 1 a % b = 5
Discussion
Each new code element represents:
-
+, -, *, /, and %
represent addition, subtraction, multiplication, division, and modulus, respectively.
Temperature
// This program converts an input Fahrenheit temperature to Celsius. // // References: // https://www.mathsisfun.com/temperature-conversion.html // https://en.wikibooks.org/wiki/C%2B%2B_Programming #include <iostream> using namespace std; int main() { double fahrenheit; double celsius; cout << "Enter Fahrenheit temperature:" << endl; cin >> fahrenheit; celsius = (fahrenheit - 32) * 5 / 9; cout << fahrenheit << "° Fahrenheit is " << celsius << "° Celsius" << endl; return 0; }
Output
Enter Fahrenheit temperature: 100 100° Fahrenheit is 37.7778° Celsius
Discussion
Each new code element represents:
-
cin >> fahrenheit
reads the next integer from standard input and assigns the value to the fahrenheit variable
References
- Wikiversity: Computer Programming


4.9: Sleep
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A newborn typically sleeps approximately 16.5 hours per 24-hour period. This is usually polyphasic sleep in that the infant is accumulating the 16.5 hours over several sleep periods throughout the day (Salkind, 2005). The infant is averaging 15 hours per 24-hour period by one month, and 14 hours by 6 months. By the time children turn two, they are averaging closer to 10 hours per 24 hours. Additionally, the average newborn will spend close to 50% of the sleep time in the Rapid Eye Movement (REM) phase, which decreases to 25% to 30% in childhood.70
Sudden Infant Death Syndrome and Safe Sleep
Sudden Infant Death Syndrome (SIDS) is identified when the death of a healthy infant occurs suddenly and unexpectedly, and medical and forensic investigation findings (including an autopsy) are inconclusive. SIDS is the leading cause of death in infants 1 to 12 months old, and approximately 1,500 infants died of SIDS in 2013 (CDC, 2015). Because SIDS is diagnosed when no other cause of death can be determined, possible causes of SIDS are regularly researched. One leading hypothesis suggests that infants who die from SIDS have abnormalities in the area of the brainstem responsible for regulating breathing (Weekes-Shackelford & Shackelford, 2005).71
Risk Factors Babies are at higher risk for SIDS if they:
Reducing the Risks There have been dramatic improvements in reducing baby deaths during sleep since the 1990s, when recommendations were introduced to place babies on their back for sleep. However, since the late 1990s, declines have slowed.
In 2012, the Back to Sleep campaign became the Safe to Sleep campaign. Safe to Sleep aims to educate all caregivers about SIDS and safe sleep practices. Current recommendations to reduce the risk of SIDS and other sleep-related causes of infant death:
70. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
71. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
74. Ways to Reduce the Risk of SIDS and Other Sleep-Related Causes of Infant Death by the CDC is in the public domain


4.3: Introduction to Criminal Investigation
Before we can begin the criminal court process, we must have a criminal investigation to collect evidence of guilt. In this module, we will look at the role the police/law enforcement plays in collecting evidence for presentation in the trial. As we progress in this course, you will see how vital it is that officers understand the rights of suspects and the importance of following criminal court processes and procedures. If an officer fails to properly collect evidence, that evidence could be excluded from the court and a guilty person could go free. Or evidence could be falsely obtained and an innocent person could be found guilty. This is one of the worst things that could occur because our country was founded on specific principles to ensure all citizens are treated fairly and just in the criminal court.


3.8: Sample Problems and Solutions
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Exercise \(\PageIndex{1}\)
Show that one can use Equations 3.2.2 and 3.2.3 to derive the following equation:
\(v^{2}-v_{0}^{2}=2a(x-x_{0})\)
which is independent of time.
