Sample Standard Deviation Example Problem

Standard deviation
public domain
By
Anne Marie Helmenstine, Ph.D.
Anne Marie Helmenstine, Ph.D.
Chemistry Expert
  • Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
  • B.A., Physics and Mathematics, Hastings College
Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. She has taught science courses at the high school, college, and graduate levels.
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Updated on May 24, 2019

This is a simple example of how to calculate sample variance and sample standard deviation. First, let's review the steps for calculating 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.
Standard deviation and variation indicate how spread out data is from its average value.
Read More
How to Calculate Population Standard Deviation
By Anne Marie Helmenstine, Ph.D.

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 population standard deviation is 3.061

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

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Your Citation
Helmenstine, Anne Marie, Ph.D. "Sample Standard Deviation Example Problem." ThoughtCo, Apr. 5, 2023, thoughtco.com/sample-standard-deviation-problem-609528. Helmenstine, Anne Marie, Ph.D. (2023, April 5). Sample Standard Deviation Example Problem. Retrieved from https://www.thoughtco.com/sample-standard-deviation-problem-609528 Helmenstine, Anne Marie, Ph.D. "Sample Standard Deviation Example Problem." ThoughtCo. https://www.thoughtco.com/sample-standard-deviation-problem-609528 (accessed April 26, 2024).