Statistics

Riddles on Averages

Welcome to statistics! Statistics is a tool for quantifying information and can reveal patterns and truth about the world in surprising ways. To get you warmed up, this quiz has some puzzles on the most common of statistical measures: that of mean, or average.

Riddles on Averages

Statistics Introduction

             

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 \( \frac{3+5+10}3 = \frac{18}3 = 6 .\)

The visualization below will automatically show the mean (as the dot above the number line) given four numbers \(a,\) \(b,\) \(c,\) and \(d.\) For example, what is the mean of \[ 5, 11, 19, 21 ?\]

Riddles on Averages

Statistics Introduction

             

Riddles on Averages

If the mean of \(a\) and \(b\) is 10 and the mean of \(c\) and \(d\) is 20, must it be the case that the mean of \(a, b, c,\) and \(d\) together is 15?

Riddles on Averages

Statistics Introduction

             

Riddles on Averages

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

Riddles on Averages

Statistics Introduction

             

Riddles on Averages

What is the average of all integers from 1 to 100 inclusive? (That is, \( 1, 2, 3, \ldots, 98, 99, 100.) \)

Riddles on Averages

Statistics Introduction

             

Riddles on Averages

According to World Bank data, if you take the percentage of energy consumption that is renewable energy between the years 1990 and 2015,

  • Of 100 countries that had their percentage of use go up, there was an average increase of 7.6% per country;
  • Of 100 countries that had their percentage of use go down, there was an average decrease of 11.5% per country.

If we look at the percentage of renewable energy use across the world (using just the countries from the data above), is it appropriate to conclude the average change is \( \frac{7.6 + (-11.5)}{2} = -1.95 \) percent?

Riddles on Averages

Statistics Introduction

             

Riddles on Averages

In reality, world renewable energy use increased from roughly 17% to 18%, although it should be noted there is volatility in the data:

This graph also includes every single country for which there is data, not just the 200 from the last question. This graph also includes every single country for which there is data, not just the 200 from the last question.

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

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

Riddles on Averages

Statistics Introduction

             
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