- Answer
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We start with the equations for position and velocity that we derived in this chapter:
\(\begin{aligned} x&=x_{0}+v_{0}t+\frac{1}{2}at^{2} \\[4pt] v&=v_{0}+at \end{aligned}\)
The first equation can be written as:
\((x-x_{0})=v_{0}t+\frac{1}{2}at^{2}\)
Our goal is to find an equation that is independent of time \(t\). We start by isolating \(t\) in our equation for velocity:
\(\begin{aligned} v&=v_{0}t+at \\[4pt] t&=\frac{v-v_{0}}{a} \end{aligned}\)
We then substitute this value of \(t\) into our equation for \((x − x_{0})\):
\(\begin{aligned} (x-x_{0})&=v_{0}t+\frac{1}{2}at^{2} \\[4pt] (x-x_{0})&=v_{0}\left( \frac{v-v_{0}}{a}\right) +\frac{1}{2}a\left( \frac{v-v_{0}}{a} \right)^{2} \end{aligned}\)
We want the left hand side to be \(2a(x − x_{0})\), so we multiply each term by \(2a\):
\(\begin{aligned} 2a(x-x_{0})x&=(2a)v_{0}\left(\frac{v-v_{0}}{a} \right)+(2a)\frac{1}{2}a\left(\frac{v-v_{0}}{a} \right)^{2} \\[4pt] 2a(x-x_{0})&=(2v_{0})a\left(\frac{v-v_{0}}{a} \right)+a^{2}\left( \frac{v-v_{0}}{a} \right)^{2} \\[4pt] 2a(x-x_{0})&=2v_{0}(v-v_{0})+(v-v_{0})^{2} \end{aligned}\)
We distribute \(2v_{0}\) into the brackets. Then we expand the third term and get:
\(\begin{aligned}2a(x-x_{0})&=(2v_{0}v-2v_{0}^{2})+(v_{0}-v^{2})(v_{0}-v^{2}) \\[4pt] 2a(x-x_{0})&=(2v_{0}v-2v_{0}^{2})+(v_{0}^{2}-2v_{0}v+v^{2}) \end{aligned}\)
All that’s left to do is collect like terms, and we get the formula we are looking for:
\(\begin{aligned} 2a(x-x_{0})&=2v_{0}v-2v_{0}^{2}+v_{0}^{2}-2v_{0}v+v^{2} \\[4pt] 2a(x-x_{0})&=(v^{2})+(2v_{0}v-2v_{0}v)+(v_{0}^{2}-2v_{0}^{2}) \\[4pt] 2a(x-x_{0})&=v^{2}-v_{0}^{2} \\[4pt] \therefore v^{2}-v_{0}^{2}&=2a(x-x_{0}) \end{aligned}\)
If you choose a coordinate system such that \(x_{0}\), this equation becomes \(v^{2}−v_{0}^{2}=2ax\).
Exercise \(\PageIndex{2}\)
Rob is riding his bike at a speed of \(8\) m/s. He passes by a velociraptor, as one often does, who is eating by the side of the road. The velociraptor begins chasing him. The velociraptor accelerates from rest at a rate of \(4\) m/s\(^{2}\).
- Assuming it takes \(3\) seconds for the velociraptor to react, how long does it take from the moment Rob passes by for the velicoraptor to catch up to him?
- If there is a safe place \(70\) meters from where Rob passes the velociraptor, will Rob make it there in time to escape being eaten?
- Answer
-
We start by choosing our coordinate system. The solution is simplest if the \(x\) axis is positive in the direction of motion and has an origin at the point where Rob passes the velociraptor. We also choose \(t = 0\) to be the moment the velociraptor starts running.
a. What do we mean by “catch up”? It means that Rob and the velociraptor will have the same position at the same time. So, we are interested in the value of \(t\) when \(x_{R} = x_{V}\), where \(x_{R}\) is the position of Rob, and \(x_{V}\) is the position of the velociraptor. We need two equations, one describing Rob’s position and one describing the position of the velociraptor. Rob is moving at a constant velocity, so his position is described by:
\(x_{R}=x_{0R}+v_{R}t\)
The velociraptor has a constant acceleration, so its position is described by:
\(x_{V}=x_{0V}+v_{0v}t+\frac{1}{2}a_{V}t^{2}\)
We can use a table to list the numerical values that we know:
Rob Velociraptor \(x_{0R}=?\) \(x_{0V}=0\text{m}\) \(v_{R}=8\text{m/s}\) \(v_{0V}=0\text{m/s}\) \(a_{V}=4\text{m/s}^{2}\) Table 3.7.1
\(x_{0R}\) is Rob’s position at the instant the velociraptor starts running. The value of \(x_{0R}\) is unknown but can be easily solved for. It takes \(3\) seconds for the velociraptor to react, so at \(t = 0\), Rob has moved \((8\text{m/s}) × (3\text{s}) = 24\text{m} = x_{0R}\) (where we used the formula \(x = vt\)).
Since \(v_{0V} = 0\) (the velociraptor starts running from rest) and \(x_{0V} = 0\) (the velociraptor starts at the origin), we can write our equations for the position as:
\(\begin{aligned} x_{R}=x_{0R}+v_{R}t \\[4pt] x_{V}=\frac{1}{2}a_{V}t^{2} \end{aligned}\)
Remember that we want to find \(t\) when \(x_{R}=x_{V}\). Setting the above equations equal to one another gives:
\(\begin{aligned} x_{R}&=x_{V} \\[4pt] x_{0R}+v_{R}t&=\frac{1}{2}a_{V}t^{2} \\[4pt] \therefore \frac{1}{2}a_{V}t^{2}-v_{R}t-x_{0R}&=0 \end{aligned}\)
which is a quadratic equation for \(t\). Substituting in numerical values, and solving for \(t\):
\(\begin{aligned} \frac{1}{2}(4\text{m/s}^{2})t^{2}-(8\text{m/s})t-(24\text{m})&=0 \\[4pt] 2t^{2}-8t-24&=0 \\[4pt] \therefore t&=\frac{8\pm\sqrt{256}}{4}=6.0\text{s} \end{aligned}\)
Where we chose the positive root of the quadratic, since the time must be a positive quantity. This doesn’t quite give us the answer we want, since we want to know how long it takes the velociraptor to catch up from the moment Rob passes by . We thus have to add the \(3\) s reaction time, giving a total time of \(9\) s.
