Measuring the Spread: Range, Variance, and Standard Deviation

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Measuring the Spread: Range, Variance, and Standard Deviation

Research Methods in Psychology · Chapter 14: Descriptive StatisticsSection 14.3 · lesson 86 of 949 min · free with a free account

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In this lesson

Why two distributions with the same center can tell completely different stories.

  • Explain why variability is essential alongside center
  • Compute range, interquartile range, variance and standard deviation
  • Explain the role of the n minus 1 divisor
  • Choose the right spread measure for a situation

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Everything said in the video, section by section (1,317 words). Read along, or search it.

Descriptive Statistics

Measuring the Spread

Two classes have the same average score. That tells you almost nothing. This section is about the second number every summary needs. By the end you will explain why measures of variability are essential complements to the center. You will compute the range, the interquartile range, the variance, and the standard deviation. You will distinguish the sample variance from the population variance. And explain the role of that n minus one divisor, which puzzles almost everyone at first. You will interpret the standard deviation as the typical distance of values from the mean. And you will choose an appropriate measure of variability for a given research situation.

The Problem

Same Mean, Different Worlds

Two distributions can have exactly the same mean and describe entirely different phenomena. Consider a class in which every student scores around seventy five. Now consider a class in which half the students score ninety five. And the other half score fifty five. Both classes have the same average. The first class is consistent. The second is bimodal, and divided. Those are not the same situation, and no teacher would treat them the same way. The same logic applies to measurement. A test producing predictable results differs fundamentally from one producing wildly varying results. Even when the average outcome is identical. Measures of variability capture that dimension. How spread out, how predictable, how diverse.

Measure One

The Range

The simplest measure of variability is the range. It is the difference between the largest and smallest values in the dataset. The range is intuitive, and easy to compute in your head. It tells the reader how far the most extreme observations are from each other. As a summary, though, it has a serious limitation. It depends entirely on those two extreme values, and ignores everything in between. Two hundred observations, and the range knows about two of them. A single outlier can multiply the range several times over. While leaving the bulk of the data essentially unchanged. So the range works best as a quick descriptive starting point. Not as a primary summary of variability.

Measure Two

The Interquartile Range

The interquartile range fixes that vulnerability by ignoring the extremes deliberately. It focuses on the middle half of the data. The first quartile is the value below which twenty five percent of observations fall. The third quartile is the value below which seventy five percent fall. And the interquartile range is simply the third quartile minus the first. That is the spread of the middle fifty percent of the data. Because it ignores the most extreme values entirely, it is robust to outliers. Robust in exactly the same way the median is. Which is why the two go together. The interquartile range pairs naturally with the median, for skewed data.

Measure Three

The Variance, in Four Steps

The variance takes a completely different approach to measuring spread. Rather than relying on two specific values, it uses every observation. Specifically, it uses each observation's deviation from the mean. The procedure has four steps. First, subtract the mean from each value to get that value's deviation. That captures how far each observation sits from the center. Second, square each deviation. Squaring removes the negative signs, and it magnifies large deviations. Third, sum all the squared deviations together. That total is the overall variability around the mean. Fourth, divide by the appropriate denominator to get an average. For a sample, divide by n minus one. For a population, divide by capital N.

The Famous Question

Why n Minus 1 for a Sample

That last step raises the most-asked question in introductory statistics. Why n minus one, rather than n? The reason has to do with where the sample mean came from. The sample mean is itself estimated from the data. And it is, by construction, the value that minimizes the sum of squared deviations. Minimizes it within that particular sample. So deviations from the sample mean are a little too small, systematically. Dividing by n would give a variance estimate slightly below the true population variance. On average, every time. That is a systematic underestimate, not random noise. Dividing by n minus one corrects the bias, producing an unbiased estimator. The correction is small for large samples and substantial for small ones. But it is always applied.

The Deeper Idea

Degrees of Freedom

There is a deeper idea hiding in that n minus one, and it is worth meeting now. It is the number of degrees of freedom. Degrees of freedom are the number of values in a sample that are free to vary. Free to vary once the mean has been determined. Think about what that means concretely. Suppose you know the mean, and you know all but one of the values. The last value is then completely fixed. It has no freedom left. So only n minus one of the values are independent pieces of information. And this is not a quirk of the variance formula. Degrees of freedom appear throughout inferential statistics, in test after test.

