How to Calculate Mean, Standard Deviation, and Standard Error

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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Obtain a set of numbers you wish to analyze. 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.

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]
X
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- 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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Calculate the standard deviation. This represents the spread of the population.
Standard deviation = σ = sq rt [(Σ((X-μ)^2))/(N)].^{[2]
X
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- 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.)

Method 4

Method 4 of 4:

The Standard Error of the Mean

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Calculate the standard error (of the mean). 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]
X
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Standard error = σ/sqrt(n)^{[4]
X
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- 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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How do you find the mean given number of observations?

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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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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?

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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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How can I find out the standard deviation of 50 samples?

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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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An easy to use online standard deviation calculator

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