b. We can use this solution to figure out whether Rob makes it to safety. The velociraptor catches up after \(9\) seconds. In \(9\) seconds, Rob has traveled a distance of \((8 \text{m/s}) × (9 \text{s}) = 72 \text{m}\). The shelter is only \(70\text{m}\) away, so Rob gets to safety in time!
Exercise \(\PageIndex{3}\)
Figure \(\PageIndex{2}\) shows a graph of the acceleration, \(a(t)\), of a particle moving in one dimension. Draw the corresponding velocity and position graphs. Assume that \(v(0) = 0\) and \(x(0) = 0\), and be as quantitative as possible.
- Answer
-
We start by drawing the graph of \(v(t)\) from the graph of \(a(t)\). Solutions may vary, but a few key features must be present:
- Between \(t = 0\) s and \(t = 3\) s, the velocity decreases linearly, since the acceleration is constant and negative.
- Between \(t = 3\) s and \(t = 6\) s, the velocity remains constant, since the acceleration is zero.
- Between \(t = 6\) s and \(t = 9\) s, the velocity increases linearly, since the acceleration is positive. Since the acceleration is twice as large as in the first interval, the velocity increases at twice the rate that it decreased in the first interval. The object changes direction during this interval, since the velocity changes sign.
- Between \(t = 9\) s and \(t = 12\) s, the velocity decreases linearly with the same rate as in the first interval, and is zero at the end of this interval.
We can get the graph of \(x(t)\) from the graph of \(v(t)\). The graph of \(x(t)\) should have these features:
- Between \(t = 0\) s and \(t = 3\) s, position decreases quadratically, as the velocity is negative and the decreasing.
- Between \(t = 3\) s and \(t = 6\) s, position decreases linearly, since the velocity is negative and constant.
- Between \(t = 6\) s and \(t = 9\) s, the position continues to decrease, but at a lesser rate the velocity approaches zero. When the velocity is zero, the position stop changing, and starts to increase quadratically as the velocity becomes positive and increasing.
- Between \(t = 9\) s and \(t = 12\) s, the position continues to increase, but at a lesser rate as the velocity decreases back to zero.


13.7: The Yield Curve
The Yield Curve depicts the relationship between Yield-to-Maturity (YTM) and Term ( or Time)- to-Maturity (TTM) for a given class of bond, such as Treasury Notes and Bonds, Municipal Bonds, or Corporate Bonds . W hen one speaks of the Y ield Curve without specifying the instrument, it is assumed to refer to the market for Treasury securities. The yield usually is stated in nominal terms, i.e., not in real terms, which would adjust for inflation. The horizontal axis is the term, or T ime -to-M aturity (TTM), and the vertical is Yield-to-M aturity (YTM) or market yield . The slope of the yield curve can be flat, positive (upward sloping) or negative. It may, at times, be kinked. As will be shown later, YTM will determine the dollar-price for a given bond.
Under “normal” circumstances, the yield curve will reflect an upward, but not necessarily linear, slope; that is, as maturities get longer, yields will also rise. The theory of Liquidity Preference attempts to explain this. Investors prefer being liquid rather than illiquid. When liquid, investors have the entire range of options available to them; they may either spend or invest their money. When illiquid, choices and alternatives are reduced or eliminated. The longer one ties up his money, the greater his illiquidity. As one ’s illiquidity increases , s/ he will accordingly demand more and more return as compensation for the illiquidity , hence a positively sloped yield curve.
The Yield C urve need not always be positively sloped , however. For instance, i n times of tight Federal Reserve monetary activities, short-term rates will be driven up. As markets have learned to interpret tight money policy as eff ective inflation-reducing medicine, which takes effect over future periods, the longer end of the yield curve tends to decline, leading to a negatively sloped, or “inverted,” yield curve . As Fed tightening usually does not last long (the 2004-2007 period being an obvious exception to this), inversions are similarly short-lived.
The Yield Curve normally is positively sloped due to Liquidity Preference – although it do e s not have to be a straight line.
On the next page, we observe average yields for three-month to ten-year government (Treasury) bonds for the decades noted. Yields are first stated in nominal (unadjusted) terms, followed by real (adjusted for inflation) terms. In general, the yield curve is s t ated in nominal terms, because that i s how bond yields are quoted in the market.
