Use this statistics calculator to find the Standard Deviation for a set of values representing a population or a sample.
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How to use the Standard Deviation Calculator?
Time needed: 1 minute.
- Enter the Data Set
Write or paste a set of at least two numbers separated by commas, spaces, tabs, or newlines into the first field.
By default the data set is considered a Population. Check the box “Sample” if it is a sample. - Press the Calculate button
Press the Calculate button below the data set field.
- Review the Output
The results will appear in the Output field.
- Copy or Save
Optionally, you can Copy the output to clipboard, or Save it as a file on your device.
Calculator features
🔢 Data set length: | Unlimited |
⚡ Speed of Computation: | Instant! |
➡️ Calculator Output: | Display, Copy, Save |
🎯 Calculator Accuracy: | 100% |
📋 Definitions and Formulas: | Available |
What is the Standard Deviation? Standard Deviation definition
Standard Deviation is a measure of dispersion or variation within a data set.
It measures by how much the values in the data set are likely to differ from the mean.
The higher the Standard Deviation, the furthest the data points tend to be to the mean (they will be more dispersed).
Conversely, a low Standard Deviation, indicates that the the data points tend to be close to the mean (they are less dispersed and more concentrated at or around the mean).
Population Standard Deviation
Population Standard Deviation is tipically denoted as σ. and it is used when is possible to measure an entire population.
How to Calculate the Standard Deviation for an entire Population
Standard deviation is the square root of variance.
Follow these steps to calculate the Standard Deviation for a population:
- Find the arithmetic mean (the average) of the numbers
- Add up all data values to get the sum.
- Count the number of values in your data set.
- Divide the sum by the count.
- For each number in the data set:
- Subtract the mean.
- Square the result.
- Calculate the variance:
- Add up all the squared differences between each number and the mean.
- Divide the sum of squared differences by the data set size (the amount of numbers).
- Calculate the square root of the variance: this is the Standard Deviation.
Example
Let’s find the standard deviation for the population 5, 11, 17, 23:
5 + 11 + 17 + 23 = 56
Find the mean: 56 / 4 = 14
Subtract the mean from each number, and we get -9, -3, 3, 9
Square each result, and we get 81, 9, 9, 81
Add up the squared differences: 81 + 9 + 9 + 81 = 180
Divide by the data set size: 180 / 4 = 45 (this is the variance)
Calculate the square root of the variance: 6.708203932
Population Standard Deviation formula
Sample Standard Deviation
Sample Standard Deviation is tipically denoted as s. It is used when it is not possible to measure the entire population, so a random sample is taken into consideration.
How to Calculate the Standard Deviation for an entire Population
Standard deviation is the square root of variance.
Follow these steps to calculate the Standard Deviation for a sample:
- Find the arithmetic mean (the average) of the numbers
- Add up all data values to get the sum.
- Count the number of values in your data set.
- Divide the sum by the count.
- For each number in the data set:
- Subtract the mean.
- Square the result.
- Calculate the variance:
- Add up all the squared differences between each number and the mean.
- Divide the sum of squared differences by the data set size (the amount of numbers) minus 1.
- Calculate the square root of the variance: this is the Standard Deviation.
Example
Let’s find the standard deviation for the sample 5, 11, 17, 23:
5 + 11 + 17 + 23 = 56
Find the mean: 56 / 4 = 14
Subtract the mean from each number, and we get -9, -3, 3, 9
Square each result, and we get 81, 9, 9, 81
Add up the squared differences: 81 + 9 + 9 + 81 = 180
Divide by the data set size: 180 / ( 4 – 1) = 60 (this is the variance)
Calculate the square root of the variance: 7.745966692