# Standard Deviation Calculator

## What is Standard Deviation?

Standard deviation is a term which measures the amount of variation or dispersion of a set of values. If the values are close to the mean of the set, it will be a low standard deviation. If the values are spread in a wider range, it will be a high standard deviation.

The concept of standard deviation was presented by KarI Pearson in 18th century. Standard deviation is the measurement of variation between given values in a group. SD is always calculated from the arithmetic mean not from median or mode. It is denoted by the symbol of sigma (σ)

## Standard Deviation formula

The standard deviation formula for population is:

$$SD=σ=\sqrt\frac{\sum(x-µ)^2}{n}$$

In the standard deviation formula the ∑ means summation value of the observation. x is value in the given data set and µ is the mean of the given data set of the population and n means the total number of items.

For every data set the sample standard deviation formula will be:

$$SD=σ=\sqrt\frac{\sum(x-x)^-2}{n-1}$$

## How to calculate standard deviation?

Follow below steps to calculate standard deviation step by step:

Step #1: Find out the mean (µ) of the given data.

Step #2: Subtract the mean (µ) from each given value (deviation from the mean).

Step #3: Take square of the each deviation of the mean.

Step #4: Find out the summation of the taken squares.

Step #5: Divide its total by the number (n) which will be called variance.

Step #6: Take the square root of variance, the result will be called the standard deviation.

Use expected variance calculator to learn the calculations of variance online.

## How to find Standard Deviation?

In order to learn how to find standard deviation lets solve an example.

The math test scores of different students are: 91, 91, 91, 41, 51.

To find standard deviation of the given class we will use standard deviation formula.

$$SD= σ =\sqrt\frac{\sum(x-µ)^2}{n}$$ $$\sqrt\frac{\sum(18+18+18-32-22)^2}{n}$$

$$\sqrt\frac{324+324+324+1024+484}{5}$$

$$\sqrt\frac{2480}{5}$$

$$SD= σ =\sqrt496$$

$$SD= σ =22.27105745132$$

Below steps will help us finding standard deviation

Step #1: Find the mean M

(91 + 91 + 91 + 41 + 51) / 5 = 73

Step #2: Calculate the deviation from the mean

91 – 73 = 18, 91 – 73 = 18, 91 – 73 = 18, 41 – 73 = -32, 51 – 73 = -22

Step #3: Square of each deviation from the mean

(18)² = 324, (18)² = 324, (18)² = 324, (-32)² = 1,024, (-22)² = 484

Step #4: Calculate the sum of all squares

324 + 324 + 324 + 1,024 + 484 = 2,480

Step #5: Divide the total of the taken squares by the number of items (n)

2,480 / 5 = 496

Step #6: Find the square root of the variance

$$\sqrt496=22$$

## How to calculate Standard Deviation of the class? Let's say a physics class took a test with scores of 90, 90, 90, 50, 50 and we are to calculate the standard deviation for the class.

$$SD= σ =\sqrt\frac{\sum(x-µ)^2}{n}$$

$$SD= σ^2 =\frac{\sum(x-µ)^2}{n}$$ $$=\frac{1920}{5}$$

$$=384$$

$$=\sqrt384$$

$$=19.595917942265$$

Our portal also has covariance formula calculator for students and teachers. You can learn about the formulas, equations and calculations covariance on our website for free.

## What is Standard Error?

The sample mean differs from the actual mean of population's data set; this deviation is called standard error of the mean. Standard error occures when we collect small sample data or too much samples of population, the variation causes difference among set of values.

## Standard Deviation vs Standard Error

Standard deviation differs from standard error. Standard deviation tells the amount of variability and dispersion from the mean of data. Standard error tells about the divergence between the sample mean and true population mean.

Standard error of mean is always smaller than standard deviation. Standard error calculator calculates the standard deviation equation and finds the standard error (SE).

## Standard Error formula

Standard error formula to calculate standard error is

$$SE=\frac{σ }{\sqrt(n)}$$

Standard error is helpful for you to accurate the mean of given data from that specific population which likely would be compared to the actual population mean.

## Standard Deviation between two sets of data

Standard deviation finds the differenence in numbers and diversity of the data set values.

If the data is 3, 7, 7, 19 vs 2, 5, 6, 7. Follow below steps to calculate SD between two sets of data. The steps are

Step #1: Collect data to create data set to calculate the standard deviation.

Step #2: Calculate the average and mean of data set by adding all the numbers and dividing the total by the number of items in data set.

(3 + 7 + 7 + 19) / 4 = 9 vs (2 + 5 + 6 + 7) / 4 = 5

Here the mean is 5

Step #3: Subtract the mean from the first number in your data set and square the differences.

3 – 9 = -6² = 36, 7 – 9 = -2² = 4, 7 – 9 = -2² = 4, 19 – 9 = 10² = 100

Vs

2 – 5 = -3² = 9, 5 – 5 = 0² = 0, 6 – 5 = 1² = 1 7 – 5 = 2² = 4

Step #4: Add squared differences and divide the total by the number of items in data set.

36 + 4 + 4 +100 = 144

144 / 4 = 36

Vs

9 + 1 + 4 = 14

Step #5: Take the square root of this mean of differences to find the standard deviation.

$$\sqrt36=6$$

Vs

$$\sqrt14=3.74$$

This is how we calculate the standard deviation between two sets of data.

## What is Standard Deviation Calculator?

Like other math concepts, finding standard deviation can be difficult if we do not have its proper concept. Calculatored has introduced an online Standard deviation calculator which takes the input and provides accurate results instantly.

## How to use Standard Deviation Calculator?

Standard deviation calculator is fast, accurate and free to use. You just need to enter the values of data set and our free standard deviation calculator will instantly calculate the values of mean, standard deviation (SD) and variance.

## External Resources: ### Shaun Murphy

Last updated: September 09, 2020

I'm graduated in biomedical and electrical engineering. Specialization in sports and medical topics but will gladly tackle everything you throw at him. He is a sailor, hiker, and motorcyclist in his free time.