What is Standard Deviation and how to calculate it?

Image Credited By M. W. Toews  / wikipedia 

 

In statistics, the standard deviation is a wide concept used to perform hypothesis testing and correlation coefficient problems accurately.

The variance of statistics is frequently used to find the standard deviation as the standard deviation is the square root of this branch of statistics.

The variance has the squared units of measurement while the standard deviation has the same units of measurement which is why it is most frequent than the variance.

In this post, we will study all the basics of the standard deviation along with a lot of solved examples.

What is standard deviation?

In statistics, a standard deviation is a form of probability distribution used to measure the spread of data values from the mean.

The difference between the expected values and the data observations is said to be the standard deviation.

The data set of observation in the standard deviation can either be sample or population. If you have to measure the spread of whole data, then the population is used while the sample data is used to measure the estimated values of the data set.

Kinds of standard deviation

There are two kinds of standard deviation used to measure the spread of data.

1)      Population standard deviation

2)      Sample standard deviation

Let’s study the basics of the kinds of standard deviation.

1.  Population standard deviation

The population standard deviation is used to measure the spread of whole data of observation of a similar kind with the division of total numbers of observations and the statistical sum of squares.

The common difference between the mean of the population and observation is said to be the deviation.

The square root of the result of variance is said to be the output of the standard deviation. Variance is the same process of the standard deviation with the squared units of the measurements.

The standard deviation formula for the population set of observations is:

Population standard deviation = σ = √ [∑ (x_i - μ)²/n]

·         In the above expression, the “σ” notation denotes the result of the population standard deviation.

·         Where the set of observations in number in a given data set is denoted by “n”.

·         x_iis the given set of comma-separated observations of population data.

·         The mean is “μ” of the given set of population observations.

·         (x_i–μ)² is the statistical sum of squares and is known as deviation.

2.  Sample standard deviation

The sample standard deviation is the process of finding the spread of sample observations from the whole to measure the statistical sum of squares divided by one decreased in the total number of observations and taking the square root of this division.

In simple words, the square root of the result of the sample variance is said to be the output of the sample standard deviation.

The sample standard deviation is the estimated values from the whole set of population observations.

The standard deviation formula for the sample set of observations is:

Sample standard deviation = s = √ [∑ (x_i - x̅)²/n – 1]

·         In the above expression, the “x̅” notation denotes the result of the sample standard deviation.

·         Where the set of observations in number in a given data set is denoted by “n”.

·         x_i is the given set of comma-separated observations of sample data.

·         The mean is “x̅” of the given set of sample observations.

·         (x_i – x̅)² is the statistical sum of squares and is known as deviation.

Use a standard deviation calculator to measure the sample and population standard deviation in a fraction of seconds.

How to calculate standard deviation?

The sample and population standard deviation problems can be solved easily by using their corresponding formulas.Follow the below steps to solve the problems of sample and population standard deviation manually.

1.      In the first step, you have to find the mean of the sample or population data observations (x̅ or μ) by taking the quotient of the sum of observations and the total number of observations.

2.      Once you find the sample or population mean, then find the deviation of the sample and population means from the observations one by one. This deviation is also said to be the subtraction of data values from the mean. After that take the square of each deviation to make all the terms positive.

3.      Find the statistical sum of squares by taking the summation of the squared subtracted values.

4.      Divide the statistical sum of squares by “n” for population standard deviation or by n – 1 for sample standard deviation.

5.      In the last step, you have to take the square root of the quotient of the statistical sum of squares and the total number of observations to get the standard deviation of population or sample data.

Example-I: For sample standard deviation

Evaluate the sample standard deviation of 5, 6, 9, 10, 13, 16, 20, 22, 29, 30.

Solution

Step-i:First of all, determine the sample mean of the comma separated observations of sample data.

