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How To Calculate Standard Deviation

Worked Example Problems

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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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This is a simple example of how to calculate sample variance and sample standard deviation. First, let's review the steps for calculation the sample standard deviation:

  1. Calculate the mean (simple average of the numbers).
  2. For each number: subtract the mean. Square the result.
  3. Add up all of the squared results.
  4. Divide this sum by one less than the number of data points (N - 1). This gives you the sample variance.
  5. Take the square root of this value to obtain the sample standard deviation.

Example Problem

You grow 20 crystals from a solution and measure the length of each crystal in millimeters. Here is your data:

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

Calculate the sample standard deviation of the length of the crystals.

  1. Calculate the mean of the data. Add up all the numbers and divide by the total number of data points.

    (9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7

     

  2. Subtract the mean from each data point (or the other way around, if you prefer... you will be squaring this number, so it does not matter if it is positive or negative).

    (9 - 7)2 = (2)2 = 4
    (2 - 7)2 = (-5)2 = 25
    (5 - 7)2 = (-2)2 = 4
    (4 - 7)2 = (-3)2 = 9
    (12 - 7)2 = (5)2 = 25
    (7 - 7)2 = (0)2 = 0
    (8 - 7)2 = (1)2 = 1
    (11 - 7)2 = (4)22 = 16
    (9 - 7)2 = (2)2 = 4
    (3 - 7)2 = (-4)22 = 16
    (7 - 7)2 = (0)2 = 0
    (4 - 7)2 = (-3)2 = 9
    (12 - 7)2 = (5)2 = 25
    (5 - 7)2 = (-2)2 = 4
    (4 - 7)2 = (-3)2 = 9
    (10 - 7)2 = (3)2 = 9
    (9 - 7)2 = (2)2 = 4
    (6 - 7)2 = (-1)2 = 1
    (9 - 7)2 = (2)2 = 4
    (4 - 7)2 = (-3)22 = 9

     

  3. Calculate the mean of the squared differences.

    (4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9) / 19 = 178/19 = 9.368

    This value is the sample variance. The sample variance is 9.368

     

  4. The population standard deviation is the square root of the variance. Use a calculator to obtain this number.

    (9.368)1/2 = 3.061

    The sample standard deviation is 3.061

Compare this with the variance and population standard deviation for the same data.

Learn More

Standard Deviation Equations
How To Calculate Standard Deviation

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