In math, **range** is a statistical measurement of dispersion, or how much a given data set is stretched out from smallest to largest. In a set of data, the range is the difference between the greatest and smallest value.

Range is one of four simple tools of statistics:

**Mean**– The arithmetic mean of a set of numbers is its average (the sum of the numbers divided by the quantity of numbers)**Median**– The statistical median is the middle number in an ordered set of numbers**Mode**– The mode is the most commonly occurring number in a data set**Range**– The range of a data set is the mathematical difference between the largest and smallest value

The first three statistical tools are measures of central tendency, or how similar the numbers are.

Only range is a measure of dispersion, highlighting how different the numbers are, and there is a simple formula to calculate range.

$Range=GreatestValue-LeastValue$

To find the range in a set of numbers, you must gather your data, organize the data from least to greatest, then subtract the smallest value from the largest value. You can find a range of positive numbers and negative numbers.

Steps for how to calculate range:

- Gather your data, so you know all the numbers to be studied.
- Arrange the data set in order from least to greatest.
- Write a subtraction sentence to subtract the smallest value from the greatest (or largest) value.

For example, if you read a biography, and wrote down how many pages you read each day, you could take the range:

- Monday – $12$ pages
- Tuesday – $9$ pages
- Wednesday – $11$ pages
- Thursday – $3$ pages
- Friday – $8$ pages

To find the range, put the number of pages in order from least to greatest:

$\left\{3,8,9,11,12\right\}$

Subtract the smallest value from the greatest value:

$R=12-3=9$

The range of this data set is $9pages$.

Let's figure the range of the following set of real numbers. Below are five of the lowest-scoring games in NBA history, listed with teams and total game points:

- Washington Capitols (49) vs. Pittsburgh Ironmen (40) –
**89 points** - Fort Wayne Pistons (19) vs. Minneapolis Lakers (18) –
**37 points** - Washington Capitols (50) vs. Detroit Falcons (33) –
**83 points** - Boston Celtics (46) vs. Pittsburgh Ironmen (44) –
**90 points** - Boston Celtics (47) vs. Washington Capitols (38) –
**85 points**

Go through the three steps to find the range of these five games:

**Gather your data points, so you know all the numbers to be studied:****Arrange the data set in order from least to highest number****Subtract the smallest value from the highest value:**

89 points, 37 points, 83 points, 90 points, 85 points

$\left\{89,37,83,90,85\right\}$

$\left\{37,83,85,89,90\right\}$

$Range=90-37=53$

$Range=53points$

The range of the five lowest-scoring games in NBA history is $53points$.

Let's try this with some more challenging numbers, like country populations.

China has an estimated population of $\mathrm{1,420,062,022}$ people. Mexico's population is $\mathrm{132,328,035}$. India has a population of $\mathrm{1,368,737,513}$. The United States has a population of $\mathrm{329,093,110}$. What is the range of this data set?

**1) Gather the data:**

- $\mathrm{1,420,062,022}$
- $\mathrm{132,328,035}$
- $\mathrm{1,368,737,513}$
- $\mathrm{329,093,110}$

**2) Arrange the data set in order from least to greatest:**

- $\mathrm{132,328,035}$
- $\mathrm{329,093,110}$
- $\mathrm{1,368,737,513}$
- $\mathrm{1,420,062,022}$

**3) Subtract the smallest value from the greatest (largest) value:**

- $Range=\mathrm{1,420,062,022}-\mathrm{132,328,038}$
- $Range=\mathrm{1,287,733,987}people$

The range is typically used to find the dispersion of values in a data set comprising several values. However, you don’t need all the other numbers to find the range between two numbers.

Finding the range between two numbers is the same as finding the range of a set of data.

Here you have a set of numbers:

{6,4,10,8}

To find the range, you take the greatest value, $10$, minus the lowest value, $4$. The range is $6$.

Now, what if you have only the two numbers $10$ and $4$ in your set:

{10,4}

The range between these two numbers is the same, $10-4=6$

The range is still $6$.

Finding the range of a data set is the same as finding the range between two numbers.

Below are several example problems where you must solve for range.

**1)** The NBA has players with oversized feet. Here are the shoe sizes of some of the NBA's notable players:

- Paul George, Size $12$
- Russell Westbrook, Size $15$
- Yao Ming, Size $18$
- Taj Gibson, Size $13$

What is the range of these four NBA players' shoe sizes?

**2)** The Mohs Hardness Scale for minerals includes these samples:

- Diamond, $10$
- Fluorite, $4$
- Talc, $1$
- Quartz, $7$
- Apatite, $5$

What is the range of the five minerals listed?

**3)** A Thoroughbred horse weighs roughly $\mathrm{1,100}$ pounds. A Shetland pony weighs around $200$ pounds. A Clydesdale horse weighs about $\mathrm{2,000}$ pounds. What is the weight range of these three horse breeds?

**4)** Marco received mathematics quiz grades of $67\%$, $101\%$ (he did the extra credit – always attempt the extra credit!), $93\%$, $81\%$, and $96\%$. What is the range of Marco's mathematics quiz grades?

Statistically speaking, most people skip to the answers without doing the math first. Don't be a statistic! Work the problems out first!

Here are the answers:

- The NBA players shoe sizes have a range of $6$, the difference between Paul George and Yao Ming.
- The Mohs Hardness Scale has a range of $9$, shown here as the difference between diamond and talc.
- The weight range of the three horse breeds is $\mathrm{1,800}$ pounds.
- Marco has a range of mathematics quiz grades of $34$ percentage points.

Range is used in real life to make mathematical calculations. Range can be used to calculate the amount of time that has passed, like when calculating your age.

The current year is $2020$, and you were born in $2005$. How old are you? Or how much time has passed?

$2020-2005=15$

$15$ years have passed, so if your $2020$ birthday has already passed, then you are $15$ years old.

Range is also used in real life to figure out the dispersion of a high school class' test scores, to determine the price *range* for a service, and much more.

After working your way through this lesson and video, you have learned:

- The definition of range in math
- How to find the range of a data set
- How to calculate the range between two numbers

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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