Statistics Basics > Center of a Distribution

## What is the Center of a Distribution?

The center of a distribution is **the middle of a distribution**. For example, the center of 1 2 3 4 5 is the number 3. Of course, it’s not usually that easy. If you’re asked to find the center of a distribution in statistics, you generally have three options:

- Look at a graph, or a list of the numbers, and see if the center is obvious.
- Find the mean, the “average” of the data set.
- Find the median, the middle number.

There’s no one correct way to define the “middle”. If you are asked to find the center of a distribution, and aren’t sure what you are supposed to be finding, the safest bet is to check with your instructor to find out which center you are supposed to be finding. If that isn’t possible, the most likely answer is to look at a graph and eyball the middle. For example, the following graph shows the central bar highlighted:

If you have actual numbers, then you’ll have to choose whether you want the mean or the median to be the center of a distribution. **In general:** use the mean if you do not have any outliers (very low or very high values) and use the median if you have outliers or if your graph is very skewed.

## Center of a Distribution Example

The following stemplot shows the percentage of residents aged 65 and older in the United States (according to the 2000 census). The stems are whole percents and the leaves are tenths of a percent. **Find the center of the distribution.**

*Stems = whole percents
Leaves = tenths of a percent*

6 8

7

8 8

9 79

10 08

11 15566

12 012223444457888999

13 01233333444899

14 02666

15 23

16 8

There are several ways to tackle this problem. First, you could simply look at the stemplot and see that the bulk of the data falls in the middle — the 12th percent column — so you could say that’s the center of this particular distribution. If you want to prove it mathematically, find the mean:

Step 1: Write the number out from the stemplot. If you aren’t sure how to do this, you might want to take a look at this article: Stemplot in Statistics.

6.8

8.8

9.7 9.9

10.0 10.8

11.1 11.5 11.5 11.6 11.6

12.0 12.1 12.2 12.2 12.2 12.3 12.4 12.4 12.4 12.4 12.5 12.7 12.8 12.8 12.8 12.9 12.9 12.9

13.0 13.1 13.2 13.3 13.3 13.3 13.3 13.3 13.4 13.4 13.4 13.8 13.9 13.9

14.0 14.2 14.6 14.6 14.6

15.2 15.3

16.8

Step 2: Calculate the mean by adding up all the numbers and divide by the number of items in the set (which is 50):

8.8+ 9.7+ 9.9+ 10.0 +10.8+ 11.1+ 11.5+ 11.5+ 11.6+ 11.6+ 12.0+ 12.1+ 12.2+ 12.2+ 12.2+ 12.3+ 12.4+ 12.4+ 12.4+ 12.4+ 12.5+ 12.7+ 12.8+ 12.8+ 12.8+ 12.9+ 12.9+ 12.9+ 13.0 +13.1 +13.2 +13.3 +13.3 +13.3 +13.3+ 13.3+ 13.4+ 13.4+ 13.4+ 13.8+ 13.9+ 13.9+ 14.0+ 14.2+ 14.6+ 14.6+ 14.6+ 15.2+ 15.3+ 16.8 = 638.3

So: 638.3 / 50 = 12.766

You could also find the median. The middle of a set of 50 numbers is the 25th number, which is 12.8.

Note that the mean, 12.766 and the median, 12.8 fall into the 12th percentile, which is where we guessed the center of the distribution was just by looking at it!

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