Math Core

A list of numbers can be hard to take in all at once. Is a basketball player who scored 12, 15, 9, 15, 11, 14 and 15 points having a good season? A single number that describes the "typical" value, plus one that describes how spread out the values are, makes the list easy to talk about. In this lesson you'll find four of those numbers and use an equation to work backward from an average.

Measures of center

A measure of center is one number that summarizes where the data values cluster. There are three common ones.

Definition

Mean, median and mode

The mean is the sum of the values divided by how many values there are. It's what most people call the "average."
The median is the middle value when the data are listed in order. If there's an even number of values, the median is the mean of the two middle values.
The mode is the value that appears most often. A data set can have one mode, more than one mode, or no mode.

The mean, written as a formula, is

\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}.

A measure of spread

Two data sets can have the same center but look very different. The scores $50$ and $100$ and the scores $74$ and $76$ both have a mean of $75$ , but the first pair is much more spread out.

Definition

Range

The range is the largest value minus the smallest value. It measures how spread out the data are.

Worked example: All four measures

A player scored these points in seven games: $12, 15, 9, 15, 11, 14, 15$ . Find the mean, median, mode and range.

First put the data in order: $9, 11, 12, 14, 15, 15, 15$ .

Mean: the sum is $9 + 11 + 12 + 14 + 15 + 15 + 15 = 91$ , and there are $7$ values, so the mean is $\dfrac{91}{7} = 13$ .
Median: with $7$ values, the middle one is the $4$ th: $14$ .
Mode: $15$ appears three times, more than any other value, so the mode is $15$ .
Range: $15 - 9 = 6$ .

Worked example: An even number of values

Six students recorded how many minutes they read last night: $18, 42, 30, 24, 60, 36$ . Find the median and the mean.

In order: $18, 24, 30, 36, 42, 60$ . The two middle values are $30$ and $36$ , so

\text{median} = \frac{30 + 36}{2} = 33.

The sum is $210$ , so the mean is $\dfrac{210}{6} = 35$ minutes.

Common mistake

Always put the data in order before finding the median. The middle of the unsorted list $18, 42, 30, 24, 60, 36$ is $30$ and $24$ , which gives the wrong median of $27$ .

Outliers

An outlier is a value that is much larger or much smaller than the rest of the data. Outliers pull the mean toward them, but they barely move the median.

Worked example: How an outlier changes the center

Five friends get weekly allowances of $10, $12, $12, $15 and $16. A sixth friend gets $73. Compare the mean and median before and after the sixth friend is included.

Before: the sum is $65$ , so the mean is $\dfrac{65}{5} = 13$ dollars. The median is the middle value, $12$ dollars.

After: the sum is $65 + 73 = 138$ , so the mean is $\dfrac{138}{6} = 23$ dollars. The data in order are $10, 12, 12, 15, 16, 73$ , so the median is $\dfrac{12 + 15}{2} = 13.5$ dollars.

The mean jumped by $10, but the median rose only $1.50. Five of the six friends get less than $23, so the median is the better description of a typical allowance here.

Choosing a measure of center

Use the mean when the data have no outliers. It uses every value.
Use the median when there are outliers, because it isn't pulled toward extreme values.
Use the mode for the most common value, or for data that aren't numbers (such as favorite colors).

Working backward with an equation

If you know the mean you want, you can find a missing value. Since mean $=$ sum $\div$ count, the sum must equal mean $\times$ count.

Worked example: What score do I need?

Maya scored $84$ , $90$ and $78$ on three tests. What does she need on the fourth test to have a mean of $85$ ?

Let $x$ be the fourth score. Write and solve an equation:

\begin{aligned} \frac{84 + 90 + 78 + x}{4} &= 85 \\ \frac{252 + x}{4} &= 85 \\ 252 + x &= 340 \\ x &= 88 \end{aligned}

Maya needs an $88$ . Check: $\dfrac{84 + 90 + 78 + 88}{4} = \dfrac{340}{4} = 85$ .

Tip

The mean is always between the smallest and largest values. If your mean for $9, 11, 12, 14, 15, 15, 15$ came out to $20$ , you'd know right away there was an error.