Measure Four

The Standard Deviation

Variance has one serious interpretive drawback. Its units are the squared units of the original variable. If exam scores are measured in points, the variance is measured in points squared. Points squared has no intuitive meaning at all. The standard deviation solves this in one step. Take the square root of the variance, and you return to the original units. It is denoted S D, or lowercase s, for samples. And the Greek letter sigma for populations. And now it means something you can actually picture. The standard deviation is the typical distance of an observation from the mean.

A Yardstick

The 68-95-99.7 Rule

On approximately normal distributions, the standard deviation does something more. It becomes a yardstick with fixed markings. About sixty eight percent of observations lie within one standard deviation of the mean. About ninety five percent lie within two. And about ninety nine point seven percent lie within three. Those benchmarks are sometimes called the empirical rule. They make the standard deviation more than a number. You can locate an individual observation within the broader distribution. And judge how unusual any particular value actually is.

Why Keep Both

Variance for Analysis, SD for Description

Here is a reasonable question at this point. If the standard deviation is more interpretable, why keep variance at all? The answer is mathematical rather than descriptive. Variance has properties the standard deviation does not. Particularly its additivity for independent variables. Variances of independent things add. Standard deviations do not. That property makes variance the foundation of analysis of variance and regression. And of many other techniques you will meet later. So researchers compute variance for analysis, and report standard deviation for description. Both appear constantly in published research, and fluency with both is essential.

Four Measures

Which One, and When

Put the four measures side by side, with the situation each one suits. The range is maximum minus minimum. Use it for quick description, reported alongside other summaries. The interquartile range is the third quartile minus the first. Use it for skewed distributions, and for data with outliers. It pairs with the median. The variance is the average squared deviation, with divisor n minus one for samples. Use it for statistical analysis, since it underlies analysis of variance and regression. And the standard deviation is the square root of the variance. It is the most common descriptive measure, and it pairs with the mean.

Pairing

Center and Spread Travel Together

Notice that the choice of spread measure mirrors the choice of center. The two choices should be made together, not one after the other. When you report the mean, the standard deviation is the natural companion. Both use all the data. Both work in the original units. And both feed directly into further analyses. When you report the median, the interquartile range is the natural companion. Both are robust to outliers, and both depend on rank rather than value. Reporting the range alongside either is informative too. It gives a quick indicator of the full extent of the data. The general rule is simple, and it is why these two ideas share a chapter. A center without a spread tells only half the story. Together they let a reader understand the distribution behind the numbers.

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FAQ

Questions students ask us

About Measuring the Spread: Range, Variance, and Standard Deviation

How do you calculate variance and standard deviation?
Variance is the sum of squared deviations from the mean divided by n - 1 for a sample, or by N for a population, and the standard deviation is its square root. For the scores 2, 4, and 6, the mean is 4, the squared deviations are 4, 0, and 4, their sum is 8, the sample variance is 8 / 2 = 4, and the standard deviation is 2, in the original units.
Why is sample variance divided by n - 1 instead of n?
Dividing by n - 1 corrects a bias: deviations from the sample mean are systematically a little too small, because the sample mean is the value that minimizes the sum of squared deviations within that sample. Dividing by n would underestimate the population variance on average, while n - 1 gives an unbiased estimate. The n - 1 is the degrees of freedom, since once the mean is known only n - 1 values are free to vary.
What is the interquartile range and when should you use it?
The interquartile range is the third quartile minus the first quartile, the spread of the middle 50 percent of the data. The first quartile is the value below which 25 percent of observations fall, and the third quartile is the value below which 75 percent fall. Because it ignores the extremes, it resists outliers in a way the range does not, and it pairs naturally with the median for skewed data.
What is the empirical rule for the standard deviation?
The empirical rule says that in an approximately normal distribution, about 68 percent of observations fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. For example, if test scores are normally distributed with a mean of 100 and a standard deviation of 15, about 95 percent fall between 70 and 130. The rule helps judge how unusual a single value is.

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