<s>[INST] Intuition for weighted average. Why $\frac{w_1}{w_1 + w_2}x_1 + \frac{w_2}{w_1 + w_2}x_2 = \frac{\sum_i w_ix_i}{\sum_i w_i}$? I know $\dfrac{w_1}{w_1 + w_2}x_1 + \dfrac{w_2}{w_1 + w_2}x_2 = \dfrac{\sum_i w_ix_i}{\sum_i w_i}$, because $\sum_i w_i$ is common denominator. I'm not asking about this algebra.
It's intuitive that $\dfrac{w_i}{w_1 + w_2}$ weighs $x_i$.
Intuitively, why's $\dfrac{\sum w_ix_i}{\sum w_i}$  Weighted Average?  You are summing $w_ix_i$ and $w_i$ separately. Thus you lost information, because the weight for $x_i$ doesn't appear. When you sum $\sum w_ix_i$ and and $\sum w_i$, these end as totals. They inform nothing about weights! And you can't recover the weights for just these sums!
Can picture explain?

 [/INST] Here is an example from statistics.
The table shows the sales of sugar (in kilograms) during $10$ days:
$$\begin{array}{c|c|c}
\text{Sales of sugar (in kg)}, x & \text{Number of days}, f & \text{Percentage of days}, P(x)\\
\hline
0&1&0.1\\
1&3&0.3\\
2&4&0.4\\
3&2&0.2\\
\hline
&10&1
\end{array}$$
On $3$ days (or during $30\%$ of the $10$-day period) $1$ kilo of sugar was sold each day. Now we need to find the average sales during the $10$-day period.
Method 1. Convert the table data to raw data. Let's assume the following sales took place each day:
$$3,0,3,2,2,1,3,1,1,2$$
So, the average sale is:
$$\frac{\sum x}{n}=\frac{3+0+2+2+2+1+3+1+1+2}{10}=1.7$$
Method 2. Let's simplify the above expression:
$$\frac{\sum x}{n}=\frac{0+1+1+1+2+2+2+2+3+3}{10}=\\
\frac{0\cdot 1+1\cdot 3+2\cdot 4+3\cdot 2}{10}=\\
0\cdot \frac{1}{10}+1\cdot \frac{3}{10}+2\cdot \frac4{10}+3\cdot \frac{2}{10}=\\
0\cdot 0.1+1\cdot 0.3+2\cdot 0.4+3\cdot 0.2=1.7$$
So, the sales figures are elements (x) and the percentage of days (P(x)) are weights. The more percentage a partiular sales figure occurs, the more its impact on the average sales figure.
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