Variability and the Range | Grade 6 Math | LevelKite

You already know how to describe a data set with a measure of center. The mean and the median each use one number to tell you what a typical value is.

A center cannot tell you everything, though. Two data sets can have the same mean and the same median and still look very different. In this lesson, you will learn to describe that difference with one number too.

Start with the table below. Each row is one student. The first column names the student. The second column lists how many minutes that student read on each of five days. The third and fourth columns give the mean and the median of those five values, in minutes.

Reading times of two students over five days

Student	Minutes read on each of 5 days	Mean (minutes)	Median (minutes)
Maya	$28$ , $29$ , $30$ , $31$ , $32$	$30$	$30$
Leo	$10$ , $20$ , $30$ , $40$ , $50$	$30$	$30$

Both students have a mean of $30$ minutes and a median of $30$ minutes. If you looked only at the center, you would say the two students read in the same way.

Now look at the lists themselves. Maya's values stay close together: every day is within $2$ minutes of $30$ . Leo's values are far apart: one day is $10$ minutes and another is $50$ minutes.

This difference is called variability. Variability is how much the values in a data set differ from one another. Values that sit close together have low variability. Values that are spread far apart have high variability.

A measure of variability is a single number that describes how spread out the values of a data set are. A measure of center answers the question "What is a typical value?" A measure of variability answers the question "How much do the values differ?"

One measure of variability is the range. To see what it means, picture every value in a data set as a point on a number line. The least value is the point farthest to the left. The greatest value is the point farthest to the right. The range is the distance between those two points.

You find the distance between two points on a number line by subtracting the lesser number from the greater number. That is why you find the range with a subtraction.

Key concept

Measures of variability and the range

A measure of variability is one number that describes how spread out the values of a data set are. The range is one measure of variability. For a data set of numbers that share the same unit, the range is the distance between the least value and the greatest value.

To find the range, subtract the least value from the greatest value. The range is a single number with the same unit as the data. It is never negative, because the greatest value is never less than the least value. It is $0$ exactly when every value in the data set is the same.

\text{range} = \text{greatest value} - \text{least value}

Go back to Maya and Leo. Their reading times are in minutes, so each range is in minutes too.

Maya: the greatest value is $32$ and the least value is $28$ , so the range is $32 - 28 = 4$ minutes.
Leo: the greatest value is $50$ and the least value is $10$ , so the range is $50 - 10 = 40$ minutes.

Now the two students no longer look the same. All of Maya's reading times fit between $28$ and $32$ , a span of only $4$ minutes. Leo's reading times stretch from $10$ to $50$ , a span of $40$ minutes. The means match and the medians match. But measured by the range, Leo's reading times vary much more than Maya's.

Worked example

Find the range of whole-number data

A basketball player scored these points in six games: $12$ , $7$ , $15$ , $9$ , $12$ , $5$ . Find the range of the points scored.

Name what each value measures. Each value is the number of points scored in one game, so the range will also be in points.
Write the values in order from least to greatest. The list follows the order of the games, not the size of the values, so ordering it makes the least and greatest values easy to find. Keep both 12s, because each one is a data value.
$5, 7, 9, 12, 12, 15$
Read the least value from the start of the ordered list and the greatest value from the end.
$\text{least} = 5, \quad \text{greatest} = 15$
Subtract the least value from the greatest value. The result is the distance between them on a number line, which is the range.
$15 - 5 = 10$

Answer

The range of the points scored is $10$ points.

Worked example

Find the range of decimal data

Seven runners finished a 50-meter sprint in these times, in seconds: $9.4$ , $8.7$ , $10.2$ , $9.1$ , $8.9$ , $9.6$ , $9.4$ . Find the range of the times.

Order the times from least to greatest so the least and greatest are easy to read. Compare the whole-number parts first, then the tenths. Then count: the problem gives seven times, so your ordered list must have seven too.
$8.7, 8.9, 9.1, 9.4, 9.4, 9.6, 10.2$
Read the least time from the start of the ordered list and the greatest time from the end.
$\text{least} = 8.7, \quad \text{greatest} = 10.2$
Subtract the least time from the greatest time to get the distance between them. Because 8.7 is 8 plus 0.7, taking away 8.7 gives the same result as taking away 8 and then taking away 0.7.
$10.2 - 8.7 = 10.2 - 8 - 0.7 = 2.2 - 0.7 = 1.5$
Check the subtraction with its inverse operation, addition. Addition undoes subtraction, so the least value plus the range must give back the greatest value. This check confirms the arithmetic only. It cannot tell you whether you picked the correct least and greatest values.
$8.7 + 1.5 = 10.2$

Answer

The range of the sprint times is $1.5$ seconds.

Worked example

Compare two data sets using the range

Two bus routes were timed on six mornings. The times are in minutes. Route A: $22$ , $25$ , $24$ , $23$ , $26$ , $24$ . Route B: $15$ , $33$ , $24$ , $18$ , $30$ , $24$ . Both routes have a median time of $24$ minutes. Measured by the range, which route's travel times vary less?

Order Route A's times from least to greatest, so the least time is first and the greatest time is last.
$22, 23, 24, 24, 25, 26$
Route A's least time is 22 and its greatest time is 26. Subtract the least from the greatest to get Route A's range, in minutes.
$26 - 22 = 4$
Order Route B's times in the same way, so its least and greatest times are easy to read.
$15, 18, 24, 24, 30, 33$
Route B's least time is 15 and its greatest time is 33. Subtract the least from the greatest to get Route B's range, in minutes.
$33 - 15 = 18$
Compare the two ranges. Both are in minutes, so you can compare them directly. The smaller range belongs to the route whose times all fit inside the narrower span.
$4 < 18$

Answer

Measured by the range, Route A's travel times vary less. Route A's range is $4$ minutes and Route B's range is $18$ minutes, even though both medians are $24$ minutes.

The range is quick to find because it uses only two values from the data set: the least and the greatest. For the same reason, it tells you nothing about how the values in between are arranged. One unusual value can also change it a lot.

Try this with Route A. Suppose a seventh morning took $40$ minutes. The greatest value becomes $40$ and the least value is still $22$ , so the range becomes $40 - 22 = 18$ minutes. That equals Route B's range, even though the other six Route A times still fit inside a span of $4$ minutes. So read the range as exactly what it is: the distance between the least value and the greatest value.

Here is the idea to keep. A measure of center tells you what a typical value is. A measure of variability tells you how much the values differ, and it does this with one number. The range is one measure of variability. To find it, find the greatest value and the least value in the data set, then subtract the least from the greatest. A small range means every value lies close to the others. A large range means the least and greatest values are far apart.

Next, you will find quartiles and the interquartile range, which is another measure of variability.