### Statistics Fundamentals Welcome to statistics! Statistics is a tool for quantifying information. It can help you better understand randomness and uncertainty in the world.

To warm up, let's look at some puzzles about means, one of the most common statistical measures.

# Riddles on Averages

The mean (or average) of a set of numbers is:

$\frac{\text{the sum of the numbers}}{\text{how many numbers are in the set}}$

For example, the mean of the numbers $3, 5,$ and $10$ is:

$\frac{3+5+10}3 = \frac{18}3 = 6$

In the visualization below, the purple dot above the line automatically shows the mean of $a,$ $b,$ $c,$ and $d.$

What is the mean of $5, 11, 19, 21$?
Hint: use the sliders above for some help!

# Riddles on Averages

If the mean of $a$ and $b$ is $10$ and the mean of $c$ and $d$ is $20,$ does the mean of $a, b, c,$ and $d$ have to be $15?$

# Riddles on Averages

If $a < b < c < d$, is it possible for the mean of the four numbers to be between $a$ and $b?$

Hint: what if $a$ is extremely small, and all other values are large?

# Riddles on Averages

What is the average of all integers from $1$ to $100,$ inclusive?

$\left \{ 1, 2, 3, \ldots , 98, 99, 100 \right \}$

# Riddles on Averages

Let's say that in the past decade:

• $100$ countries in the world have had their energy consumption increase by $8\%$ per country.
• $100$ countries in the world have had their energy consumption decrease by $10\%$ per country. The International Energy Association is interested in the percent change across all $200$ countries. Is it accurate to say that energy consumption decreased by $\left | \frac{8 + (-10)}{2} \right | =1\%$ per country?

# Riddles on Averages

As the previous question illustrates, statistics can be deceptive when presented in a particular way. One of the themes of this course will be grappling with this issue and bolstering your ability to spot bad statistics.

First, though, let's look at something very important: the difference between statistics and probability.

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