How to Calculate Mean, Standard Deviation, and Standard Error

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After collecting data, oftentimes the first thing you need to do is analyze it. This usually entails finding the mean, the standard deviation, and the standard error of the data. This article will show you how it's done.

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Method 1
Method 1 of 4:

The Data

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  1. This information is referred to as a sample.
    • For example, a test was given to a class of 5 students, and the test results are 12, 55, 74, 79 and 90.

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Method 2
Method 2 of 4:

The Mean

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  1. Calculate the mean . Add up all the numbers and divide by the population size: [1]
    • Mean (μ) = ΣX/N, where Σ is the summation (addition) sign, x i is each individual number, and N is the population size.

    • In the case above, the mean μ is simply (12+55+74+79+90)/5 = 62.

Method 3
Method 3 of 4:

The Standard Deviation

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  1. This represents the spread of the population.
    Standard deviation = σ = sq rt [(Σ((X-μ)^2))/(N)]. [2]
    • For the example given, the standard deviation is sqrt[((12-62)^2 + (55-62)^2 + (74-62)^2 + (79-62)^2 + (90-62)^2)/(5)] = 27.4. (Note that if this was the sample standard deviation, you would divide by n-1, the sample size minus 1.)

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Method 4
Method 4 of 4:

The Standard Error of the Mean

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  1. This represents how well the sample mean approximates the population mean. The larger the sample, the smaller the standard error, and the closer the sample mean approximates the population mean. Do this by dividing the standard deviation by the square root of N, the sample size. [3]
    Standard error = σ/sqrt(n) [4]
    • So for the example above, if this were a sampling of 5 students from a class of 50 and the 50 students had a standard deviation of 17 (σ = 21), the standard error = 17/sqrt(5) = 7.6.

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  • Question
    How do you find the mean given number of observations?
    Community Answer
    To find the mean, add all the numbers together and divide by how many numbers there are. e.g to find the mean of 1,7,8,4,2: 1+7+8+4+2 = 22/5 = 4.4.
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  • Question
    The standard error is calculated as 0.2 and the standard deviation of a sample is 5kg. Can it be said to be smaller or larger than the standard deviation?
    Community Answer
    The standard error (SE) must be smaller than the standard deviation (SD), because the SE is calculating by dividing the SD by something - i.e. making it smaller.
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  • Question
    How can I find out the standard deviation of 50 samples?
    Community Answer
    The results of all your figures (number plus number plus number etc.) divided by quantity of samples 50 =SD.
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      • Calculations of the mean, standard deviation, and standard error are most useful for analysis of normally distributed data. One standard deviation about the central tendency covers approximately 68 percent of the data, 2 standard deviation 95 percent of the data, and 3 standard deviation 99.7 percent of the data. The standard error gets smaller (narrower spread) as the sample size increases.
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      Warnings

      • Check your math carefully. It is very easy to make mistakes or enter numbers incorrectly.
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      References

      1. ↑ https://www.radford.edu/~biol-web/stats/standarderrorcalc.pdf
      2. ↑ https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/a/calculating-standard-deviation-step-by-step
      3. ↑ https://sites.radford.edu/~biol-web/stats/standarderrorcalc.pdf
      4. ↑ https://handbook-5-1.cochrane.org/chapter_7/7_7_3_2_obtaining_standard_deviations_from_standard_errors_and.htm

      About This Article

      Article Summary X

      The mean is simply the average of a set of numbers. You can work it out by adding up all the numbers and dividing the total by the amount of numbers. For example, if you wanted to find the average test score of 3 students who scored 74, 79, and 90, you'd add the 3 numbers together to get 243, then divide it by 3 to get 81. The standard error represents how well the sample mean approximates the population mean. All you need to do is divide the standard deviation by the square root of the sample size. For instance, if you were sampling 5 students from a class of 50 and the 50 students had a standard deviation of 17, you'd divide 17 by the square root of 5 to get 7.6. For more tips, including how to calculate the standard deviation, read on!

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