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See also: What is a Sample Range?
What is a range in statistics?
In statistics, the range is the difference between the largest and smallest values in a data set. For example, if a data set contains the values 1, 5, 6, 8, and 9, the range is calculated as 9 – 1 = 8.
In mathematics, particularly in functions, the range refers to the set of output values (y-values) corresponding to a given domain.
Despite the difference in definitions, calculus and statistics share the term “range” because, in both contexts, it represents the extent or spread of values.
- In elementary statistics, the range is found by subtracting the smallest value from the largest.
- In calculus and functions, the range consists of all possible output values.
Watch the video below for a brief overview of the range in statistics or read on:
Can’t see the video? Click here to watch it on YouTube.
How to Find a Range in Statistics
The same two steps are used to find a range whether you are dealing with positive numbers, negative numbers, or time (e.g. seconds or minutes).
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?
- Sort the numbers in order, from smallest to largest: 7, 10, 21, 33, 43, 45, 45, 65, 67, 87, 98, 99
- 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
- Sort the numbers in order, from smallest to largest: -12, -11, -8, -5, 0, 7, 14, 17, 19
- 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.
- Sort the numbers in order, from smallest to largest: 2.7, 3.5, 4.9, 5.1, 8.3
- 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 of How to Find a Range
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:
- Order your scores from smallest to largest: 73, 79, 84, 87, 88, 91, 94.
- Subtract the smallest number from the highest = 94 – 73 = 21. Answer: 21.
When it Might be Misleading
The range in statistics is sensitive to outliers or extreme values, as it only considers the two most extreme data points (the minimum and the maximum). Outliers are data points significantly outside the typical range of values. When an outlier exists, it can distort the range, causing it to seem larger or smaller than it truly is.
For instance, consider a dataset containing heights for a group of individuals. The average height is 5’10” with a standard deviation of 2″. The range of the dataset would be 5’6″ to 6’2″. However, if one individual in the dataset is 7′ tall, the range would suddenly become 5’6″ to 7′. This would make the heights in the dataset appear more varied than they actually are.
Other measures of dispersion, such as the interquartile range (IQR) or standard deviation, can provide a clearer understanding of the data’s variability.
Additionally, a range can be misleading if the dataset does not follow a normal distribution. A normal distribution is symmetrical, with equal mean, median, and mode values. If the dataset is not normally distributed, the range might not accurately represent variability.
For example, consider a dataset of test scores for a group of students. The average test score is 80, and the standard deviation is 10. The range of the dataset would be 70 to 90. However, if the dataset is right-skewed, the range may be misleading. This is because the outliers on the right side of the distribution will distort the range, making it seem larger than it really is.
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 range using the rule of thumb:
- Find the standard deviation (e.g. 8).
- Divide Step 1 by four (e.g. 8 / 4 = 2).
The rule of thumb doesn’t work that well to find a range 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.
More info: Range rule of thumb.
Find a Range in Excel 2013-2016
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
- Click the “Data” tab and then click “Data Analysis.”
- Click “Descriptive Statistics” and then click “OK.”
- 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
- Click the radio button for Rows or Columns, depending on how your data is laid out.
- Click the “Labels in first row” box if your data has column headers.
- Click the “Descriptive Statistics” check box.
- Select a location for your output. For example, click the “New Worksheet” radio button. 8th: Click “OK.”
History of the domain and range in statistics
The origin of the term range in statistics remains uncertain. However, some early instances of its usage in a statistical context can be traced back to 1848, in H. Lloyd’s paper titled “On Certain Questions Connected with the Reduction of Magnetical and Meteorological Observations,” published in the Proceedings of the Royal Irish Academy, 4, 180-183. Later, in 1865, the term appeared in a calculus book called The Differential Calculus by John Spare, which states: “…in respect to the range of values which the function and its variable may sustain, and to their mutual dependence” [1]. Although this reference pertains to calculus rather than statistics, the concept of range in both fields shares a similar meaning, essentially denoting the spread from the smallest to the largest value.
According to MacTutor [1], the earliest use of the word domain was used in 1886 by Arthur Cayley in “On Linear Differential Equations” in the Quarterly Journal of Pure and Applied Mathematics: “… for points x within the domain of the point a“.
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