Sum of sample values = 5 + 6 + 9 + 10 + 13 + 16 + 20 + 22 + 29 + 30

 = 160

Total number of observations = n = 10

Sample mean of data set = x̅= 160/10 = 80/5

                                                    = 16

Step-ii:Now find the difference of each data value from the sample mean.

x₁ - x̅ = 5–16= -11

x₂ - x̅ = 6–16= -10

x₃ - x̅ = 9–16= -7

x₄ - x̅ = 10–16= -6

x₅ - x̅ = 13–16= -3

x₆ - x̅ = 16–16= 0

x₇ - x̅ = 20–16= 4

x₈ - x̅ = 22–16= 6

x₉ - x̅ = 29–16= 13

x₁₀ - x̅ = 30–16 = 14

Step-iii:Find the deviation squares to make all the entries positive.

(x₁ - x̅)² = (-11)² = 121

(x₂ - x̅)² = (-10)² = 100

(x₃ - x̅)² = (-7)² = 49

(x₄ - x̅)² = (-6)² = 36

(x₅ - x̅)² = (-3)² = 9

(x₆ - x̅)² = (0)² = 0

(x₇ - x̅)² = (4)² = 16

(x₈ - x̅)² = (6)² = 36

(x₉ - x̅)² = (13)² = 169

(x₉ - x̅)² = (14)² = 196

Step iv:Now find the statistical sum of squares.

∑ (x_i - x̅)² = 121 + 100 + 49 + 36 + 9 + 0 + 16 + 36 + 169 + 196

                  = 732

Step-v:Now divide the statistical sum of squares by n – 1.

∑ (x_i - x̅)² / n - 1 = 732 / 10 – 1

                            = 732 / 9

                            = 81.33

Step vi:Take the square root of the division of the statistical sum of squares by n – 1 to get the sample standard deviation.

√ [∑ (x_i - x̅)² / n – 1] = √81.33

                                  = 9.018

Example-II: For population standard deviation

Calculate the population standard deviation of 2, 6, 9, 12, 16, 21, 25, 28, 30, 31.

Solution

Step-i:First of all, determine the population mean of the comma separated observations of sample data.

Sum of population values = 2 + 6 + 9 + 12 + 16 + 21 + 25 + 28 + 30 + 31

                                               = 180

Total number of observations = n = 10

Mean of population data set = μ= 180/10 = 90/5

                                                           = 18

Step-ii:Now find the difference of each data value from the sample mean.

x₁ - μ = 2–18= -16

x₂ - μ = 6–18= -12

x₃ - μ = 9–18= -9

x₄ - μ = 12 - 18= -6

x₅ -μ = 16–18= -2

x₆ - μ= 21–18= 3

x₇ -μ = 25–18= 7

x₈ - μ = 28–18= 10

x₉ - μ = 30–18= 12

x₁₀ - μ = 31–18 = 13

Step-iii:Find the deviation squares to make all the entries positive.

 (x₁ -μ)² = (-16)² = 256

(x₂ -μ)² = (-12)² = 144

(x₃ - μ)² = (-9)² = 81

(x₄ -μ)² = (-6)² = 36

(x₅ - μ)² = (-2)² = 4

(x₆ - μ)² = (3)² = 9

(x₇ -μ)² = (7)² = 49

(x₈ - μ)² = (10)² = 100

(x₉ - μ)² = (12)² = 144

(x₁₀ - μ)² = (13)² = 169

Step iv:Now find the statistical sum of squares.

∑ (x_i - μ)² = 256 + 144 + 81 + 36 + 4 + 9 + 49 + 100 + 144 + 169

                   = 992

Step-v:Now divide the statistical sum of squares by n.

∑ (x_i -μ)² / n = 992 / 10

                       = 496 / 5

                       = 99.2

Step vi:Take the square root of the division of the statistical sum of squares by n to get the population standard deviation.

√ [∑ (x_i - μ)² / n] = √99.2

                             = 9.96

Summary

Now you are witnessed that this topic is not difficult just a little effort is required. Once you grab the basics of the standard deviation, you can easily solve any problem of sample or population standard deviation.

 

Ad Reference ( ? ) 

Use a standard deviation calculator  to Measure the sample and population standard deviation in a fraction of seconds by Lisa Johnson   @  Standard Deviation Calculator Website 

Disclaimer

See Also

• What is Dispresion?