How to Calculate Standard Deviation

Figuring It out by Hand

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Standard deviation (usually denoted by the lowercase Greek letter σ) is the average or means of all the averages for multiple sets of data. Standard deviation is an important calculation for math and sciences, particularly for lab reports. Scientists and statisticians use standard deviation to determine how closely sets of data are to the mean of all the sets. Fortunately, it's an easy calculation to perform. Many calculators have a standard deviation function. However, you can perform the calculation by hand and should understand how to do it.

Different Ways to Calculate Standard Deviation

There are two main ways to calculate standard deviation: population standard deviation and sample standard deviation. If you collect data from all members of a population or set, you apply the population standard deviation. If you take data that represents a sample of a larger population, you apply the sample standard deviation formula. The equations/calculations are nearly the same with two exceptions: for the population standard deviation, the variance is divided by the number of data points (N), while for the sample ​standard deviation, it's divided by the number of data points minus one (N-1, degrees of freedom).

Which Equation Do I Use?

In general, if you're analyzing data that represents a larger set, choose the sample standard deviation. If you gather data from every member of a set, choose the population standard deviation. Here are some examples:

  • Population Standard Deviation—Analyzing test scores of a class.
  • Population Standard Deviation—Analyzing the age of respondents on a national census.
  • Sample Standard Deviation—Analyzing the effect of caffeine on reaction time on people ages 18 to 25.
  • Sample Standard Deviation—Analyzing the amount of copper in the public water supply.

Calculate the Sample Standard Deviation

Here are step-by-step instructions for calculating standard deviation by hand:

  1. Calculate the mean or average of each data set. To do this, add up all the numbers in a data set and divide by the total number of pieces of data. For example, if you have four numbers in a data set, divide the sum by four. This is the mean of the data set.
  2. Subtract the deviance of each piece of data by subtracting the mean from each number. Note that the variance for each piece of data may be a positive or negative number.
  3. Square each of the deviations.
  4. Add up all of the squared deviations.
  5. Divide this number by one less than the number of items in the data set. For example, if you had four numbers, divide by three.
  6. Calculate the square root of the resulting value. This is the sample standard deviation.

Calculate the Population Standard Deviation

  1. Calculate the mean or average of each data set. Add up all the numbers in a data set and divide by the total number of pieces of data. For example, if you have four numbers in a data set, divide the sum by four. This is the mean of the data set.
  2. Subtract the deviance of each piece of data by subtracting the mean from each number. Note that the variance for each piece of data may be a positive or negative number.
  3. Square each of the deviations.
  4. Add up all of the squared deviations.
  5. Divide this value by the number of items in the data set. For example, if you had four numbers, divide by four.
  6. Calculate the square root of the resulting value. This is the population standard deviation.