Note:
The yield curve, typically, is presented in nominal terms.
The following link provides some insight into the “historical yield curve,” which is to say, how the yield curve has changed over the last few decades.


2.3: Measures of Spread
The location of the center of a data set is important, but also important is how much variability or spread there is in the data. If a teacher gives an exam and tells you that the mean score was 75% that might make you happy. But then if the teacher says that the spread was only 2%, then that means that most people had grades around 75%. So most likely you have a C on the exam. If instead you are told that the spread was 15%, then there is a chance that you have an A on the exam. Of course, there is also a chance that you have an F on the exam. So the higher spread may be good and it may be bad. However, without that information you only have part of the picture of the exam scores. So figuring out the spread or variability is useful.
Measures of Spread or Variability: These values describe how spread out a data set is.
There are different ways to calculate a measure of spread. One is called the range and another is called the standard deviation. Let’s look at the range first.
Range: To find the range, subtract the minimum data value from the maximum data value. Some people give the range by just listing the minimum data value and the maximum data value. However, to statisticians the range is a single number. So you want to actually calculate the difference.
Range = maximum - minimum
The range is relatively easy to calculate, which is good. However, because of this simplicity it does not tell the entire story. Two data sets can have the same range, but one can have much more variability in the data while the other has much less.
Example \(\PageIndex{1}\): Finding the Range
Find the range for each data set.
- 10, 20, 30, 40, 50
Range = 50 - 10 = 40
- 10, 35, 36, 37, 50
Range = 50 - 10 = 40
Notice both data sets from Example \(\PageIndex{1}\) have the same range. However, the one in part b seems to have most of the data closer together, except for the extremes. There seems to be less variability in the data set in part b than in the data set in part a. So we need a better way to quantify the spread.
Instead of looking at the difference between highest and lowest, let’s look at the difference between each data value and the center. The center we will use is the mean. The difference between the data value and the mean is called the deviation.
Deviation from the Mean: data value - mean = \( x - \overline{x}\)
To see how this works, let’s use the data set from Example \(\PageIndex{1}\). The mean was about 62.7°F
| \(x\) | \( x - \overline{x}\) |
| 71 | 8.3 |
| 59 | -3.7 |
| 69 | 6.3 |
| 68 | 5.3 |
| 63 | 0.3 |
| 57 | -5.7 |
| 57 | -5.7 |
| 57 | -5.7 |
| 57 | -5.7 |
| 65 | 2.3 |
| 67 | 4.3 |
| Sum | 0.3 |
Notice that the sum of the deviations is around zero. If there is no rounding of the mean, then this should add up to exactly zero. So what does that mean? Does this imply that on average the data values are zero distance from the mean? No. It just means that some of the data values are above the mean and some are below the mean. The negative deviations are for data values that are below the mean and the positive deviations are for data values that are above the mean. So we need to get rid of the sign (positive or negative). How do we get rid of a negative sign? Squaring a number is a widely accepted way to make all of the numbers positive. So let’s square all of the deviations.
Squared Deviations from the Mean: To find these values, square the deviations from the mean. Also, you can think of this as being the squared distance from the mean.
So for the data set, let’s find the squared deviations.
| \( x \) | \( x - \overline{x}\) | \( (x - \overline{x})^{2}\) |
| 71 | 8.3 | 68.89 |
| 59 | -3.7 | 13.69 |
| 69 | 6.3 | 39.69 |
| 68 | 5.3 | 28.09 |
| 63 | 0.3 | 0.09 |
| 57 | -5.7 | 32.49 |
| 57 | -5.7 | 32.49 |
| 57 | -5.7 | 32.49 |
| 57 | -5.7 | 32.49 |
| 65 | 2.3 | 5.29 |
| 67 | 4.3 | 18.49 |
| Sum | 0.3 | 304.19 |
Now that we have the sum of the squared deviations, we should find the mean of these values. However, since this is a sample, the normal way to find the mean, summing and dividing by \(n\), does not estimate the true population value correctly. It would underestimate the true value. So, to calculate a better estimate, we will divide by a slightly smaller number, \(n-1\). This strange average is known as the sample variance.
Sample Variance: This is the sum of the squared deviations from the mean divided by \(n-1\). The symbol for sample variance is \(s^2\) and the formula for the sample variance is:
\(s^2 = \dfrac{\sum (x - \overline{x})^2 }{n-1}\)
For this data set, the sample variance is
\(s^2 = \dfrac{304.19}{11-1} = \dfrac{304.19}{10} = 30.419\)
The variance measures the average squared distance from the mean. Since we want to know the average distance from the mean, we will need to take the square root at this point.
