## Definition of Standard Deviation

What is standard deviation? The definition of standard deviation is that a specific quantity expressing that how much members of a given group are different from the mean value for the group.

First time, the concept of standard deviation was presented by KarI Pearson in last decade of 18th century. Actually it is a 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)

## Formula of Standard Deviation

Formula of Standard Deviation

The standard deviation formula for population is:

$$SD=σ=\sqrt\frac{\sum(x-µ)^2}{n}$$

In this formula ∑ means summation of value of the variable of observation, while x is each 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.

If you are dealing with any sample data set the sample standard deviation formula will be:

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

## How to calculate standard deviation?

Method of calculate standard deviation step by step

- Find out the mean (µ) of the given data.
- Subtract the mean (µ) from each given value the result will be called the deviation from the mean.
- Take square of the each deviation of the mean.
- Find out the summation of the taken squares
- Then divide its total by the number (n) which will be called variance.
- Then take the square root of the variance the result will be called the standard deviation.

Standard deviation calculator works in the same manner as given above.

## Example of Standard Deviation

Here is an example that will clear that how to calculate standard deviation.

### Example 1:

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

To calculate the standard deviation of the given class we will follow the above given steps.

$$SD= σ =\sqrt\frac{\sum(x-µ)^2}{n}$$

$$SD= σ =\sqrt\frac{\sum(91-73)^2+(51-73)^2+(51-73)^2}{5}$$

$$\sqrt\frac{\sum(18+18+18-32-22)^2}{n}$$

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

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

$$SD= σ =\sqrt496$$

$$SD= σ =22.27105745132$$

Now let us watch it according to above given steps

- Find the mean µ

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

- Calculate the deviation from the mean

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

- Square of each deviation from the mean

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

- Calculate the sum of all squares

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

- Divide the total of the taken squares by the number of items (n)

2,480 / 5 = 496

- Find the square root of the variance

$$\sqrt496=22$$

## Example 2:

Different physics class took the same test with five test scores>

90, 90, 90, 50, 50

Calculate the standard deviation for the class

$$SD= σ =\sqrt\frac{\sum(x-µ)^2}{n}$$

$$SD= σ^2 =\frac{\sum(x-µ)^2}{n}$$

$$SD= σ^2 =\frac{(90-74)^2+...+(50-74)^2}{5}$$

$$=\frac{1920}{5}$$

$$=384$$

$$=\sqrt384$$

$$=19.595917942265$$

## Standard Error

In statistics, a sample mean differs from the actual mean of the given data set of population; this deviation is called standard error of the mean. Standard error occurs when we conduct any research and mostly collect only small sample data of that specific population just because of it we find a slight difference among set of values each time. Another side if we take enough samples of the population, the mean will be automatically arranged in to the distribution of true population mean.

There is a slight difference between standard deviation and standard error that is a SD tells the amount of variability and dispersion from the mean for given set of data but SE of the mean tells about the divergence between the sample mean of data and true population mean. Standard error of mean will always be smaller than standard deviation.

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 Units

Does standard deviation has units like other statistics values? The answer is, standard deviation is stated in the same units as the original data.

## Calculation of Standard Deviation between Two Sets of Data

Standard deviation quantifies how to diverse the values of your data set are, and is useful in determining how different your numbers are from each other.

The data is 3, 7, 7, 19 vs 2, 5, 6, 7

- Collect your data to create the data set from which you want to calculate the standard deviation.

- Calculate the average, or mean of the data set by adding all of the numbers of the set and dividing the total by the number of items in your set.

(3 + 7 + 7 + 19) / 4 = 9

vs

(2 + 5 + 6 + 7) / 4 = 5

Here the mean is 5

- 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

- Add squared differences and then divide the total by the number of items in data set.

36 + 4 + 4 +100 = 144

144 / 4 = 36

Vs

9 + 1 + 4 = 14

- Take the square root of this mean of differences to find the standard deviation.

$$\sqrt36=6$$

Vs

$$\sqrt14=3.74$$

Hope this article will be enough helpful for you now you should be able to calculate standard deviation.