Mean, Median & Mode

These are the three common ways to describe the middle, or typical value, of a list of numbers.

mean = sum of valueshow many values

For 2, 4, 4, 5, 10: the mean is 5, the median (middle) is 4, and the mode (most common) is 4.

What each symbol means

When you have a list of numbers, you often want a single number that sums up "what is typical". There are three standard ways to do that, and they can give different answers. Knowing all three, and when each is useful, is the whole skill.

Mean (the average)

Add all the values, then divide by how many values there are. This is what most people mean by "average".

Median (the middle)

Put the numbers in order from smallest to largest, then find the one in the middle. If there are two middle numbers (an even count), take their mean.

Mode (the most common)

The mode is simply the value that shows up most often. A list can have one mode, more than one, or none at all.

When the mean can mislead: One very large or very small value pulls the mean toward it. In "1, 2, 2, 3, 100" the mean is over 21, but the median is 2, which describes the group much better. That is why the median is often used for things like house prices and incomes.

A worked example

Find the mean, median, and mode of 2, 4, 4, 5, 10

  1. Mean: add them, 2 + 4 + 4 + 5 + 10 = 25, then divide by 5, giving 25 ÷ 5 = 5.
  2. Median: the numbers are already in order, and the middle one (the 3rd of 5) is 4.
  3. Mode: the value that appears most is 4 (it shows up twice).
  4. So mean = 5, median = 4, mode = 4.

Why the mean sits at the balance point

  1. Imagine the numbers as equal weights placed along a ruler at their values.
  2. The mean is the exact spot where the ruler would balance on a fingertip.
  3. Adding up all the values and sharing that total equally among the positions is what finds that balance point.
  4. That is why "add them all, divide by how many" gives the number that best represents the group as a single fair share.

Practice questions

Question 1

Find the mean of 3, 5, 7, 9.

  1. The mean is the sum of the values divided by how many there are.
  2. Add them all up: 3 + 5 + 7 + 9 = 24.
  3. Count the values: there are 4 of them.
  4. Divide: 24 ÷ 4 = 6. The mean is 6.

Answer: 6

Question 2

Find the median of 8, 3, 5, 9, 1.

  1. The median is the middle value once the numbers are in order.
  2. Put them in order from smallest to largest: 1, 3, 5, 8, 9.
  3. There are 5 numbers, so the middle one is the 3rd.
  4. The 3rd value is 5, so the median is 5.

Answer: 5

Question 3

Find the mode of 2, 7, 7, 4, 7, 1.

  1. The mode is the value that appears most often.
  2. Count how many times each value shows up.
  3. The value 7 appears three times, more than any other.
  4. So the mode is 7.

Answer: 7

Question 4

Find the median of 4, 6, 8, 10 (an even count).

  1. The median is the middle value once the numbers are in order.
  2. These are already in order: 4, 6, 8, 10.
  3. With an even count there are two middle numbers, 6 and 8.
  4. Take their mean: (6 + 8) ÷ 2 = 14 ÷ 2 = 7. The median is 7.

Answer: 7

Question 5

A student scores 80, 90, 100, 90, 90. What is the mode?

  1. The mode is the value that appears most often.
  2. Count each score: 80 once, 100 once, and 90 three times.
  3. The score 90 shows up more than any other.
  4. So the mode is 90.

Answer: 90