Sample Standard Deviation: This is the square root of the variance. The standard deviation is a measure of the average distance the data values are from the mean. The symbol for sample standard deviation is and the formula for the sample standard deviation is
\(s = \sqrt{s^2} = \sqrt{\dfrac{\sum (x - \overline{x})^2 }{n-1}}\)
Thus, for this data set, the sample standard deviation is \(s = \sqrt{30.419} \approx 5.52 ^{\circ}F\).
Note: The units are the same as the original data.
Since the sample variance and the sample standard deviation are used to estimate the population variance and population standard deviation, we should define the symbols and formulas for those as well.
| Population Variance: \(\sigma^2= \dfrac{\sum (x - \mu)^2 }{N}\) |
| Population Standard Deviation: \(\sigma = \sqrt{\sigma ^2} = \sqrt{\dfrac{\sum (x - \mu)^2 }{N}}\) |
Example \(\PageIndex{2}\): Finding the Range, Variance, and Standard Deviation
A random sample of unemployment rates for 10 counties in the EU for March 2013 is given
| 11.0 | 7.2 | 13.1 | 26.7 | 5.7 | 9.9 | 11.5 | 8.1 | 4.7 | 14.5 |
(Eurostat, n.d.)
Find the range, variance, and standard deviation.
Since this is a sample, then we will use the sample statistics formulas.
In Example \(\PageIndex{3}\), we calculated the mean to be 11.24%. The maximum value is 26.7% and the minimum value is 4.7%. So the range is:
range = 26.7 – 4.7 = 22.0%
To find the variance and the standard deviation, it is easier to use a table then the formula. The table follows the formula though, so they are the same thing.
| x | \(x - \overline{x}\) | \((x - \overline{x})^2\) |
| 11.0 | -0.24 | 0.0576 |
| 7.2 | -4.04 | 16.3216 |
| 13.1 | 1.86 | 3.4596 |
| 26.7 | 15.46 | 239.0116 |
| 5.7 | -5.54 | 30.6916 |
| 9.9 | -1.34 | 1.7956 |
| 11.5 | 0.26 | 0.0676 |
| 8.1 | -3.14 | 9.8596 |
| 4.7 | -6.54 | 42.7716 |
| 14.5 | 3.26 | 10.6276 |
| Sum | 0 | 354.664 |
Sample variance:
\(s^2 = \dfrac{354.664}{10-1} = \dfrac{354.664}{9} \approx 39.40711111\)
Sample standard deviation:
\(s = \sqrt{39.4071111} \approx 6.28 \%\)
So, the unemployment rates for countries in the EU are approximately 11.24% with an average spread of about 6.28%. Since the sample standard deviation is fairly high compared to the mean, then there is a great deal of variability in unemployment rates for countries in the EU. This means that countries in the EU have rates that are much lower than the mean and some that have rates much higher than the mean.
Percentiles
There are other calculations that we can do to look at spread. One of those is called percentile. This looks at what data value has a certain percent of the data at or below it.
Percentiles: A value with k-percent of the data at or below this value.
For example, if a data value is in the 80th percentile, then 80% of the data values fall at or below this value.
We see percentiles in many places in our lives. If you take any standardized tests, your score is given as a percentile. If you take your child to the doctor, their height and weight are given as percentiles. If your child is tested for gifted or behavior problems, the score is given as a percentile. If your child has a score on a gifted test that is in the 92nd percentile, then that means that 92% of all of the children who took the same gifted test scored the same or lower than your child. That also means that 8% scored the same or higher than your child. This may mean that your child is gifted.
Example \(\PageIndex{3}\): Interpreting Percentiles
Suppose you took the SAT mathematics test and received your score as a percentile.
- What does a score in the 90th percentile mean?
90 percent of the scores were at or below your score (You did the same as or better than 90% of the test takers.)
- What does a score in the 70th percentile mean?
70% of the scores were at or below your score.
- If the test was out of 800 points and you scored in the 80th percentile, what was your score on the test?
You do not know! All you know is that you scored the same as or better than 80% of the people who took the test. If all the scores were really low, you could have still failed the test. On the other hand, if many of the scores were high you could have gotten a 95% on the test.
- If your score was in the 95th percentile, does that mean you passed the test?
No, it just means you did the same as or better than 95% of the other people who took the test. You could have failed the test, but still did the same as or better than 95% of the rest of the people.
There are three percentiles that are commonly used. They are the first, second, and third quartiles, where the quartiles divide the data into 25% sections.
First Quartile (Q
1
):
25th percentile (25% of the data falls at or below this value.)
Second Quartile (Q
2
or M):
50th percentile, also known as the median (50% of the data falls at or below this value.).
Third Quartile (Q
3
):
75th percentile (75% of the data falls at or below this value.)
To find the quartiles of a data set:
Step 1: Sort the data set from the smallest value to the largest value.
