Free Statistics Guide
Learn how to find the mean, median, mode, and range of a data set, understand what each statistic measures, handle special cases, and avoid the most common mistakes.
Mean, median, mode, and range are four common ways to describe a set of numerical data. They are often taught together, but they do not all tell us the same thing.
The mean and median describe the center of a data set, the mode describes frequency, and the range describes spread. Understanding those differences is just as important as knowing the calculations.
Each statistic answers a different question about the same data.
| Statistic | What It Tells You | Main Calculation | Memory Word |
|---|---|---|---|
| Mean | Average value | Sum ÷ number of values | Average |
| Median | Center of ordered data | Find the middle | Middle |
| Mode | Most common value | Compare frequencies | Most |
| Range | Overall spread | Maximum − minimum | Spread |
The mean is what we usually mean when we use the word average.
Suppose the data set is:
Imagine redistributing the total equally among all the data values. The mean is the equal-share amount each position would have.
The median is the middle value when the data is arranged from least to greatest.
You cannot reliably identify the median from an unordered data set.
Start with:
Put the values in order:
There are five values, so there is one value directly in the center.
If a data set contains an even number of values, there is no single middle value. Instead, there are two middle values.
The two middle values are 7 and 9.
The mode is the value that occurs more often than the other values.
The number 5 appears three times. Every other value appears only once.
The mode is not necessarily the largest number. It is the number that appears most frequently.
Every value appears exactly once, so no value occurs more frequently than the others. This data set has no mode.
Both 3 and 7 occur twice, which is the highest frequency. The modes are 3 and 7.
The number of times a value appears is its frequency. The mode must be an actual value from the data set.
The range measures the distance from the smallest value to the largest value.
The largest value is the maximum. The range is the difference between the maximum and minimum.
Consider the ordered data set:
| Statistic | Calculation | Answer | What It Describes |
|---|---|---|---|
| Mean | (2 + 4 + 4 + 6 + 9) ÷ 5 | 5 | Average |
| Median | Middle value | 4 | Center |
| Mode | Most frequent value | 4 | Frequency |
| Range | 9 − 2 | 7 | Spread |
Mean, median, mode, and range measure different characteristics of the data.
Divide by the number of data values, not by the largest value or the number of different values.
The median is the middle of the ordered data set.
With an even number of data values, find the two middle values and average them.
The mode is determined by frequency, not by size.
A data set can have no mode, one mode, or more than one mode.
Range = maximum − minimum. You must subtract.
If you need to find all four statistics, this order works well because sorting the data helps with several calculations at once.
Practice What You Learned
Practice each statistic individually or analyze a complete data set. The interactive tool walks through sorting, sums, counts, middle values, frequencies, minimums, maximums, and final calculations one step at a time.
Start Mean, Median, Mode & Range PracticeData analysis uses many of the same arithmetic skills involved in fractions, decimals, percentages, and multi-step calculations.
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