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# Statistics

## 1.1 Sales for the past week was the following amounts: [3505, 2400, 3027, 2798, 3700, 3250, 2689]. Without using library functions, what is the mean, variance, and standard deviation of of sales from last week? (for extra bonus points, write your own function that can calculate these two values for any sized list)


```
def sales_mean (lst):
  return sum(lst)/len(lst)

```


```
lst = [3505,2400,3027,2798,3700,3250,2689]
```


```
sales_mean(lst)
```




    3052.714285714286




```
m = sales_mean(lst)
```


```
var = sum((xi-m)**2 for xi in lst)/len(lst)
```


```
var
```




    183761.06122448976




```
std = var**(1/2)
```


```
std
```




    428.67360686714756



## 1.2 Find the covariance between last week's sales numbers and the number of customers that entered the store last week: [127, 80, 105, 92, 120, 115, 93] (you may use librray functions for calculating the covariance since we didn't specifically talk about its formula)


```
import numpy as np
```


```
lst2 = [127,80,105,92,120,115,93]
```


```
np.cov(lst)
```




    array(214387.9047619)



## 1.3 Find the standard deviation of customers who entered the store last week. Then, use the standard deviations of both sales and customers to standardize the covariance to find the correlation coefficient that summarizes the relationship between sales and customers. (You may use library functions to check your work.)


```
from statistics import stdev
```


```
stdev(lst)
```




    463.02041505953576



## 1.4 Use pandas to import a cleaned version of the titanic dataset from the following link: [Titanic Dataset](https://raw.githubusercontent.com/Geoyi/Cleaning-Titanic-Data/master/titanic_clean.csv)

## Calculate the variance-covariance matrix and correlation matrix for the titanic dataset's numeric columns. (you can encode some of the categorical variables and include them as a stretch goal if you finish early)


```
import pandas as pd
```


```
df = pd.read_csv('https://raw.githubusercontent.com/Geoyi/Cleaning-Titanic-Data/master/titanic_clean.csv')
```


```
df.cov()
```




<div>
<style scoped>
    .dataframe tbody tr th:only-of-type {
        vertical-align: middle;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }

    .dataframe thead th {
        text-align: right;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>Unnamed: 0</th>
      <th>pclass</th>
      <th>survived</th>
      <th>age</th>
      <th>sibsp</th>
      <th>parch</th>
      <th>fare</th>
      <th>body</th>
      <th>has_cabin_number</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>Unnamed: 0</th>
      <td>143117.500000</td>
      <td>284.357034</td>
      <td>-53.967125</td>
      <td>-1442.939812</td>
      <td>25.828746</td>
      <td>1.172783</td>
      <td>-9410.735123</td>
      <td>591.579132</td>
      <td>-95.438885</td>
    </tr>
    <tr>
      <th>pclass</th>
      <td>284.357034</td>
      <td>0.701969</td>
      <td>-0.127248</td>
      <td>-3.954605</td>
      <td>0.053090</td>
      <td>0.013287</td>
      <td>-24.227788</td>
      <td>-2.876653</td>
      <td>-0.249992</td>
    </tr>
    <tr>
      <th>survived</th>
      <td>-53.967125</td>
      <td>-0.127248</td>
      <td>0.236250</td>
      <td>-0.314343</td>
      <td>-0.014088</td>
      <td>0.034776</td>
      <td>6.146023</td>
      <td>0.000000</td>
      <td>0.061406</td>
    </tr>
    <tr>
      <th>age</th>
      <td>-1442.939812</td>
      <td>-3.954605</td>
      <td>-0.314343</td>
      <td>165.850021</td>
      <td>-2.559806</td>
      <td>-1.459378</td>
      <td>114.416613</td>
      <td>81.622922</td>
      <td>1.463138</td>
    </tr>
    <tr>
      <th>sibsp</th>
      <td>25.828746</td>
      <td>0.053090</td>
      <td>-0.014088</td>
      <td>-2.559806</td>
      <td>1.085052</td>
      <td>0.336833</td>
      <td>8.641768</td>
      <td>-8.708471</td>
      <td>-0.003946</td>
    </tr>
    <tr>
      <th>parch</th>
      <td>1.172783</td>
      <td>0.013287</td>
      <td>0.034776</td>
      <td>-1.459378</td>
      <td>0.336833</td>
      <td>0.749195</td>
      <td>9.928031</td>
      <td>4.237190</td>
      <td>0.013316</td>
    </tr>
    <tr>
      <th>fare</th>
      <td>-9410.735123</td>
      <td>-24.227788</td>
      <td>6.146023</td>
      <td>114.416613</td>
      <td>8.641768</td>
      <td>9.928031</td>
      <td>2678.959738</td>
      <td>-179.164684</td>
      <td>10.976961</td>
    </tr>
    <tr>
      <th>body</th>
      <td>591.579132</td>
      <td>-2.876653</td>
      <td>0.000000</td>
      <td>81.622922</td>
      <td>-8.708471</td>
      <td>4.237190</td>
      <td>-179.164684</td>
      <td>9544.688567</td>
      <td>3.625689</td>
    </tr>
    <tr>
      <th>has_cabin_number</th>
      <td>-95.438885</td>
      <td>-0.249992</td>
      <td>0.061406</td>
      <td>1.463138</td>
      <td>-0.003946</td>
      <td>0.013316</td>
      <td>10.976961</td>
      <td>3.625689</td>
      <td>0.174613</td>
    </tr>
  </tbody>
</table>
</div>



