Standard Deviation Formulas

What is Standard Deviation?

Standard Deviation is the measure of the dispersion of data from its mean. It measures the absolute variability of a distribution. The higher is the dispersion or variability of data, the larger will be the standard deviation and the larger will be the magnitude of the deviation of value from the mean whereas the lower is the dispersion or variability of data, the lower will be the standard deviation and the lower will be the magnitude of the deviation of value from the mean. The standard deviation formula is used to find the values of a specific data that is dispersed from the mean value. It is important to observe that the value of standard deviation can never be negative.


There Are Two Types of Standard Deviation

  • Population Standard Deviation

  • Sample Standard Deviation


Standard Deviation formula to calculate the value of standard deviation is given below:

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Standard Deviation Formulas For Both Sample and Population

Population Standard Deviation Formula

σ = \[\sqrt{\sum (X-\mu)^{2/n}}\]

Sample Standard Deviation Formula

s = \[\sqrt{X-\bar{X}^{2/n-1}}\]


Notations For the Sample Standard Deviation Formula and Population Standard Deviation Formula

 σ = Standard Deviation

Xi = Terms given in the data

N = Total number of terms

\[\bar{X}\] = Mean of the data


What is Variance and Standard Deviation?

Variance -  The variance is a numerical value that represents how broadly individuals in a group may change. The variance will be larger if the individual observations change largely from the group mean and vice versa.

It is important to notice similarities between the variance of sample and variance population. They have different representations and are calculated differently. The variance of a population is represented by σ² whereas the variance of a sample is represented by s².

Standard Deviation - Standard deviation is a measure of dispersion in statistics. It gives an estimation how individuals in data are dispersed from the mean value. Standard deviation is defined as the square root of the mean of a square of the deviation of all the values of a series derived from the arithmetic mean. It is also known as root mean square deviation.The symbol used to represent standard deviation is Greek Letter sigma (σ 2).


Variance and Standard Deviation Formula

Variance, σ 2 = \[\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n}\]

Standard Deviation, σ = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n}}\]

In the above variance and standard deviation formula:

x =  Data set values

\[\bar{x}\] = Mean of the data

With the help of the variance and standard deviation formula given above, we can observe that variance is equal to the square of the standard deviation.


Mean and Standard Deviation Formula

The sample mean is the average and is calculated as the addition of all the observed outcomes from the sample divided by the total number of events. Sample mean is represented by the symbol \[\bar{x}\]. In Mathematical terms, sample mean formula is given as:

\[\bar{x}\]= 1/n \[\sum_{i=1}^{n}x\]

In the above sample mean formula

N is the sample size and 

X is the correspond observed values


Standard Deviation - On the other hand, standard deviation perceives the significant amount of dispersion of observations when comes up close with data. In Mathematical terms, standard dev formula is given as:

Standard Deviation, σ = \[\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n}}\]


Standard Error of Mean Formula

The standard error of the mean is a procedure used to assess the standard deviation of a sampling distribution. It is also known as standard deviation of the mean and is represented as SEM. Generally, the population mean approximated  value is the sample mean, in a sample space. But, if we select another sample from the same population, it may obtain a different value.

Therefore, a population of the sampled means will appear to have different variance and mean values. The standard error of the mean can be determined as the standard deviation of such a sample means including all the possible samples drawn from the same population. SEM is basically an approximation of standard deviation, which has been evaluated from the sample.

The standard error of the mean formula is equal to the ratio of the standard deviation to the root of the sample size.

SEM = SD/√N

In the above standard error of mean formula,

‘SD’ is the standard deviation 

N is the number of observations.


Variance and Standard Deviation Formula for Grouped Data 

σ2  = \[\sum f(m-\mu)^{2}/N\]

And

s2= \[\sum f(m-\bar{x})^{2}/n-1\]

The calculation of standard deviation can be done by taking the square root of the variance. Hence, the standard deviation is calculated as 

Population Standard Deviation - σ 2= \[\sqrt{\sigma^{2}}\]

Sample Standard Deviation - s = \[\sqrt{s^{2}}\]

Here in the above variance and std deviation formula,

σ2 is the population variance, s2 is the sample variance, m is the midpoint of a class.


Standard Deviation Formula for Discrete Frequency Distribution

For the discrete frequency distribution of the type.

y : y₁, y₂, y₃,y₄

f : f₁, f₂, f₃, f₄

The formula for standard deviation becomes

\[\sqrt{1/N \sum_{i=1}^{n}f_{i}(x_{i}-\bar{x})^{2}}\]

In the above formula, N is the total number of observations.


