Standard Deviation Formula

Before we derive the standard deviation formula let us first understand the meaning of standard deviation. For a set of data, the measure of dispersion, about mean, when expressed as the positive square root of the variance, is called standard deviation. It measures the dispersion or spread of data.

The formula for calculating the standard deviation of given data is:

Standard Deviation Formula

The population standard deviation formula is given as:

\(\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2}\)

Here,

σ = Population standard deviation

N = Number of observations in population

Xi = ith observation in the population

μ = Population mean

Similarly, the sample standard deviation formula is:

\(s =\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\overline{x})^2}\)

Here,

s = Sample standard deviation

n = Number of observations in sample

xi = ith observation in the sample

\(\overline{x}\) = Sample mean

Variance Formula:

The population variance formula is given by:

\(\sigma^2 =\frac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2\)

The sample variance formula is given by:

\(s^2 =\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\overline{x})^2\)

This is the basic formula to find the standard deviation for a given set of data.

Standard Deviation For Discrete Data

For discrete frequency distribution of the type:

Frequency Distribution

The standard deviation is given as:

Standard Deviation Formula

where   is the mean of the distribution and

Standard Deviation Formula

Standard Deviation For Continuous Frequency Distribution

For continuous frequency distribution, the mid-point of each class is considered for calculating the standard deviation. If the frequency distribution of n classes is defined by its mid-point xi with frequency fi, the standard deviation is given by:

Standard Deviation Formula

where xi are the midpoints of the classes and fi are their respective frequencies.

And  is the mean of the distribution and

Mean

Derivation

Now let us try to obtain another standard deviation formula:

Mean of the Distribution

Or,

Standard Deviation Formula

This gives us the standard deviation (σ) as:

Standard Deviation Formula

This is the standard deviation formula for a given set of observations. But sometimes it happens that the value xi in a given data set or the midpoints of classes in a given frequency distribution is very enormous. In such cases, determination of mean or median or variance becomes lengthy and time-consuming. To solve this problem we make use of step deviation method for simplifying the procedure.

Simple Method to Find Standard Deviation

Let us discuss this shortcut method for determining variance and standard deviation.

Consider the assumed mean as ‘A’ and the width of the class interval be h. Let the step deviation be yi.

Standard Deviation Formula

The mean of a data set is given by:

Standard Deviation Formula

Substituting the values of xi from equation (1) into equation (2), we get

Standard Deviation Formula

Variance of the variable x,

Standard Deviation Formula

Substituting the values from equation (1) and (3), we have

Standard Deviation Formula

σx=h2 × variance of variable yi

⇒σx2=h2σy2

⇒σx=hσy—(4)

From equation (3) and (4), we can conclude that:

Standard Deviation Formula

Solved Examples

Let us look into an example for a better insight.

Example 1: Find out the mean, variance and standard deviation for the following data representing the age group of employees working in XYZ Company.

Solution: Let the assumed mean A = 30 and h = 10. From the table given above, we can obtain.

mean, variance and standard deviation -solution

Thus,

The variance of the above data can be calculated as:

= (10/100)2 [100 × 160 – (60)2] = (1/100) [16000 – 3600] = (100/100) [160 – 36] = 124

The standard deviation can be given as:

σ = √124 = 11.136

Example 2: Find the mean, standard deviation and variance for the following data:

57, 64, 43, 67, 49, 59, 44, 47, 61, 59

Solution:

Given,

57, 64, 43, 67, 49, 59, 44, 47, 61, 59

Number of data values = 10

Mean \((\overline{x})\) = (57 + 64 + 43 + 67 + 49 + 59 + 61 + 59 + 44 + 47)/10

= 550/10

= 55

Standard variance (σ) \(=\sqrt{\frac{(x_i-\overline{x})2}{n}}\)

= \(=\sqrt{\frac{[(57 – 55)^2+(64 – 55)^2 + (43 – 55)^2 + (67 – 55)^2 + (49 – 55)^2 + (59 – 55)^2 + (44 – 55)^2 + (47 – 55)^2 + (61 – 55)^2 + (59 – 55)^2]}{10}}\)

= \(=\sqrt{\frac{4 + 81 + 144 + 144 + 36 + 16 + 36 + 16 + 121 + 64}{10}}\)

= √(662/10)

= √66.2

= 8.13

Variance = σ2 = 66.2

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