In statistics, the mean summarizes an entire dataset with a single number representing the data’s center point or typical value. It is also known as the arithmetic average, and it is one of several measures of central tendency. It is likely the measure of central tendency with which you’re most familiar! Learn how to calculate the mean, and when it is and is not a good statistic to use!
How Do You Find the Mean?
Finding the mean is very simple. Just add all the values and divide by the number of observations—the formula is below.
For example, if the heights of five people are 48, 51, 52, 54, and 56 inches, their average height is 52.2 inches.
48 + 51 + 52 + 54 + 56 / 5 = 52.2
When Do You Use the Mean?
Ideally, the mean indicates the region where most values in a distribution fall. Statisticians refer to it as the central location of a distribution. You can think of it as the tendency of data to cluster around a middle value. The histogram below illustrates the average accurately finding the center of the data’s distribution.
However, the mean does not always find the center of the data. It is sensitive to skewed data and extreme values. For example, when the data are skewed, it can miss the mark. In the histogram below, the average is outside the area with the most common values.
This problem occurs because outliers have a substantial impact on the mean. Extreme values in an extended tail pull the it away from the center. As the distribution becomes more skewed, the average is drawn further away from the center.
In these cases, the mean can be misleading because because it might not be near the most common values. Consequently, it’s best to use the average to measure the central tendency when you have a symmetric distribution.
For skewed distributions, it’s often better to use the median, which uses a different method to find the central location. Note that the mean provides no information about the variability present in a distribution. To evaluate that characteristic, assess the standard deviation.
Relate post: Measures of Central Tendency: Mean, Median, and Mode
Using Sample Means to Estimate Population Means
In statistics, analysts often use a sample average to estimate a population mean. For small samples, the sample mean can differ greatly from the population. However, as the sample size grows, the law of large numbers states that the sample average is likely to be close to the population value.
Hypothesis tests, such as t-tests and ANOVA, use samples to determine whether population means are different. Statisticians refer to this process of using samples to estimate the properties of entire populations as inferential statistics.
Related post: Descriptive Statistics Vs. Inferential Statistics
In statistics, we usually use the arithmetic mean, which is the type I focus on this post. However, there are other types of means, including the geometric mean. Read my post about the geometric mean to learn more.