# Orthogonality

## 2.1 Plot two vectors that are orthogonal to each other. What is a synonym for orthogonal?


```
import matplotlib.pyplot as plt
```


```
red = [0,3]
blue = [3,0]
```


```
plt.arrow(0,0, blue[0], blue[1], head_width=0.05, head_length=0.05, color='blue')
plt.arrow(0,0, red[0], red[1], head_width=0.05, head_length=0.05, color='red')
plt.xlim(-1,4)          
plt.ylim(-1,4)
```


```
# Perpendicular
```

## 2.2 Are the following vectors orthogonal? Why or why not?

\begin{align}
a = \begin{bmatrix} -5 \\ 3 \\ 7 \end{bmatrix}
\qquad
b = \begin{bmatrix} 6 \\ -8 \\ 2 \end{bmatrix}
\end{align}


```
a = np.array([
              [-5],
              [3],
              [7]
])
b = np.array([
              [6],
              [-8],
              [2]
])
```


```
np.vdot(a,b)
# No because thed dot product is not zero
```




    -40




```
plt.arrow(0,0, a[0], a[1], head_width=0.05, head_length=0.05, color='blue')
plt.arrow(0,0, b[0], b[1], head_width=0.05, head_length=0.05, color='red')
```

## 2.3 Compute the following values: What do these quantities have in common?

## What is $||c||^2$? 

## What is $c \cdot c$? 

## What is $c^{T}c$?

\begin{align}
c = \begin{bmatrix} 2 & -15 & 6 & 20 \end{bmatrix}
\end{align}


```
c = np.array([2,-15,6,20])
```


```
square_c =([
            [2,-15],
            [6,20]
])
```


```
square_c2 = np.linalg.det(square_c)
```


```
c**2
```




    array([  4, 225,  36, 400])




```
np.dot(c,c)
```




    665




```
np.dot(c.T,c)
# These two are equal because c's tranpsotion doesn't change anything
# as it is a 1 d list
```




    665



# Unit Vectors

## 3.1 Using Latex, write the following vectors as a linear combination of scalars and unit vectors:

\begin{align}
d = \begin{bmatrix} 7 \\ 12 \end{bmatrix}
\qquad
e = \begin{bmatrix} 2 \\ 11 \\ -8  \end{bmatrix}
\end{align}

\begin{align}
d =  7\hat{i} + 12\hat{j}
\end{align}
\begin{align}
e = 2\hat{i} + 11\hat{j} - 8\hat{i}
\end{align}

## 3.2 Turn vector $f$ into a unit vector:

\begin{align}
f = \begin{bmatrix} 4 & 12 & 11 & 9 & 2 \end{bmatrix}
\end{align}


```

```

# Linear Independence / Dependence 

## 4.1 Plot two vectors that are linearly dependent and two vectors that are linearly independent (bonus points if done in $\mathbb{R}^3$).

