Statistics and Distributions

How to calculate variance and standard deviation?

Quick answer

You calculate variance by finding the mean of your data set, squaring each point's difference from that mean, summing those squares, and dividing by the data point count for a population or n minus 1 for a sample. Taking the positive square root of this variance then gives you the standard deviation.

The topic explained

Understanding Spread and the Two Measures

Measuring variance and standard deviation helps you evaluate the spread or dispersion of a data set. Two classes might both average 75 percent on a math exam, but one class might have scores clustered tightly between 72 and 78, while the other spans from 50 to 100. The mean shows the center, but variance and standard deviation describe how far individual scores scatter around that center.

Variance measures average squared distance from the mean. Squaring each distance eliminates negative values so distances do not cancel out to zero. Because variance ends up in squared units, such as square test points or square inches, we take its positive square root to produce the standard deviation. This brings the measure back into the original units of your data.

Step-by-Step Calculation for a Data Set

Consider a population of five quiz scores: 4, 6, 7, 9, and 14. First, calculate the mean by adding the numbers and dividing by 5, which equals 8. Second, subtract the mean from every score to find each deviation: 4 minus 8 is -4, 6 minus 8 is -2, 7 minus 8 is -1, 9 minus 8 is 1, and 14 minus 8 is 6.

Third, square each deviation to get 16, 4, 1, 1, and 36. Fourth, add these squared values together: 16 plus 4 plus 1 plus 1 plus 36 equals 58. Fifth, divide by 5 to find the population variance, which is 11.6. Finally, take the square root of 11.6 to get the standard deviation, approximately 3.41 points.

Frequency Tables and Data Transformations

When data appears in a frequency table, multiply each squared deviation by its corresponding frequency before summing, then divide by the total frequency. For example, if a deviation of 2 occurs 3 times, its contribution to the sum of squares is 3 times 4, which is 12.

Watch what happens when you transform data. Adding a constant to every value shifts the entire distribution along the number line, changing the mean but leaving the spread and standard deviation unchanged. In contrast, multiplying every value by a constant multiplies both the mean and the standard deviation by that factor, while multiplying the variance by the square of that factor.

Free printable worksheet

Variance and Standard Deviation Practice

This practice worksheet is designed for 11th grade math students. Calculate the mean, variance, and standard deviation for each problem using the step-by-step deviation method.

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Variance and Standard Deviation PracticeName:

Find the mean, population variance, and population standard deviation of the data set: 2, 4, 6, 8, 10.
Two runners have the same average race time of 50 seconds. Runner A has a standard deviation of 1.2 seconds, while Runner B has a standard deviation of 4.5 seconds. Which runner is more consistent?
A data set has a mean of 20 and a standard deviation of 4. If you add 5 to every data point, what are the new mean and the new standard deviation?
A data set has a mean of 15 and a standard deviation of 3. If you multiply every data point by 2, what are the new mean, standard deviation, and variance?
A frequency table lists values (x) and frequencies (f): x = 1 (f = 2), x = 3 (f = 5), x = 5 (f = 3). Calculate the mean and the population standard deviation.

Common questions

What is the difference between sample variance and population variance?

Population variance divides the sum of squared deviations by N, the total number of items in the entire population. Sample variance divides by n minus 1. Dividing by n minus 1 corrects for bias, giving a better estimate of the true population spread when using sample data.

Why do we square the deviations instead of taking their average?

Deviations from the mean always sum to zero because positive and negative differences cancel each other out. Squaring turns every deviation positive. It also penalizes larger outliers more heavily, which is useful in advanced probability and statistical modeling.

Can standard deviation ever be a negative number?

No. Because variance is the average of squared real numbers, it is always zero or positive. Taking the principal square root of variance yields a non-negative value. A standard deviation of zero means every value in the data set is identical.

What topics usually follow variance and standard deviation in 11th grade?

Students typically study standard scores, also known as z-scores, to compare values across different distributions. That is followed by the normal distribution curve and calculating empirical rule probabilities within one, two, or three standard deviations of the mean.

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