Statistics: Mean, Median, and Variance

1.Three Values That Describe the Middle

The mean is the sum of all the values divided by how many there are. It tells you how much each would get if everything were shared out equally. The median is the value in the very middle when the values are lined up in order of size; if there is an even number of values, you use the mean of the two middle values. The mode is the value that appears most often.

The three can differ even for the same data, and you should pick one based on what you want to know. The mode tells a shoe store which size to stock the most of; the median better represents "a typical person's income"; and the mean is right for calculations that divide up a total (such as usage per person).

2.The Mean of Means Is Not the Mean: Weighted Averages

When you know the means of two groups and want the overall mean, adding the two means and dividing by 2 is usually wrong. That only works when the two groups have the same number of people. If the sizes differ, the mean of the larger group should have more influence on the overall mean.

The right method is to go back to the definition of the mean. Recover each group's total as "mean × number of people," add them all up, and divide by the total number of people. A mean found this way, by multiplying each value by its weight (number of people, mass, time, and so on), is called a weighted average. A grade point average where courses carry different credits, and an average purchase price where you bought different quantities at different times, are both weighted averages.

3.Spread: The Same Mean Doesn't Mean the Same Data

Here are test scores from two classes. Class A's five students scored 70, 70, 70, 70, 70, and Class B's five students scored 50, 60, 70, 80, 90. Both classes have a mean of 70, but they are completely different classes. In Class A everyone is similar, while Class B has big differences in ability. Variance and standard deviation put this difference into numbers.

Here is how. Find how far each value is from the mean (its deviation). Simply adding the deviations always gives 0, so you square them to make them all positive. The mean of the squared deviations is the variance. Variance is in squared units (points²), which makes it hard to interpret, so you take its square root to return to the original units: that is the standard deviation.

In this lesson, we treat all the data we have as a single population and use the population variance, dividing by n. Class B's deviations are −20, −10, 0, 10, 20; squared, they are 400, 100, 0, 100, 400, which sum to 1,000. 1,000 ÷ 5 = 200 is the variance, and √200 ≈ 14.1 points is the standard deviation. In Class A, all deviations are 0, so both the variance and the standard deviation are 0.

4.Calculating Variance and Standard Deviation by Hand

Let's work through the earlier data 2, 4, 4, 4, 5, 5, 7, 9 from start to finish. Making a table and filling in the cells one by one greatly reduces mistakes.

• Population variance (divide by n): when the data you have is the entire group you care about. In this example, 4.
• Sample variance (divide by n − 1): when you estimate the variance of a whole group from a sample drawn from it. In this example, 32 ÷ 7 ≈ 4.57, with a standard deviation ≈ 2.14.
• When you calculate from a sample, the mean also comes from the sample, so the spread tends to come out a little smaller than it really is; dividing by n − 1 corrects for this.
• Check first which one Excel, your calculator, or an AI is using. The same data can give different values.

5.Outliers: One Person Who Shakes the Mean

A value far away from the rest is called an outlier. Because the mean adds up every value, a single outlier pulls it a long way, while the median looks only at order and barely moves. That's why, for data with a few very large values mixed in, such as incomes or house prices, the median represents "typical" better than the mean.

Deleting outliers automatically isn't the answer either. If one is a data entry error, it should be fixed, but if it is a value that really occurred, it may be important information in itself. Look at the mean and median together, and if they differ a lot, take that as a sign of an outlier or a lopsided distribution.