A measure of spread is a number that describes how stretched out or clustered together a set of data values is. While averages tell you the center of your data, measures of spread tell you how far the individual values stray from that center. The most common measures of spread are range, interquartile range, variance, and standard deviation, and each one captures a different aspect of how your data is distributed.
Range: The Simplest Measure
The range is the difference between the largest and smallest values in a dataset. If a class of students scored between 55 and 98 on an exam, the range is 43 points. It’s the quickest way to get a sense of how widely your data stretches, but it has a major weakness: it only uses two data points. A single extreme value, whether it’s a measurement error or a genuine outlier, can make the range misleadingly large. For that reason, the range works best as a first glance rather than a final answer.
Interquartile Range
The interquartile range (IQR) solves the outlier problem by focusing on the middle 50% of your data. To find it, you sort all your values from lowest to highest and split them into four equal groups called quartiles. The first quartile (Q1) is the value that separates the bottom 25% from the rest. The third quartile (Q3) separates the top 25%. The IQR is simply Q3 minus Q1.
Because the IQR ignores the extreme highs and lows, it gives you a much more stable picture of spread when your data contains unusual values. It’s also the basis for a widely used rule for spotting outliers: any data point more than 1.5 times the IQR below Q1, or more than 1.5 times the IQR above Q3, is flagged as an outlier. If your Q1 is 30 and your Q3 is 50, the IQR is 20. Any value below 0 or above 80 would be considered an outlier under this rule.
Variance and Standard Deviation
Variance and standard deviation measure how far each data point sits from the mean, on average. They’re the most commonly used measures of spread in statistics, and they’re closely related: standard deviation is just the square root of variance.
Here’s how the calculation works. First, subtract the mean from every data point. Some of these differences will be positive (above the mean) and some negative (below it). To keep them from canceling each other out, you square each difference. Then you average those squared differences. That average is the variance. Finally, taking the square root brings the number back into the same units as your original data, and that result is the standard deviation.
Consider a group of nine people whose heights in inches are 56, 65, 74, 75, 76, 77, 80, 81, and 91. The mean height is 75 inches. After subtracting the mean from each value, squaring those differences, and averaging them, the variance comes out to 87.1. The square root of 87.1 is 9.3 inches. That standard deviation tells you the typical distance between any one person’s height and the group average.
A small standard deviation means the data points are clustered tightly around the mean. A large one means they’re spread out. A standard deviation close to zero means nearly every value is practically identical.
The 68-95-99.7 Rule
When data follows a bell-shaped (normal) distribution, the standard deviation becomes especially powerful because of a predictable pattern. Roughly 68% of all values fall within one standard deviation of the mean. About 95% fall within two standard deviations. And 99.7% fall within three. So in the height example above, if the data were normally distributed with a mean of 75 and a standard deviation of 9.3, you’d expect about 68% of people to measure between 65.7 and 84.3 inches, and about 95% to fall between 56.4 and 93.6 inches. This rule gives you an intuitive way to judge whether a specific value is typical or unusual.
Mean Absolute Deviation
Mean absolute deviation (MAD) takes a simpler approach than standard deviation. Instead of squaring the differences from the mean, it uses their absolute values, treating every gap as positive regardless of direction, and then averages those gaps. The result is always equal to or smaller than the standard deviation, because squaring exaggerates larger differences more than taking absolute values does.
The gap between the two measures grows wider when outliers are present. In one example comparison, a dataset with an extreme outlier had a standard deviation of 63.27 but a mean absolute deviation of only 41.75. The MAD was far less influenced by the single extreme value. Despite this advantage, standard deviation remains the default in most statistical work because it has mathematical properties that make it easier to use in formulas and probability models.
Comparing Spread Across Different Datasets
Standard deviation works well when you’re looking at a single dataset, but it can be misleading when you want to compare spread between two datasets measured in different units or at very different scales. The heights of adult humans might have a standard deviation of 3 inches, while household incomes might have a standard deviation of $20,000. Those two numbers can’t be meaningfully compared.
The coefficient of variation (CV) handles this by dividing the standard deviation by the mean and expressing the result as a percentage. This creates a dimensionless number, one with no units attached, that represents spread as a proportion of the average. A CV of 10% means the typical deviation is 10% of the mean, regardless of whether you’re measuring inches, dollars, or kilograms. It’s the go-to tool whenever you need to ask “which of these datasets is relatively more variable?”
Visualizing Spread With Box Plots
A box plot (sometimes called a box-and-whisker plot) is the most common way to see spread at a glance. It displays five key numbers: the minimum, Q1, the median, Q3, and the maximum. The box itself stretches from Q1 to Q3, showing the interquartile range. A vertical line inside the box marks the median. Whiskers extend from each edge of the box out to the minimum and maximum values.
Each of the four sections created by the five-number summary contains roughly 25% of the data. A tall, stretched box means the middle half of your data is widely spread. A short, compact box means those values are tightly grouped. When one whisker is much longer than the other, the data is skewed in that direction. Box plots are especially useful when you place several of them side by side to compare the spread of different groups.
Choosing the Right Measure
Your choice depends on your data and your goal. The range gives you a quick snapshot but is easily distorted. The IQR is better when outliers are a concern, or when your data is skewed rather than symmetric. Standard deviation is the most widely used and most mathematically versatile, making it the default for most analyses, but it’s also sensitive to extreme values. Mean absolute deviation offers a more intuitive and outlier-resistant alternative when you don’t need the mathematical conveniences of standard deviation. And the coefficient of variation is essential when you’re comparing variability across datasets with different scales.
In practice, reporting more than one measure often gives the clearest picture. A dataset’s standard deviation tells you how spread out the values are overall, while its IQR tells you how spread out the typical values are, ignoring the extremes. Together, the two reveal whether your spread is being driven by the bulk of your data or by a few unusual points at the edges.