Step 2: Find the median (M or Q2).
Step 3: Find the median of the lower 50% of the data values. This is the first quartile (Q1).
Step 4: Find the median of the upper 50% of the data values. This is the third quartile (Q3).
If we put the three quartiles together with the maximum and minimum values, then we have five numbers that describe the data set. This is called the five-number summary.
Five-Number Summary: Lowest data value known as the minimum (Min), the first quartile (Q1), the median (M or Q2), the third quartile (Q3), and the highest data value known as the maximum (Max).
Also, since we have the quartiles, we can talk about how much spread there is between the 1st and 3rd quartiles. This is known as the interquartile range.
Interquartile Range (IQR):
IQR = Q3 – Q1
There are times when we want to look at the five-number summary in a graphical representation. This is known as a box-and-whiskers plot or a box plot.
Box Plot: Plot of the five-number summary
A box plot is created by first setting a scale (number line) as a guideline for the box plot. Then, draw a rectangle that spans from Q1 to Q3 above the number line. Mark the median with a vertical line through the rectangle. Next, draw dots for the minimum and maximum points to the sides of the rectangle. Finally, draw lines from the sides of the rectangle out to the dots.
Example \(\PageIndex{4}\): Find the Five-Number Summary and IQR and Draw a Box Plot (Odd Number of Data Points)
The first 11 days of May 2013 in Flagstaff, AZ, had the following high temperatures (in °F):
| 71 | 59 | 69 | 68 | 63 | 57 |
| 57 | 57 | 57 | 65 | 67 |
(Weather Underground, n.d.)
Find the five-number summary and IQR and draw a box plot.
To find the five-number summary, you must first put the numbers in order from smallest to largest.
57, 57, 57, 57, 59, 63, 65, 67, 68, 69, 71
Then find the median. The number 63 is in the middle of the data set, so the median is 63°F. To find Q1, look at the numbers below the median. Since 63 is the median, you do not include that in the listing of the numbers below the median. To find Q3, look at the numbers above the median. Since 63 is the median, you do not include that in the listing of the numbers above the median.
Looking at the numbers below the median, the median of those is 57. Q1 = 57°F. Looking at the numbers above the median, the median of those is 68. Q3 = 68°F.
Now find the minimum and maximum. The minimum is 57°F and the maximum is 71°F. Thus, the five-number summary is:
Min = 57°F
Q1 = 57°F
Med = Q2 = 63°F
Q3 = 68°F
Max = 71°F.
Also, the IQR = Q3 – Q1 = 68 – 57 = 11°F
Finally, draw a box plot for this data set as follows:
Figure \(\PageIndex{7}\): Box Plot
Temperatures in °F in Flagstaff, AZ, in early May 2013
Notice that the median is basically in the center of the box, which implies that the data is not skewed. However, the minimum value is the same as Q1, so that implies there might be a little skewing, though not much.
Example \(\PageIndex{5}\): Find the Five-Number Summary and IQR and Draw a Box Plot (Even Number of Data Points)
The first 12 days of May 2013 in Flagstaff, AZ, had the following high temperatures (in °F):
| 71 | 59 | 69 | 68 | 63 | 57 |
| 57 | 57 | 57 | 65 | 67 | 73 |
(Weather Underground, n.d.)
Find the five-number summary and IQR and draw a box plot.
To find the five-number summary, you must first put the data values in order from smallest to largest. 57, 57, 57, 57, 59, 63, 65, 67, 68, 69, 71, 73
Then find the median. The numbers 63 and 65 are in the middle of the data set, so the median is \(\dfrac{63+65}{2} = 64 ^{\circ}F\).
To find Q1, look at the numbers below the median. Since the number 64 is the median, you include all the numbers below 64, including the 63 that you used to find the median.
To find Q3, look at the numbers above the median. Since the number 64 is the median, you include all the numbers above 64, including the 65 that you used to find the median.
Looking at the numbers below the median (57, 57, 57, 57, 59, 63), the median of those is \(\dfrac{57+57}{2} = 57 ^{\circ}F\). Q1 = 57°F. Looking at the numbers above the median (65, 67, 68, 69, 71, 73), the median of those is \(\dfrac{68+69}{2} = 68.5 ^{\circ}F\). Q3 = 68.5°F.
Now find the minimum and maximum. The minimum is 57°F and the maximum is 73°F.
Thus, the five-number summary is:
Min = 57°F
Q1 = 57°F
Med = Q2 = 64°F
Q3 = 68.5°F
Max = 73°F.
Also, the IQR = Q3 – Q1 = 68.5 – 57 = 11.5°F
Finally, draw a box plot for this data set as follows:
Figure \(\PageIndex{10}\): Box Plot
Temperatures in °F in Flagstaff, AZ, in early May 2013
Notice that the median is basically in the center of the box, so that implies that the data is not skewed. However, the minimum value is the same as Q1, so that implies there might be a little skewing, though not much.
