Math Core

The mean is one way to describe a typical value, but it isn't always the best one. One very large or very small value can pull the mean far away from most of the data. The median is a measure of center that doesn't get pushed around so easily, and the mode tells you the most common value.

The median

The median is the middle value when the data is in order. Half of the data is at or below it, and half is at or above it.

Definition

Median

The median is the middle value of a data set listed in order from least to greatest.

If there is an odd number of values, the median is the one in the middle.
If there is an even number of values, there are two middle values. The median is the mean of those two: add them and divide by 2.

Worked example: An odd number of values

Find the median of $7, \ 3, \ 12, \ 5, \ 9, \ 4, \ 10$ .

Step 1: put the values in order. $3, \ 4, \ 5, \ 7, \ 9, \ 10, \ 12$

Step 2: find the middle. There are 7 values, so the middle one is the 4th, with 3 values on each side:

3, \ 4, \ 5, \ \boxed{7}, \ 9, \ 10, \ 12

The median is 7.

Worked example: An even number of values

Six students recorded how many minutes they spent reading: $25, \ 40, \ 15, \ 30, \ 20, \ 45$ . Find the median.

Step 1: order. $15, \ 20, \ 25, \ 30, \ 40, \ 45$

Step 2: find the middle. With 6 values, the two middle values are the 3rd and 4th: 25 and 30.

Step 3: find the number halfway between them. $\dfrac{25 + 30}{2} = \dfrac{55}{2} = 27.5$

The median is 27.5 minutes. Notice it isn't one of the data values, and that's fine.

Common mistake

The most common mistake is finding the middle of the list without putting it in order first. In the first example, the middle of the unordered list $7, 3, 12, \mathbf{5}, 9, 4, 10$ is 5, but the median is 7. Always order the data first.

The mode

Definition

Mode

The mode is the value that appears most often. A data set can have one mode, more than one mode, or no mode at all (if every value appears the same number of times).

$2, \ 5, \ 5, \ 6, \ 8$ : the mode is 5.
$1, \ 1, \ 3, \ 4, \ 4, \ 7$ : there are two modes, 1 and 4.
$3, \ 6, \ 8, \ 10$ : every value appears once, so there is no mode.

On a dot plot, the mode is the value with the tallest stack.

Worked example: Median and mode from a dot plot

The dot plot shows the shoe sizes of 11 students.

Shoe sizes

Mode: The tallest stack is above 7, so the mode is 7.

Median: A dot plot already puts the data in order from left to right. There are 11 values, so the median is the 6th value. Count marks from the left: size 5 is the 1st, size 6 covers the 2nd and 3rd, and size 7 covers the 4th through 7th. The 6th value is 7.

Mean or median?

Worked example: When an outlier pulls the mean

Five people work at a small shop. Their weekly pay, in dollars, is

400, \ 420, \ 450, \ 480, \ 1{,}750

The 1,750 is the owner's pay.

Mean: $\dfrac{400 + 420 + 450 + 480 + 1750}{5} = \dfrac{3500}{5} = 700$
Median: The values are already in order. The middle value is 450.

Four of the five people earn less than $700, so the mean doesn't describe a typical worker very well. The one very large value pulled the mean up. The median, $450, is much closer to what most people earn.

Choosing a measure of center

The median is not affected much by outliers or lopsided (skewed) data. Use it when the data has values far away from the rest.
The mean uses every value, so an outlier pulls it toward that outlier. It works well when the data is roughly symmetric with no outliers.

Tip

To find the middle position in a long ordered list of $n$ values, compute $\dfrac{n + 1}{2}$ . For 11 values, that's position 6. For 20 values, it's 10.5, which tells you to average the 10th and 11th values.

Practice

Practice 1

Find the median of $3, \ 9, \ 4, \ 7, \ 12$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find the median of $15, \ 8, \ 21, \ 10, \ 13, \ 18$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find the mode of $4, \ 6, \ 6, \ 7, \ 8, \ 8, \ 8, \ 9$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Find all the modes of $2, \ 3, \ 3, \ 5, \ 7, \ 7, \ 9$ . Separate them with a comma.

Separate answers with commas, e.g. 2, -5

Practice 5

The dot plot shows how many times 13 students went to the library last month. What is the median?

Library visits last month

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Here are the prices, in thousands of dollars, of five houses on a street: $210, \ 225, \ 230, \ 245, \ 1{,}340$ . Which measure of center best describes a typical house price on this street?

Practice 7

The data set $4, \ 6, \ 9, \ 11$ gets one more value, $x$ . After that, the median of the five values is 8. What is $x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.