# Span

## 5.1 What is the span of the following vectors?

\begin{align}
g = \begin{bmatrix} 1 & 2 \end{bmatrix}
\qquad
h = \begin{bmatrix} 4 & 8 \end{bmatrix}
\end{align}


```

```

## 5.2 What is the span of $\{l, m, n\}$?

\begin{align}
l = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}
\qquad
m = \begin{bmatrix} -1 & 0 & 7 \end{bmatrix}
\qquad
n = \begin{bmatrix} 4 & 8  & 2\end{bmatrix}
\end{align}


```

```

# Basis

## 6.1 Graph two vectors that form a basis for $\mathbb{R}^2$




```

```

## 6.2 What does it mean to form a basis?



# Rank

## 7.1 What is the Rank of P?

\begin{align}
P = \begin{bmatrix} 
1 & 2 & 3 \\
 -1 & 0 & 7 \\
4 & 8  & 2
\end{bmatrix}
\end{align}

## 7.2 What does the rank of a matrix tell us?



# Linear Projections

## 8.1 Line $L$ is formed by all of the vectors that can be created by scaling vector $v$ 
\begin{align}
v = \begin{bmatrix} 1 & 3 \end{bmatrix}
\end{align}

\begin{align}
w = \begin{bmatrix} -1 & 2 \end{bmatrix}
\end{align}

## find $proj_{L}(w)$

## graph your projected vector to check your work (make sure your axis are square/even)


```

```

# Stretch Goal

## For vectors that begin at the origin, the coordinates of where the vector ends can be interpreted as regular data points. (See 3Blue1Brown videos about Spans, Basis, etc.)

## Write a function that can calculate the linear projection of each point (x,y) (vector) onto the line y=x. run the function and plot the original points in blue and the new projected points on the line y=x in red. 

## For extra points plot the orthogonal vectors as a dashed line from the original blue points to the projected red points.


```
import pandas as pd
import matplotlib.pyplot as plt

# Creating a dataframe for you to work with -Feel free to not use the dataframe if you don't want to.
x_values = [1, 4, 7, 3, 9, 4, 5 ]
y_values = [4, 2, 5, 0, 8, 2, 8]

data = {"x": x_values, "y": y_values}

df = pd.DataFrame(data)

df.head()

plt.scatter(df.x, df.y)
plt.show()
```


```

```

88bc62a14c31f36848b7dda6ec230ad9e4a968f7
ipynb
Jupyter Notebook
DS-Unit-1-Sprint-3-Linear-Algebra/module2-intermediate-linear-algebra/Copy_of_LS_DS_132_Intermediate_Linear_Algebra_Assignment.ipynb
Cal-0/Cal-0-dashboard.github.io
520ba74b593b8b27e7adbca37326f1dd23ba2d40
MIT
DS-Unit-1-Sprint-3-Linear-Algebra/module2-intermediate-linear-algebra/Copy_of_LS_DS_132_Intermediate_Linear_Algebra_Assignment.ipynb
Cal-0/Cal-0-dashboard.github.io
520ba74b593b8b27e7adbca37326f1dd23ba2d40
MIT
DS-Unit-1-Sprint-3-Linear-Algebra/module2-intermediate-linear-algebra/Copy_of_LS_DS_132_Intermediate_Linear_Algebra_Assignment.ipynb
Cal-0/Cal-0-dashboard.github.io
520ba74b593b8b27e7adbca37326f1dd23ba2d40
MIT
Qwen/Qwen-72B
1. YES
2. YES