Range Formula

The range is the difference between the maximum and minimum values in a data set, giving the simplest measure of overall spread.

The Formula

range=maximum−minimum\text{range} = \text{maximum} - \text{minimum}

When to use: Range tells you how spread out your data is from end to end. If the tallest kid is 5 feet and the shortest is 4 feet, the range is 1 foot - that's the 'stretch' of heights.

Quick Example

Quiz scores: 72, 85, 90, 68, 95. Range=95−68=27 points\text{Range} = 95 - 68 = 27 \text{ points}

Notation

RR denotes the range. xmax⁡x_{\max} is the maximum value and xmin⁡x_{\min} is the minimum value in the dataset.

What This Formula Means

The range is the difference between the maximum and minimum values in a data set, giving the simplest measure of overall spread. It tells you the total span of the data from lowest to highest in a single number.

Range tells you how spread out your data is from end to end. If the tallest kid is 5 feet and the shortest is 4 feet, the range is 1 foot - that's the 'stretch' of heights.

Formal View

For a dataset {x1,x2,…,xn}\{x_1, x_2, \ldots, x_n\}, the range is R=xmax⁡−xmin⁡=max⁡i(xi)−min⁡i(xi)R = x_{\max} - x_{\min} = \max_i(x_i) - \min_i(x_i).

Worked Examples

Example 1

medium
Data set {5,9,x,14,20}\{5, 9, x, 14, 20\} has range 1818. Find all possible values of xx.

Answer

x=2 or x≤2 (if max stays 20)x=2 \text{ or } x\le 2 \text{ (if max stays } 20\text{)}

First step

1
Range == max −- min =18= 18.

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Example 2

hard
A class has 55 test scores with mean 8080 and range 2020. What are the possible values of the maximum if all five scores are distinct integers?

Example 3

challenge
Six positive integers have mean 1010 and range 1414. If the smallest is 44, can all six be distinct? Justify with an example or counterexample.

Common Mistakes

  • Forgetting to subtract - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Confusing with number of values - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Ignoring that outliers inflate range - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing range from a keyword alone - Keywords like spread, variation, consistent are only clues; the data structure must match the concept.

Why This Formula Matters

Range prevents students from treating equal centers as equal data sets. The spread tells how predictable the values are, whether a summary is stable, and whether a comparison hides important variation.

Frequently Asked Questions

What is the Range formula?

The range is the difference between the maximum and minimum values in a data set, giving the simplest measure of overall spread. It tells you the total span of the data from lowest to highest in a single number.

How do you use the Range formula?

Range tells you how spread out your data is from end to end. If the tallest kid is 5 feet and the shortest is 4 feet, the range is 1 foot - that's the 'stretch' of heights.

What do the symbols mean in the Range formula?

RR denotes the range. xmax⁡x_{\max} is the maximum value and xmin⁡x_{\min} is the minimum value in the dataset.

Why is the Range formula important in Statistics?

Range prevents students from treating equal centers as equal data sets. The spread tells how predictable the values are, whether a summary is stable, and whether a comparison hides important variation.

What do students get wrong about Range?

Students often know a procedure related to range but skip the recognition step: Do I need to describe how far the data values extend or vary, rather than where the middle is? That leads to a calculation or graph that looks reasonable but answers a different question.

What should I learn before the Range formula?

Before studying the Range formula, you should understand: spread vs center.

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