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How To Calculate Standard Deviation

Calculate Standard Deviation By Hand

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This is the formula for the sample standard deviation.

This is the formula for the sample standard deviation, which is used when data is drawn from a larger set. It takes the degrees of freedom into account, using what is known as Bessel's correction.

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Standard deviation is an important calculation for math and sciences, particularly for lab reports. Standard deviation usually is denoted by the lowercase Greek letter σ. Here are step by step instructions for calculating standard deviation by hand.

What Is Standard Deviation?

Standard deviation is the average or mean of all the averages for multiple sets of data. Scientists and statisticians use the standard deviation to determine how closely sets of data are to the mean of all the sets. Standard deviation is an easy calculation to perform. Many calculators have a standard deviation function, but you can perform the calculation by hand, and should understand how it is done.

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, except the variance is divided by the number of data points (N) for the population standard deviation, but is divided by the number of data points minus one (N-1, degrees of freedom) for the sample standard deviation.

Which Equation Do I Use?

In general, if you are 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 age of respondents on a national census.

Sample Standard Deviation - Analyzing the effect of caffeine on reaction time on people age 18-25.

Sample Standard Deviation - Analyzing the amount of copper in the public water supply.

Calculate the Sample Standard Deviation

  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 found numbers in a data set, divide the sum by 4. 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 4 numbers, divide by 3.

     

  6. Calculate the square root of the resulting value. This is the sample standard deviation.
See a worked example of how to calculate sample variance and 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 found numbers in a data set, divide the sum by 4. 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 4 numbers, divide by 4.

     

  6. Calculate the square root of the resulting value. This is the population standard deviation.
See an example worked problem for variance and population standard deviation.

Learn More

How To Calculate the Mean or Average
Calculate Mean, Median, Mode, and Range
Accuracy and Precision

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