Standard Deviation Calculator

Enter the comma separated values in the box to find standard deviation using standard deviation calculator.

Why do we use it?

Finding standard deviation was never so simple, thanks to the sample standard deviation calculator. The calculations offered by sample mean calculator are:

Sample Variance
Sample Standard Deviation
Population Variance
Population Standard Deviation
Differences, and
Square of Differences

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Let’s explore a few key points such as standard deviation definition, how to find standard deviation without using a relative standard deviation calculator, and the standard deviation equation.

What is the standard deviation?

A measure of the extent to which numbers are spread out.
Standard Deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is denoted by the Greek symbol sigma σ.

Below, You Can Find The Plot Of Normal Distribution With the Width Of 1 Band.

normal distribution plot

Standard Deviation Formula

The standard deviation formula can be expressed by taking the square root of the variance.

Sample Standard Deviation Formula

formula image

s refers to sample standard deviation
N is the number of observations
x_i is the observed values of the sample item, and
x̄ is the mean value of the sample

Population Standard Deviation Formula

formula image

σ refers to population standard deviation
N is the size of the population
x_i is the observed values from the population, and
μ is the population mean

How To Calculate Standard Deviation?

Standard Deviation Calculation can be carried out using the mean and standard deviation calculator above. However, we will explain the method to calculate SD with an example

Example:

Find the standard deviation of the given sample:
30, 20, 28, 24, 11, 17

Step 1: Calculate the mean value of sample data:

N = 6

$$ \frac{\sum X}{N}\:=\:\frac{\left(30+20+28+24+11+17\right)}{6}\:=\:\frac{130}{6} $$

Step 2: Calculate $(x i - x̄)$ by subtracting mean value from each value of data set.

$x 1 - x̅ = 30 - 21.67 = 8.33$

$x 2 - x̅ = 20 - 21.67 = -1.67$

$x 3 - x̅ = 28 - 21.67 = 6.33$

$x 4 - x̅ = 24 - 21.67 = 2.33$

$x 5 - x̅ = 11 - 21.67 = -10.67$

$x 6 - x̅ = 17 - 21.67 = -4.67$

Step 3: Now, take the square of $(x i - x̄)$ for each value to find $(x i - x̄) 2$ .

$(x 1 - x̅) 2 = 8.33 2 = 69.4$

$(x 2 - x̅) 2 = -1.67 2 = 2.78$

$(x 3 - x̅) 2 = 6.33 2 = 40$

$(x 4 - x̅) 2 = 2.33 2 = 5.43$

$(x 5 - x̅) 2 = -10.67 2 = 113.85$

$(x 6 - x̅) 2 = -4.67 2 = 21.80$

Step 4: Get the sum of all values for $(x i - x̄) 2$ .

∑ $(x i - x̅) 2$
$69.4$
$2.78$
$40$
$5.43$
$113.85$
$21.80$
∑ $(x i - x̅) 2 = 253.26$

Step 5: Divide ∑ $(x i - x) 2$ with $(N-1)$ .

$$ \frac{∑(x_i - \overline x)^2 }{N - 1} = \frac{253.26}{5} = 50.65 $$

Variance = $50.65$

Step 6: Take the square root of $$ \frac{∑(x_i - \overline x)^2 }{N - 1} $$ to get the standard deviation.

$$ \sqrt{\frac{\sum \left(x_i\:-\:\overline{\:}x\right)^2\:}{N-1}}\:=\:\sqrt{\frac{253.26\:}{6-1}}=\:\sqrt{\frac{253.26\:}{5}}$$

$$\:\sqrt{\frac{\sum \left(x_i\:-\:\overline{\:}x\right)^2\:}{N-1}}\:=\:\sqrt{50.65}=7.1169$$

$s = 7.1169$

Use the population standard deviation calculator above to cross-check the values for SD calculations.

References:

Standard deviation. Investopedia. Retrieved September 1, 2021
Understanding and calculating standard deviation. Scribbr. Retrieved September 1, 2021
Standard deviation: Calculating step by step. Khan Academy Retrieved September 1, 2021