Standard Deviation (SD): An introduction with properties and examples

 

Standard deviation is a statistical term that measures how much the observed value is far from their means. If the difference between the actual value and means increases then the standard deviation also increases. First, we should know the difference between grouped data and ungrouped data to move the standard deviation.

Grouped data is in the form of class interval and have frequency. Like, 0-5, 6-10, 11-15, and so on. However, ungrouped data refers to raw information that is collected randomly and not sorted into classified. For example 14,16,89,35.

We will discuss standard deviation formulas for both grouped and ungrouped data. Means, Variance, and standard deviation formulas are quite related. We will cover all terms that relate to standard deviation.

What is Standard deviation (SD) in statistics?

Standard deviation refers to how much each observed value deviates from its means. If the observed value deviates from its means then the standard deviation is less. It can be defined as the square root of the means of the square of the deviation value from the means. In other words, we can say the under root of the variance is the standard deviation.

Different Standard deviation formulas

Let’s discuss the different formulas of standard deviation

Population standard deviation (σ) and sample standard deviation (s) formulas

The population is the entire group or event in which we are interested. However, a sample is any subgroup of the population. Population standard deviation is denoted by the Greek letter sigma (lower case) “σ” and “S” indicates sample standard deviation.

Population standard deviation	sample standard deviation
σ = Sqrt [å (x – μ) 2 / N] And, μ = Sqrt [å x / N] Here, μ = population means x = observed values N = total number of observed values	S = Sqrt [å (x – x̄) 2 / (n – 1)] And,  x̄ = Sqrt [å x / n] Here, x̄ = sample means x = observed values n = total number of observed values

A standard deviation calculator (https://www.standarddeviationcalculator.io/) is a help source to obtain the standard deviation of the given set of data according to the formula of sample and population STD.

SD for ungrouped data

·         Direct method: S = Sqrt [å (x – x̄) ² / n]

·         Shortcut method :S = Sqrt [(å x ² / n) - (å x / n) ²]

SD for grouped data

There are two conditions. If only class intervals are given but their frequency is not known then we find the midpoints (x) by adding lower-class interval and upper-class interval and then dividing it by 2. We solve it by ungrouped method.

If we have class intervals as well as their frequency. Then we have

S.D = Sqrt [å f(x – x̄) ² / åf]

Standard deviation properties

Some important properties of standard deviation are given below,

·         Standard deviation of identical observed values is zero.

·         Standard deviation is independent of his region i.e. S.D (x ± a) = S.D (x)

·         If a is any constant then S.D (ax) = |a| S.D (x) and S.D (x / a) = |1 / a| S.D(x)

·         If x and y are independent variables then S.D (x ± y) = S.D (x) + S.D (y)

Method of finding the standard deviation by a direct method

Let’s learn how we can find standard deviation by a direct method.

·         First, we calculate arithmetic means (average).

·         Find (x – x̄) for each given observation, then calculate the square of these values.

·         Find the Sum of all (x – x̄)²

·         To get the standard deviation, divide å (x – x̄) ² by the total number of observations.

Solved Examples of Standard deviation

Example1.

Calculate the standard deviation for the following observation

2, 4, 6, 8

First, we will find arithmetic means (x̄)

Arithmetic means = sum of all observations / total number of observation

x̄ = 2 + 4 + 6 + 8 / 4 = 20

x	(x – x̄)	(x – x̄)2
2	2 – 20 = -18	(-18)2 = 324
4	4 – 20 = -16	(-16)2  = 256
6	6 – 20 = -14	(-14)2 = 196
8	8 – 20 = -12	(-12)2 = 144
		å(x – x̄)2 = 920

 

å(x – x̄) ² = 1738.996

Put it in the standard deviation formula i.e. S = Sqrt [å (x – x̄) ² / n], we have

S = Sqrt [920 / 4] = 230

S = 230

So, the standard deviation of 2, 4, 6, and 8 is 230

Example 2

Find the standard deviation for the following values,

65, 46, 56, 75, 54, 64, 79, 95

Solution

Arithmetic Means = x̄ = 66.75

 

x	(x – x̄)	(x – x̄)2
65	65 – 66.75 = – 1.75	3.062
46	46 – 66.75 = – 20.75	430.562
56	56 – 66.75 = – 10.75	115.562
75	75 – 66.75 = 8.25	68.062
54	54 – 66.75 = – 12.75	162.562
64	64 – 66.75 = – 2.75	7.562
79	79 – 66.75 = 12.75	162.562
95	95 – 66.75 = 28.25	789.062
		å(x – x̄)2 = 1738.996

 

As we know that

S.D = Sqrt [å (x – x̄) ² / n]

Now put the values, we have

S.D = Sqrt [1738.996 / 8] = 217.3745

S.D = Sqrt [217.3745] = 14.743

The standard deviation of the given observation is 14.743.

Conclusion

In this article, we have discussed the standard deviation in detail. Then we described the definition of standard deviation. Then we discussed the formula of population and sample standard deviation, also discussed the formula for grouped and ungrouped data. Some properties of standard deviation covered in this article.

After this, we learned how to calculate standard deviation by the direct method. Then we solved the example of standard definition in a good manner. After reading this article, you can easily solve any question related to standard deviation.