# Range (of Data)

In statistics, we are frequently working with large amounts of data. When we do so, we need different ways to categorize and describe the data that we are looking at. Some of the ways we describe the data we're looking at are to find the mean, median, mode, and range of the data set. Before we get into the range in detail, let's take a look at the mean, median, and mode methods of analyzing a data set.

## Analyzing data using the mean

The mean is another way of saying the average of the numbers in a data set. We find the mean by adding up the sum of all the numbers in the data set and then dividing it by the number of numbers in the data set.

## Analyzing data using the median

The median is the central number in a data set or series. To find the median, we must first line up the numbers in the set from least to greatest or from greatest to least. If there are an odd number of data points, the number in the middle is the median.

So the median of the data set $\{1,2,3,4,5\}$ is $3$. Note that while in this case, the mean and median are the same, but that is not always the case. However, it is not uncommon for the two to be near each other in a normal set of data points.

## Analyzing data using the mode

The mode of a data set is the data point that appears most frequently. If two numbers appear the most, there are two modes, and the data set is considered bimodal. More than two numbers can appear most frequently in data which is called multimodal. If all the numbers in a data set appear with equal frequency, there is no mode.

## Analyzing data using the range

The range is a measure of dispersion or spread in a data set. It gives us an idea of how much variation there is within the data. Understanding the range can be useful for various reasons:

**Comparing data sets:**The range helps us compare different data sets to see which one has more variability. For example, if you are comparing the test scores of two classes, a larger range indicates that the scores are more spread out, while a smaller range suggests that the scores are more consistent.**Identifying outliers:**The range can help identify extreme values or outliers in the data set. If the range is large, it may be due to one or more outliers affecting the overall spread of the data. Outliers can sometimes indicate errors in data collection or unusual circumstances.**Decision-making:**Knowing the range of a data set can be helpful in making informed decisions. For example, if a manager knows the range of employee performance, they can take appropriate actions such as offering support to low-performing employees or rewarding high-performing employees.**Assessing data quality:**A large range may indicate inconsistency in the data collection process or the presence of errors. By examining the range, researchers can assess the quality of the data and decide if further investigation is needed.

**Example 1**

Let's find the range of a small data set. We have the set $\{1.36,1.78,1.45,1.89,1.55,1.37,1.83,1.64,1.28\}$.

The best first step in finding the range is to order the data from least to greatest. So our new data set is $\{1.28,1.36,1.37,1.45,1.55,1.64,1.78,1.83,1.89\}$.

Now it's easy to find the range. We simply subtract the least number from the greatest.

$1.89-1.28=0.61$

Therefore, the range of the data set is 0.61.

**Example 2**

Let's find the range of a larger, more practical set of data. Below is a list of students and the scores they received on an exam. First, put the scores in order and then find the range of their scores.

- Gillian – 86
- Lucas – 61
- Marietta – 63
- Bella – 88
- Lisa – 93
- Jocelyn – 80
- Mario – 97
- Carmen – 55
- Carla – 91
- Robert – 82
- Vance – 86
- Cyndi – 88
- Michael – 45
- Marcus – 89
- Ashley – 37
- Victor – 77

First, we will put the scores in order. We don't need to attach the names at this point because we are only looking at the numerical data to find the range.

The data set in order is $\{37,45,55,61,63,77,82,86,86,88,88,89,91,93,97\}$. We then subtract the lowest score from the highest score to find the range.

$97-37=60$

The range of test scores is 60.

## Topics related to the Range (of Data)

## Flashcards covering the Range (of Data)

Common Core: High School - Statistics and Probability Flashcards

## Practice tests covering the Range (of Data)

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

## Get help learning about the range of data

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