Measures of Central Tendency: Mean

What is the Mean and How Do I Find It?

We've covered other measures of central tendency, like median and mode. We know that median is the middle number, and mode is the frequent number in a set of numbers, and we can do word problems with this. In later lessons, we'll cover standard deviation (both the concept and how to calculate it), range, and statistical significance.

The arithmetic mean of a set of values is the same thing as the average.

We find the mean in two steps.

1. Add up all of the values in the data set.
2. Divide the sum by the total number of values.

To find the mean, you find the sum of all the values, and divide it by the number of values.

$$\frac{\text{sum of values}}{\text{number of values}} = \frac{1704}{81} \approx 81$$

So, $81$ is the mean of the above dataset!

Let's do another example.

$$\frac{\text{sum of values}}{\text{number of values}} = \frac{371.29}{9} \approx 41.25$$

So, $\$41.25$ is the mean of the above dataset!

So, now you know mean! To summarize, we know that the middle number is the median and the most is the most common number in a list of numbers.

(Video) How to Find the Average

by Math with Mr. J

This video by Math with Mr. J defines the average of a data set.

Summary

Mean is the same as average. Average represents the typical value or critical value for a dataset.

There are two steps to finding the average of a dataset.

1. Add up all of the numbers.

1. Divide by how many numbers there are.

Example 1: $$[8, 9, 12, 8, 7, 11, 3, 9, 5]$$ The sum of all numbers in this dataset is: $72$

There are $9$ numbers in this dataset. $$\text{Mean}=\frac{72}{9}$$ $$\text{Mean}=8$$

Example 2: $$[80, 75, 96, 82, 91, 87, 70]$$ The sum of all numbers in the dataset: $581$

There are $7$ numbers in this dataset. $$\text{Mean}=\frac{581}{7}$$ $$\text{Mean}=83$$

If you calculate an average that is a decimal, that is okay! Sometimes you may divide two numbers and end up with a decimal value.

Mean

You can find the mean, also called the average, $\bar{x}$ (sample mean), or $\mu$ (population mean) by adding up all of the values in the data set, then dividing that sum by the total number of values.

In other words, the mean of the data set $x_1, x_2, x_3, ... x_n$ is: $$\text{mean}=\frac{x_1+x_2+x_3+ ... +x_n}{n}$$

Let's find the mean of $2, 4, 6, 8, 10.$ $$\text{mean}=\frac{2+4+6+8+10}{5} = \frac{30}{5} = \boxed{6}$$

In the future, you may need to cover weighted means (using weighted arithmetic), so mastering this concept is really important!