Mean, median, and mode are measures of central tendency used to summarize numbers in a data set. Mean, median, and mode help you approximate the center or central number(s) of a data set. The range is the measure of dispersion in a data set.

**Mean**– The arithmetic average of the numbers in a data set.**Median**– The central value in a data set when the numbers are arranged least to greatest.**Mode**– The most commonly appearing value in a data set**Range**– The difference between the largest and smallest number in a data set.

For most math problems, the mean, median, mode, and range provide plenty of summary data.

To find the mean, median, mode, and range, you need a set of numbers and the ability to do simple addition, subtraction, and division.

Data sets can be tightly clustered around a repeated value, like these numbers:

$\left\{1,2,2,2,3\right\}$

Or they can be dispersed, like these values:

$\left\{1,2,3,6,9,14,21\right\}$

In both cases, you can summarize the data by noting which values tend toward the middle of the set of numbers by finding the mean, median, mode, or range.

The **arithmetic mean** of a data set is an average. The mean is revealed by summing the individual numbers in the set, then dividing the sum by the number of numbers in the set, like this:

For this set:

$\left\{1,2,2,2,3\right\}$

First, we sum the numbers:

$1+2+2+2+3=10$

And then divide the sum by how many numbers are in the set, which is $5$:

$\frac{10}{5}=\mathbf{2}$

The mean of our data set is $2$.

Here is another example data set. What is the mean of this set?:

$\left\{1,2,3,6,9,14,21\right\}$

We sum the numbers:

$1+2+3+6+9+14+21=56$

And then divide by the number of numbers in the set, which is $7$:

$\frac{55}{7}=\mathbf{8}$

The mean or average of this data set is $8$

Whole numbers are not the only set of numbers for which you can find a mean:

$\left\{\frac{1}{2},\frac{3}{4},1\frac{1}{2},\frac{7}{8}\right\}$

$\frac{1}{2}+\frac{3}{4}+1\frac{1}{2}+\frac{7}{8}=6\frac{5}{8}(or6.625)$

$\frac{6\frac{5}{8}}{4}=\mathbf{1.65625}$

You can use these steps to calculate the mean of whole numbers, fractions, and decimals.

The order of the list of numbers matters when calculating the median. The **median** is the middle value in a data set when the data are arranged from least to greatest, like this:

$\left\{1,2,2,2,3\right\}$

Or like this:

$\left\{1,2,3,6,9,14,21\right\}$

Or like this:

$\left\{7,35,91,104,298,502\right\}$

The first two data sets easily show their medians or middle numbers. For the first data set, with five values, the center value is $2$.

For the second data set, with seven values, the middle value is $6$. So, the two medians for the two data sets are $2$ and $6$.

Finding the middle number of a data set with an odd number of values is straight forward.

The last data set, however, has an even number of values, six. To calculate the median from an even quantity of numbers, take the mean of the center two numbers.

The two middle numbers are $91$ and $104$. We sum them, and then divide by $2$:

$91+104=195$

$\frac{195}{2}=\mathbf{97.5}$

The median value or middle number of our third data set is $97.5$

Almost always, when finding the median of an even quantity of numbers, the calculated median will * not* be a number in the set.

It could happen, though, like in this example:

$\left\{24,37,37,49\right\}$

$37+37=74$

$\frac{74}{2}=\mathbf{37}$

The median here is $37$, which * is* a number in the original data set.

Median is especially useful when your data set has outliers or numbers far away from the middle of the group of numbers. Outliers can drastically skew your central tendency causing your summary data to be less accurate.

The effect of outliers can be diminished by paying more attention to the median than to the outliers.

If you have navigated through the first two measures of central tendency, we have great news for you; the other two measures are far easier to understand and calculate.

The **mode** of a data set is the quantity appearing the most number of times. Unlike the other measures of central tendency, a mode is not a requirement of a data set. On the other hand, a data set can have multiple modes.

Here is a set of numbers with an obvious mode:

$\left\{1,2,2,2,3\right\}$

The mode is $2$. It appears more than the other quantities. Here is a data set with no mode:

$\left\{1,2,3,6,9,14,21\right\}$

This data set has no mode because no number appears more than any other.

And here is a set of data points with multiple modes:

$\left\{7,\mathbf{35},\mathbf{}\mathbf{35},91,\mathbf{104},\mathbf{}\mathbf{104},298,\mathbf{502},\mathbf{}\mathbf{502},617\right\}$

We immediately see that $35$, $104$, and $502$ all appear twice. They are the three modes of the data set.

You do not have to arrange the data from the lowest number to the highest number, but it makes finding the mode(s) easier.

The range of a data set is the difference between the largest value and smallest value:

$\left\{1,2,2,2,3\right\}$

Here the range is easily calculated; just subtract the lowest value from the highest value:

$3-1=\mathbf{2}$

In this data set, the range is $2$.

For this more populated data set, the range is still easily calculated:

$\left\{7,35,35,91,104,104,298,502,502,617\right\}$

You subtract the smallest number from the larest number in the set:

$617-7=\mathbf{610}$

The range is $610$.

When finding the range, you do not have to arrange your data set in numerical order, but if you have a large number of data points, it will make it easier to see which numbers you need to subtract.

Here are six mean, median, and mode example problems for you to solve:

- For the given data set $\left\{11,11,14,14,14,16,17\right\}$, identify the mode.
- For the data set $\left\{1.1,6.7,9.5,17.8\right\}$, identify the range.
- For the data set $\left\{5,5,23,891,892\right\}$, calculate the median.
- For that same data set, $\left\{5,5,23,891,892\right\}$, calculate the mean. Were the mean and median close or not close at all?
- For the data set $\left\{1,3,7,9,9,13,17,99\right\}$, find or calculate the mean, median, mode, and range.
- For the data set $\left\{1,2,3,4,5\right\}$, what is the mode?

Before you peek, check your answers. Take your time!

The mode is $14$. The range is $16.7$. The median is $23$. The mean is $363.2$; the median and mean are not close at all! The mean is $19.75$, the median is $9$, the mode is $9$, and the range is $98$. No mode exists. Each number appears only once.

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

- What mean, median, mode, and range are in math.
- What a data set is.
- How to find the mean, median, mode, and range of a data set.
- How the mean, median, and mode are used to find a data set's central tendency.

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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