Standard Deviation V/S Variance

The list of standard deviation v/s variance is given below in tabulated from

Variance

Standard Deviation

Variance is simply stated as the numerical value, which mentions how variable in the observation are.

Standard deviation is simply stated as the observations that are measured through a given data set.

Variance is nothing but average taken out from the standard deviation.

Standard deviation is stated as the root of the mean square deviation.

It is defined using squared units

It is defined using the same units of the data available

Mathematically, variance is denoted as (σ2)

Mathematically, variance is denoted as (σ)

Variance is the accurate estimate of the individuals spread out in the group

Variance is the accurate estimate of the observations in a  given data set.


Solved Examples

  1. During a survey, 6 students were asked the number hours per day they give time to their studies on an average? The answers of the students are as follows: 2, 6, 5, 3, 2, 3.Calculate the standard deviation.

Solution: 

Step 1: Calculate the mean value of the given data

2 + 6 + 5 + 3 + 2 + 3/6

= 21/6 

= 3.5


Step 2: Construct a table for the above given data

x₁

(x₁ - \[\bar{x}\])

(x₁ - \[\bar{x}\])²

2

-1.5

2.25

6

2.5

6.25

5

1.5

2.25

3

-0.5

-0.25

2

-1/5

2.25

3

-0.5

0.25


Step 3 : Now, use the standard dev formula.

Sample Standard Deviation Formula = s = \[\sqrt{\sum (X-\bar{X})^{2/n-1}}\]

= (13.5/[6-1])\[\sqrt{(13.5)/(6-1)}\]

= \[\sqrt{2.7}\]

= 1.643


  1. A Hen lays eight eggs. The weight of each egg laid by hen is given below.

Weight of an Egg (X)

60 gms

56 gms

61 gms

68 gms

51 gms

53 gms

69 gms

54 gms


Solution:

Step 1: Let us first calculate the mean of the above data

Mean = \[\sum X/N\]

60+ 56 + 61+ 68+ 51+ 53 + 69 + 54 /8

= 472/8

= 59


Step 2: Construct a table for the above - given data

x₁

(x₁ - \[\bar{x}\])

(x₁ - \[\bar{x}\])²

60

1

1

56

-3

9

61

2

4

68

9

81

51

-8

64

53

-6

36

69

10

100

54

-5

25

472


320


Step 3 :  Now, use the standard dev fromula

Standard Deviation Formula =  \[\sqrt{\sum (X-\bar{X})^{2/n}}\]

\[\sqrt{(320/8)}\]

= 6.32 grams


Quiz Time

  1. Find the Standard Deviation for the Given Data

5,10,7,12,10,20,15,22,8.2

  1. 6.89

  2. 10.01

  3. 7.26

  4. 9


2. Which of the Following Is the Measure of Variability?

  1. Mean

  2. Median

  3. Mode

  4. Standard Deviation

FAQ (Frequently Asked Questions)

1. What Are the Different Properties of Standard Deviation?

Some different properties of standard deviation are given below:

  • Standard deviation is used to compute spread or dispersion around the mean of a given set of data.

  • The value of standard deviation is always positive. It can never be negative.

  • Standard deviation is speedily affected outliers. A single outlier can increase the standard deviation value and in turn, misrepresent  the picture of spread.

  • For data with almost the similar mean, the larger  the spread, the greater the value of standard deviation.

  • If all values in a given set are similar, the value of standard deviation becomes zero (because each value is equivalent to the mean).

2. Mention Some Basic Points on Difference Between Standard Deviation and Variance?

The difference between standard deviation and variance is given below in tabulated form:

Variance

Standard Deviation

Variance is simply stated as the numerical value, which mentions how variable in the observation are.

Standard deviation is simply stated as the observations that are measured through a given data set.

Variance is nothing but average taken out from the standard deviation.

Standard deviation is stated as the root of the mean square deviation.

It is defined using squared units

It is defined using the same units of the data available

Mathematically, variance is denoted as (σ2)

Mathematically, variance is denoted as (σ)

Variance is the accurate estimate of the individuals spread out in the group

Variance is the accurate estimate of the observations in a  given data set.

3. What is the Relative Standard Deviation?

Relative standard deviation is one of the measures of deviation of a set of numbers dispersed from the mean and is computed as the ratio of stand deviation to the mean for a set of numbers. Larger the deviation, further the numbers are dispersed away from the mean. Lower the deviation, the close the numbers are dispersed from the mean. It is also called a coefficient of variation.

The formula for the relative standard deviation is given as:

RSD =  s * 100 / x bar

In the above relative standard deviation formula.

RSD = Relative standard deviation

S =Standard deviation 

X bar = Mean of the data