  # Range of a Set of Data in Math and Statistics

## Definition of a Range (Statistics)

In statistics, the range is a measure of spread: it’s the difference between the highest value and the lowest value in a data set. Note: In some areas of math, the range can also mean the entire range of numbers — for example, the range of cell phone prices might be \$40 to \$550. In calculus, the range is defined differently. It is all of the output values of a function. See: How to Find the Domain and Range of a Function.

## How to Find a Range in Statistics

Watch the video, or read the article below:

The same two steps are used whether you are dealing with positive numbers, negative numbers, or time (e.g. seconds or minutes).

## How to Find a Range

Example question 1: What is the range for the following set of numbers? 10, 99, 87, 45, 67, 43, 45, 33, 21, 7, 65, 98?

Step 1: Sort the numbers in order, from smallest to largest:
7, 10, 21, 33, 43, 45, 45, 65, 67, 87, 98, 99

Step 2: Subtract the smallest number in the set from the largest number in the set:
99 – 7 = 92
The range is 92

That’s it!

Example question 2: What is the range of these integers?
14, -12, 7, 0, -5, -8, 17, -11, 19
Step 1: Sort the numbers in order, from smallest to largest:
-12, -11, -8, -5, 0, 7, 14, 17, 19

Step 2: Subtract the smallest number in the set from the largest number in the set:
19 – -12 = 19 + 12 = 31
The range is 31.

That’s it!

Example question 3: What is the range of the following times?
2.7 hrs, 8.3 hrs, 3.5 hrs, 5.1 hrs, 4.9 hrs
Step 1: Sort the numbers in order, from smallest to largest:
2.7, 3.5, 4.9, 5.1, 8.3

Step 2: Subtract the smallest number in the set from the largest number in the set:
8.3 hr – 2.7 hr = 5.6 hr
The range is 5.6 hr.

That’s how to find a range!

### Another Example.

Problem: You take 7 statistics tests over the course of a semester. You score 94, 88, 73, 84, 91, 87, and 79. What is the range of your scores?
Solution:
Step 1: Order your scores from smallest to largest:
73, 79, 84, 87, 88, 91, 94.
Step 2: Subtract the smallest number from the highest = 94 – 73 = 21.

## When it Might be Misleading

The range only uses the smallest and the largest number in a set; The rest of the values are ignored. That could lead to a misleading result. Take the above test scores. Let’s say you had the flu one test day and scored a 10. Assuming your highest score on another test was 94, then:
94 – 10 = 84!
That’s not a good reflection of your overall test performance at all.

The score of 10 in the example above is what we call an outlier. It’s an extremely high or low value that can throw off stats. That’s why other measures of spread are sometimes preferred, like the mean.

## Rule of Thumb

The rule of thumb says that the range is about four times the standard deviation. The standard deviation is another measure of spread in statistics. It tells you how your data is clustered around the mean. What the rule of thumb tells you in most cases is that the bulk of the data can be found pretty close to the mean (within a couple of standard deviations); The result is that those erroneous “outliers” should have very little effect on your final statistic.

Procedure for finding a standard deviation using the rule of thumb:
Step 1: Find the range.
Step 2: Divide Step 1 by four.

The rule of thumb doesn’t work that well for small data sets. And it doesn’t work at all if you don’t have data that fits a normal distribution. That’s why you’ll rarely see it used in statistics. See: Range rule of thumb.

## Range in Excel 2013-2016

Watch the video or read the steps below:

To find a range in Excel, you have two options: you can use the MAX and MIN functions to find the largest and smallest numbers in a data set and then you can subtract the two. For example, if you had a data set in cells A1 to A10, you’d need three formulas in three blank cells. Lastly the format (assuming you put these formulas into cells B1:B3) would be:
B1 = MAX(A1:A10)
B2 = MIN(A1:A10)
B3 =(B1-B2)
A much easier way is to use Data Analysis, where in just a couple of clicks (with no entering formulas) you can display a variety of summary statistics, including the range (How to load the Data Analysis Toolpak).

### Range in Excel: Data Analysis Steps

1st: Click the “Data” tab and then click “Data Analysis.”

2nd: Click “Descriptive Statistics” and then click “OK.”

3rd: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box

4th: Click the radio button for Rows or Columns, depending on how your data is laid out.

5th: Click the “Labels in first row” box if your data has column headers.

6th: Click the “Descriptive Statistics” check box.

7th: Select a location for your output. For example, click the “New Worksheet” radio button.

8th: Click “OK.”

## Origins

The origin of the word “Range” in mathematics is unknown, but a few early uses of the word as it’s used in statistics can be found as far back as 1848, in H. Lloyd, “On Certain Questions Connected with the Reduction of Magnetical and Meteorological Observations,” Proceedings of the Royal Irish Academy, 4, 180-183 (David, 1995). Following this, the word was later used in a book on Calculus in 1865: The Differential Calculus by John Spare mentions: “…in respect to the range of values which the function and its variable may sustain, and to their mutual dependence” [University of Michigan Digital Library]. Although technically not statistics, the range in calculus has practically the same meaning (the spread from the smallest value to the largest).

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