Mean, Median, and Mode

Step 1: Let's Learn

Read it, or press Listen and follow the words.

The mean adds and divides, the median is the middle value in order, and the mode is the most frequent value.

Resistance

The median is resistant: one extreme value barely moves it. The mean is not, because every value enters the total.

Why it matters

One billionaire in a small town raises the mean income enormously and the median hardly at all. The median describes the typical resident.

Skew pulls the mean

In a right-skewed distribution the mean sits above the median, dragged by the tail. In a left-skewed one it sits below.

Choosing

Use the mean for roughly symmetric data with no outliers. Use the median when the data is skewed or has extremes.

The mode

The mode is the only measure that works for categorical data. The most common eye colour is a mode.

Three centres

The mean is the fair-share value, the median the middle value, the mode the most common. They coincide for symmetric data and diverge for skewed data, sometimes dramatically.

The mean is dragged by outliers

One extreme value moves the mean and barely moves the median. That is why incomes and house prices are reported as medians — the mean would describe nobody in the distribution.

Skew pulls the mean towards the tail

In right-skewed data the mean exceeds the median. Comparing the two is a quick numerical test for skew that needs no graph, and it works reliably for unimodal distributions.

Choosing one is a claim

Someone arguing that pay is high will quote the mean; someone arguing it is low will quote the median. Both may be accurate. Noticing which was chosen is how you read a statistical claim critically.

Step 2: Try It Yourself

Tap and try it out.

Push one value far to the right. Watch how much it would move a mean, and how little it moves the middle.

12345678

Score

12 things were measured in all.

The tallest column is at 3, with 4 marks.

Score 1

Score 2

Score 3

Score 4

Score 5

Score 8

Step 3: Watch an Example

One step at a time.

Watch Diego Compare Two Centres

Five salaries: 30, 32, 35, 38 and 400 thousand.

Step 1
The total is 535, so the mean is 535 ÷ 5 = 107 thousand.

Step 4: Your Turn

Practice makes it stick.

The Average

Problem 1 of 2

Data 4, 6, 8, 10, 12. What is the mean?

The Middle

Problem 2 of 2

Data 3, 5, 9, 11, 100. What is the median?

Find the Centre

1 of 8

Data 2, 4, 6. Mean?

2 of 8

Data 1, 3, 3, 7. Median?

3 of 8

Data 2, 5, 5, 5, 9. Mode?

4 of 8

Data 10, 20, 30, 40. Mean?

5 of 8

A distribution is skewed right. Is the mean above or below the median? 1 above, 2 below.

6 of 8

Which measure works for categorical data? 1 mean, 2 median, 3 mode.

7 of 8

Sort each situation by which centre reports it best.

Tap something to move it.

Empty

Empty

8 of 8

Data 5, 5, 5, 5. Mean?

Step 5: Quick Check

Show what you know.

Question 1 of 2

Data 3, 7, 11. What is the mean?

Question 2 of 2

Which measure of centre resists outliers?

What You Learned

The mean averages, the median takes the middle, and the mode is most frequent.
The median resists outliers; the mean does not.
A skewed distribution pulls the mean toward its tail.