In the theory of statistics and probability for data analysis, standard deviation is a widely used method to measure the variability or dispersion value or to estimate the degree of dispersion of the individual data of sample or entire population. This solved example problem along with formulas & step by step calculation illustrates how the data values are being used in the formula to find standard deviation.

Standard Deviation (σ) In statistics, standard deviation often abbreviated to SD, is a common measure of dispersion or deviation always calculated from the mean or center value of the population data, is often denoted by the Greek symbol σ or by the alphabet s. The two methods of finding the data dispersion from the expected value are sample and population standard deviations.

The widely used method is sample standard deviation (to measure the SD for samples of population data) as getting the entire data population data for every experiments is practically not possible. The method of (N - 1) based formula often being used in such experiments instead of entire population data N. Whereas the N based formula used to find the population standard deviation (to measure the SD for entire population data). For both methods, it is a Bell Shaped Curve in a graph of normal distribution illustrates how much deviation or spread (how widely individual values are dispersed from the average value (the mean)) is occurred or can be occurred from the expected value or mean of data set.

Variance Variance is defined as the square of the standard deviation and it is denoted by σ^{2}

Standard Deviation Formulas

Formulas to estimate sample standard deviation

The below three formulas are used to find mean, SD and variance for any random samples of a population respectively. Refer the solved example for step by step calculation. The SD is always calculated from the mean, so the mean can be calculated by using the below formula

Formula to estimate standard deviation based on the random samples of a population :

Formula to calculate SD variance:

Formulas to estimate population standard deviation

The below formulas can be used to calculate the population mean, PSD and PSD variance.

Formula to find the mean of a population :

Formula to estimate standard deviation of the entire population data :

Formula to estimate PSD variance :

Solved Example Problem

1. Calculate the Mean, Standard Deviation, Variance and population SD for the given input data x = {3,4,6,8,10,15}

Solution: Inputs ={3,4,6,8,10,15}
Total Inputs (N)=6

(i) To Find Mean: Mean(x_{m})= (x_{1}+x_{2}+x_{3}...x_{n})/N
= 46/6
= 7.6667

(ii) Find Standard Deviation: SD = sqrt(1/(N-1)*((x_{1}-x_{m})^{2}+(x_{2}-x_{m})^{2}+..+(x_{n}-x_{m}^{2}))
= sqrt(1/(6-1)((3-7.667)^{2}+(4-7.667)^{2}+(6-7.667)^{2}+(8-7.667)^{2}+(10-7.667)^{2}+(15-7.667)^{2}))
= sqrt(1/5((-4.667)^{2}+(-3.667)^{2}+(-1.667)^{2}+(0.333)^{2}+(2.333)^{2}+(7.333)^{2}))
= sqrt(1/5((21.778)+(13.445)+(2.778)+(0.111)+(5.444)+(53.777)))
= sqrt(19.467)
= 4.4121

Example Problem: 2. The Standard Deviation of 10 values is 4. If each value is increased by 3, find the standard deviation and the variance of the new set of values.

Solution: Given SD of given 10 set of data is 4;
Increment in each value = 3
SD is unchanged by the increments in the values.
So the Standard Deviation σ = 4
and Variance σ^{2} = 4^{2} = 16

Standard Deviation Worksheet or Practice Problems

1. The following are the bowling rate per over of the player in 12 cricket matches. 6.5,5.0,5.2,5.3,5.5,4.7,4.6,6.3,3.0, 4.0 and 9.0 Find the standard deviation?

2. Find the Standard Deviation for the following values: 30, 80, 60, 70, 20, 40, 60.

3. The marks of 5 students scored out of 50 are 20, 25, 30, 40 and 35. Find the standard deviation of the marks. When we convert the marks to 100 find the new standard deviation.

4. The Variance of 5 value is 9. If each value is doubled then find the standard deviation of new values.

The above formulas, solved example and step by step calculation to calculate standard deviation, variance for the given population data may makes you understand how the data values are being used in the formulas. When try such calculations on your own, use this standard deviation calculator to verify your results.