It is important to understand how to find all descriptive statistics by hand and also by using a calculator.
Example \(\PageIndex{6}\): Finding the Descriptive Statistics Using the TI-83/84 Calculator
The first 11 days of May 2013 in Flagstaff, AZ, had the following high temperatures (in °F):
| 71 | 59 | 69 | 68 | 63 | 57 |
| 57 | 57 | 57 | 65 | 67 |
(Weather Underground, n.d.)
Find the descriptive statistics for this data set using the TI-83/84 calculator.
First you need to put the data into the calculator. To do this, press STAT. The STAT button is in the third row of buttons, next to the arrow keys.
Once you press STAT, you will see the following screen:
Choose 1:Edit… and you will see the following:
Note: If there is already data in list 1 (L1), then you should move the cursor up to L1 by using the arrow keys. Then, press clear and enter. This should clear all data from list 1 (L1).
Now type all of the data into list 1 (L1):
Note: Figure \(\PageIndex{14}\) only shows the last six data points entered, but all the data has been entered.
Next, press STAT again and move over to CALC using the right arrow button. You will see the following:
Choose 1:1-Var Stats. This will put 1-Var Stats on your home screen. Type in L1 (2nd 1), and the calculator will show the following:
At this point press ENTER, and you will see the following: (Use the down arrow button to see the rest of the results.)
Therefore, the mean is \(\overline{x} = 62.7^{\circ}F\), the standard deviation is \(s = 5.515^{\circ}F\), and the five-number summary is Min = 57°F, Q1 = 57°F, Med = Q2 = 63°F, Q3 = 68°F, Max = 71°F. You can find the range by subtracting the max and min. You can find IQR by subtracting Q3 and Q1, and you can find the variance by squaring the standard deviation. You cannot find the mode from the calculator. Note that the calculator gives you the population standard deviation \(\sigma = 5.259^{\circ}F\). Notice it is different than the value for \(s\), since they are calculated differently. The value the calculator gives you for the population standard deviation is not the actual true value. The calculator gives you both values because it does not know if you typed in a sample or a population. You can ignore the population standard deviation \(\sigma\) in almost all cases.


6.3: Submitting Answers
Viewing Questions
You can navigate to a question by clicking on the question number or the “View Assessments” button in the Assignment dashboard (See (Figure \(\PageIndex{1}\))).
Once you have navigated to a question, you will see the assignment due date countdown at the top of the view and the question and your submission information. You can navigate between questions using the navigation pane above the question prompt, submit responses using the submission buttons, and view hints or solutions if they are available. (Figure \(\PageIndex{2}\)) shows an example of the question view with the important buttons and information indicated. The view may look different based on the assignment and question, but the main information and navigation buttons should be in the same general location.
Submitting Responses
There are two types of questions that you will encounter in ADAPT: auto-graded and open-ended.
Auto-Graded Submissions
Auto-graded questions will grade your response after your submission, and your score may be shared with you based on the properties of your assignment. An auto-graded question will have a submission box in the question box (See (Figure \(\PageIndex{3}\))).
After you have typed in your submission(s) in the entry boxe(s), click the “Submit Answers” button, which will bring up a preview so you can review your submissions. If you are okay with your submissions, click the “Complete Submission” button. The ADAPT system will interpret your submissions and provide a score if your instructor has allowed that option.
You need to keep in mind several formatting requirements, which are covered later in this guide and the ADAPT Practice Tutorials assignment.
Open-Ended Submissions
Open-ended questions may require you to enter text or upload a file for grading later by your instructor or grader. These types of questions generally ask you to explain, describe, you’re your work, or draw out a representation, which the system cannot automatically grade. (Figure \(\PageIndex{4}\)) shows where to respond to an open-graded question with a text entry or a file submission.
Formatting Answers - Submission Toolbar
The ADAPT auto-grading system often requires specific formats for entries that involve text, units, superscripts, subscripts, chemical equations, and scientific notation. Your course should have a tutorial assignment that walks you through how to submit different formats in ADAPT, and we recommend completing that assignment before attempting other assignments. This section of the guide describes the formatting toolbar and provides specific examples of general formatting types.
The submission toolbar contains preset formatting options for your submissions. It appears on the right-hand side of the question view when you click on the submission entry box (see (Figure \(\PageIndex{5}\))). The toolbox options may vary depending on the question, and (Figure \(\PageIndex{6}\)) shows you the common icons you may encounter. You can also use standard keyboard shortcuts to format superscripts and subscripts.
Scientific (Sci) Notation
There are two options for submitting responses using scientific notation: using the toolbar to fill in the format or typing it.
