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# Measures of Central Tendency and Dispersion

## Mean, median, mode, range

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Mean, Median, Mode and Range

Have you ever been to an amusement park? Take a look at this dilemma.

An amusement park is designing a new section for children over 3 years old and under 8 years old. As part of their research, they used a survey of the heights and weights of a thousand children in that age group. Which measure of central tendency should they use to accommodate the greatest number of children on a roller coaster?

In this Concept, you will learn all about how to identify and find each different measure. By the end of this Concept, you will be able to answer this question.

### Guidance

In the real world, there are many situations in which a large group of data is collected. In order to make sense of the data, we use a number of statistical measures . These measures help us to generalize a group of data, make inferences about it, and compare it with other groups of data.

Statistical measures include mean, median, mode and range. Depending on the situation, certain measures may be more helpful than others in interpreting data.

Let’s look at these statistical measures.

The mean , median , and mode are three common measures of central tendency ; they are three mathematical tools frequently used to analyze data.

The mean , commonly referred to as the average, is the sum of all the data items divided by the number of data items. The median is the middle number in a set of data that is ordered from lowest to highest. If there is an even number of data, we take the average of the middle two numbers to find the median. Finally, the mode is the number that occurs most often.

Take a look at this dilemma.

A manager at a small movie theater was analyzing the number of people who came to the movies during the week. Over nine days, he found the following data: 81, 89, 92, 85, 93, 62, 85, 105, and 90. Find the mean, median, and mode of the data.

First, let’s find the mean. Remember that the mean is the same as the average.

Mean: add all of the data items and divide by the number of items.

The average or mean is 86.8 which could be rounded up to 87.

Next, let’s find the median.

Median: the middle number when the data is ordered from lowest to highest.

First reorder the data from least to greatest:

\begin{align*} & 62, 81, 85, 85, 89, 90, 92, 93, 105 \\ & \qquad \qquad \quad \ \ \uparrow \\ & \text{The middle number, 89, is the median.}\end{align*}

The median is 89.

Finally, let’s find the mode.

The mode is the number that occurs most often. In this case, 85 occurs two times and all of the other number only once. The number 85 is the mode.

The mode is 85.

#### Example A

Find the mean of the data set.

Solution: \begin{align*}21.8\end{align*}

#### Example B

Find the median of the data set.

Solution: \begin{align*}20\end{align*}

#### Example C

Find the mode of the data set.

Solution: There isn't a mode.

Now let's go back to the dilemma from the beginning of the Concept.

In order to attract the more customers, they should accommodate as many children as possible. For this reason, they should use the range which will include heights of children from the tallest to the shortest.

### Vocabulary

Statistics
Statistics is a branch of mathematics that involves collecting, analyzing and displaying data.
Measures of Central Tendency
In statistics, a measure of central tendency of a data set is a central or typical value of the data set.
Mean
The mean of a data set is the average of the data set. The mean is found by calculating the sum of the values in the data set and then dividing by the number of values in the data set.
Median
The median of a data set is the middle value of an organized data set.
Mode
The mode of a data set is the value or values with greatest frequency in the data set.
Range
The range of a data set is the difference between the smallest value and the greatest value in the data set.

### Guided Practice

Here is one for you to try on your own.

Find the mean, median, mode and range of the following data set.

\begin{align*}{12, 13, 15, 18, 22, 25, 30, 31, 32, 34, 40}\end{align*}

Solution

First, we find the mean by adding up all of the values in the data set and then we divide by the number of values in the set.

The mean is \begin{align*}24.72\end{align*} .

The median is the middle score. Since these values are already in order from least to greatest, we can simply find the middle value.

The median is \begin{align*}25\end{align*} .

There isn't a mode.

To find the range, we find the difference between the greatest value and the smallest value.

\begin{align*}40 - 12 = 28\end{align*}

The range is \begin{align*}28\end{align*} .

### Explore More

Directions: Find the mean, median, mode and range Round all answers to the nearest tenths place. Notice that each answer has four answers.

\begin{align*}{13, 18, 24, 21, 16, 24, 14, 17, 24}\end{align*}

1. Mean

2. Median

3. Mode

4. Range

\begin{align*}{116, 137, 120, 75, 98, 98, 137, 139, 139}\end{align*}

5. Mean

6. Median

7. Mode

8. Range

\begin{align*}{22, 24, 25, 30, 32, 34, 37, 22, 22, 38, 40}\end{align*}

9. Mean

10. Median

11. Mode

12. Range

\begin{align*}{123, 150, 163, 150, 163, 150, 180, 200, 201}\end{align*}

13. Mean

14. Median

15. Mode

16. Range

### Vocabulary Language: English

arithmetic mean

arithmetic mean

The arithmetic mean is also called the average.
descriptive statistics

descriptive statistics

In descriptive statistics, the goal is to describe the data that found in a sample or given in a problem.
inferential statistics

inferential statistics

With inferential statistics, your goal is use the data in a sample to draw conclusions about a larger population.
measure of central tendency

measure of central tendency

In statistics, a measure of central tendency of a data set is a central or typical value of the data set.
Median

Median

The median of a data set is the middle value of an organized data set.
Mode

Mode

The mode of a data set is the value or values with greatest frequency in the data set.
multimodal

multimodal

When a set of data has more than 2 values that occur with the same greatest frequency, the set is called multimodal    .
Outlier

Outlier

In statistics, an outlier is a data value that is far from other data values.
Population Mean

Population Mean

The population mean is the mean of all of the members of an entire population.
Range

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.
resistant

resistant

A statistic that is not affected by outliers is called resistant.
Sample Mean

Sample Mean

A sample mean is the mean only of the members of a sample or subset of a population.
Statistics

Statistics

Statistics is a branch of mathematics that involves collecting, analyzing and displaying data.