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. 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.

## Types of “center”

Defining the “middle” can vary, but the mean and median are the two main numerical measures for the center of a distribution [1]:

- The mean (the average) is suitable for numerical (interval or ratio level) data. It is calculated by summing the values and dividing by the number of items in the set. For example, the mean of 3, 3, 5, 7, 9 is (3 + 3 + 5 + 7 + 9)/5 = 5.4. It should only be used with reasonably symmetric distributions with few or no outliers, otherwise use the median.
- The median (the middle of the set of data) is suitable for ordinal (ordered) data and data that is highly skewed. For instance, in the sequence 1, 3, 5, 7, 9 the number 5 is the median.

The mean can be distorted by outliers, which are observations that deviate significantly from the norm. Chalrles Wheelan illustrates this in the following example from his book *Naked Statistics: Stripping the Dread from the Data* [4]. Imagine ten individuals sitting on bar stools in a middle-class watering hole in Seattle. Each person earns $35,000 annually, resulting in a mean group income of $35,000. Now, let’s spice things up with the entrance of Bill Gates, who has an annual income of $1 billion. As Bill takes a seat on the eleventh bar stool, the mean annual income for the bar patrons skyrockets to about $91 million.

“If I were to describe the patrons of this bar as having an average annual income of $91 million, the statement would be both statistically correct and grossly misleading [Note: the median would remain unchanged]. This isn’t a bar where multimillionaires hang out; it’s a bar where a bunch of guys with relatively low incomes happen to be sitting next to Bill Gates and his talking parrot.” ~ Charles Wheelan [2].

Less common measures of the center of a distribution include:

**Geometric mean:**defined as the*n*th root of the product of*n*values.**Harmonic mean:**the reciprocal of the arithmetic mean of the reciprocals of n values.**Midrange**: the average of the highest and lowest data values.**Trimmed mean (**the mean of a distribution after the lowest and highest values have been removed).**Winsorized mean:**the lowest and highest values are replaced by the next highest and next lowest values.

If you’re uncertain about the intended “center,” ask your instructor for clarification. If that’s not possible, a practical approach — assuming you are given a graph or chart — is to look at the graph or chart and estimate the middle.

## 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 dividing 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! Check out our YouTube channel for hundreds of statistics help videos!

## References

- Measures of Center. Retrieved August 11, 2023 from: https://bolt.mph.ufl.edu/6050-6052/unit-1/one-quantitative-variable-introduction/measures-of-center/
- Charles Wheelan. (2013). Naked Statistics: Stripping the Dread from the Data.