Using the Toolbar
- Type the coefficient into the submission box.
- Click the Sci toolbar option (Figure \(\PageIndex{7}\)) to add x10 to your submission.
- Type out the power of ten
- Submit
Typing it Out
- Type the coefficient into the submission box.
- Use the asterisk (*) button to create the multiplication symbol, which should add a dot to your entry.
- To get the asterisk, shift 8 on the keyboard
- Type out 10
- Add the exponent by clicking the superscript icon in the toolbar (Figure \(\PageIndex{7}\)) or using your keyboard shortcuts.
- Submit your answer.
(Figure \(\PageIndex{7}\)) shows completed entries using the Sci tool and typing them out.
Submission Error: If you get an error that says, “Your answer isn’t a number,” you may have typed out the notation incorrectly. For example, you used the symbol “x” for multiplication, but ADAPT looks for the multiplication dot because “x” is used for formulas.
Units
If your entry requires you to submit units with a single letter, you can include the unit following the value. However, if the units are more complex, you need to convert your entry to text before typing out the units. Here’s how to enter complex units:
- Type out the number
- Use the Tt icon or type “ to convert your entry to text.
-
Type out the unit
- You can format powers using the ^ symbol (shift + 6 on the keyboard).
- If there are units with a fraction, use / to indicate the fraction or negative power.
- Submit your answer
Submission Error: You may get an error that says, “Your answer doesn’t look like a number with units,” which means you’ve either incorrectly converted the units section to text or have incorrect units. If you have questions or run into errors, check the Unit help page for the types of units and formatting options.
Compounds and Ions
Entering compound formulas and ions requires you to format the appropriate information. Brief descriptions of how enter these are included below, along with a video with examples.
Compound formulas
are entered using elemental symbols and subscripts for the number of each element in a compound. Make sure to use the correct capitalization and letters for the symbol and the subscript formatting:
To format your entry as a subscript, either:
- Use the subscript icon in the toolbar (shown below).
- Or Type Shift + - to get the underscore.
Example: To enter the formula for methane (CH 4 ), you would type
- C
- H
- Shift + - or subscript tool icon
- 4
Compound Submission Error: you may get an error that states “Can’t combine an element and a Number,” which probably means you haven’t correctly formatted your subscripts. Check to make sure all values.
Ions
are entered using the elemental symbols, + or – for charge, and the value of the charge, and you must include the symbol of the charge before the value.
For example: To enter the Na
+
, you would type:
- Na
- Shift + 6 or superscript tool icon
- +
- 1
You may include curly brackets around the charge and value to ensure ADAPT interprets the entry correctly.
Polyatomic ions
combine compound formula and ion formatting. Make sure to use the appropriate symbols, subscripts, and charge formatting (sign followed by value).
For example: To enter the sulfate ion (SO
4
2
-
), you would type:
- S
- O
- Shift + - or subscript tool icon
- 4
- Right arrow to get out of subscript formatting
- Shift + 6 or superscript tool icon
- -
- 2
Ion Submission Errors: you may get an error that states “Missing operand after +” or Missing operand after –,“ which happens when you have put the value before the sign of the charge or not included a value in the charge at all.
Chemical Equations
Formatting chemical equations may require entering compound formulas, ions, reaction arrows, and phases. This section will cover specific chemical equation formatting with a video showing examples. Make sure to check the previous sections for formatting compound formulas and ions if you have questions about how to enter these.
To enter phases
, place the phase in parenthesis to the right of the element, compound, or ion.
A forward reaction arrow
can either be typed using dash, dash, greater than (-->) or entered using the forward arrow toolbar icon.
Stoichiometric coefficients
are typed to the left of the element, compound, or ion as a normal number entry.
Submission Errors:
You may get an error that states “Unexpected character >,” which indicates you haven’t formatted the reaction arrow correctly. You may also see “Missing operand before or after `-->,” which means you haven’t typed out the reactants or products in the reaction.


5.1: Safety Introduction
- Indicate correct identification of patient prior to performing any patient care measures
- Identify safety considerations for adults of all ages
- Include industry standards and regulations regarding microbiological, physical, and environmental safety
- Apply decision-making related to measures to minimize use of restraints
- Identify evidence-based practices
A national focus on reducing medical errors has been in place since 1999 when the Institute of Medicine (IOM) released a report titled To Err is Human: Building a Safer Health System . This historic report broke the silence surrounding health care errors and encouraged safety to be built into the processes of providing patient care. It was soon followed by the establishment of several safety initiatives by The Joint Commission, including the release of annual National Patient Safety Goals. Additionally, the Quality and Safety Education for Nurses (QSEN) Institute was developed to promote emphasis on high-quality, safe patient care in nursing. This chapter will discuss several safety initiatives that promote a safe health care environment.