Calculate the Standard Deviation of a population or sample!

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?

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🔢 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:

1. Find the arithmetic mean (the average) of the numbers
   1. Add up all data values to get the sum.
   2. Count the number of values in your data set.
   3. Divide the sum by the count.
2. For each number in the data set:
   1. Subtract the mean.
   2. Square the result.
3. Calculate the variance:
   1. Add up all the squared differences between each number and the mean.
   2. Divide the sum of squared differences by the data set size (the amount of numbers).
4. 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

$$\sigma =\sqrt{{\displaystyle \frac{\sum _{i=1}^n(x_i-\mu {)}^2}{n}}}$$

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:

1. Find the arithmetic mean (the average) of the numbers
   1. Add up all data values to get the sum.
   2. Count the number of values in your data set.
   3. Divide the sum by the count.
2. For each number in the data set:
   1. Subtract the mean.
   2. Square the result.
3. Calculate the variance:
   1. Add up all the squared differences between each number and the mean.
   2. Divide the sum of squared differences by the data set size (the amount of numbers) minus 1.
4. 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

Sample Standard Deviation formula

$$s=\sqrt{{\displaystyle \frac{\sum _{i=1}^n(x_i-\overline{x}{)}^2}{n-1}}}$$