1. What is Standard Deviation?

The quantitative degree by which every value in a given data varies from a Measure of Central Tendency is known as Standard Deviation (S.D.) It is denoted by Greek small letter σ. It is the positive square root of the Arithmetic mean of squares of deviations of a given data set from their Arithmetic mean.

The term ‘Standard Deviation’ was first used by Karl Pearson in 1894.

Question:


What is Measure of Central Tendency?

Answer:
A single value which defines the properties of a given distribution is known as Measure of Central Tendency. Mean, mode and median are three main Measure of Central Tendency.  Mean is the most frequently used Measure of Central Tendency.

Question:

Why is mean used as a Measure of Central Tendency?

Answer:

Mean is the sum of all the values of an observation divided by the total number of observations in a given data.

Formula of Arithmetic Mean 


sd


For the Frequency Distribution

sd2

Arithmetic Mean is used as a Measure of Central Tendency quite frequently because of the following merits:-

  • It covers all the values of an observation.
  • It allows further mathematical treatment.
  • The error is minimized by using Arithmetic Mean for every value.
  • Mean is the only value for which the sum of deviations for an observation is zero.

In case of Standard Deviation, Mean is represented by x-bar and in case of population observations mean is denoted as Greek letter mu (µ). Unlike Standard Deviation, the formula of mean remains the same in both the cases.

2. Formula for Standard Deviation

Population Standard Deviation

sd3

Sample Standard Deviation

sd4

Question:

Why are there two different methods of Standard Deviation S.D.?

Answer:

S is an unbiased estimator for Population Standard Deviation. However, Sigma (σ) is a biased estimator for Population S.D. For large samples, the bias starts tending to zero as n increases. Hence, sigma is also considered as an equally important S.D.

For understanding the concept of Standard Deviation, the understanding of Variance is important

Question:

What is Variance?

Answer:

Given a data set, the values vary about a measure of central tendency (mean, median, mode etc) and these measures are known as measures of variation or dispersion. Variance is one of the frequently used measures of dispersion where,

For Discrete Data:

sd5


Finding Standard Deviation through Variance:

For finding Standard Deviation in case of ungrouped data (Discrete Data) or grouped data (Continuous Data) one can use, methods (1) and (2) given above respectively.

S.D. = √Variance


3. How to Find Out Standard Deviation

In case of Population Values, procedural steps to find out Standard Deviation are as follows:

Step 1: Find µ.

Step 2: For each value, subtract the mean from the value and square it.

Step 3: Add all the values obtained through step 1.

Step 4: Divide the values obtained through 3 by total number of observations. (N in case of grouped data and ∑ frequencies in case of ungrouped data). We obtain the variance after this.

Step 5: Take the square root. This is S. D. of the observation.


4. Example of Standard Deviation

Question:

For Discrete Data, consider marks obtained in a class of 10 students (out of 50): 10, 20, 45, 45, 24, 8, 19, 45, 23, 35.

Solution:

Here, N = 10
The mean of the given Data set is (10 + 20 + 45 + 45 + 24 + 8 + 19 + 45 + 23 + 35)/10 = 27.4

Xi (Observations)

(xi - µ)

(xi - µ)2

10

-17.4

302.76

20

-7.4

54.76

45

17.6

309.76

45

17.6

309.76

24

-3.4

11.56

8

-19.4

376.36

19

-8.4

70.56

45

17.6

309.76

23

-4.4

19.36

35

7.6

57.76

1.N = 10

2.∑(xi - µ) = 0

3.∑(xi - µ)2 = 1822.4


Here, N=10, variance= (3/N) = 182.4

S.D. = √variance → Standard Deviation = 13.49963

Here, S.D. is very high which indicates that there is a very high variation in marks of students in the class.

Question:

Given no. of people belonging to different age groups. Find the S.D. for grouped data?


Age group

Mid-value

Frequency

d = x - A/h

fd

fd2

20-30

25

3

-3

-9

27

30-40

35

61

-2

-122

244

40-50

45

132

-1

-132

132

50-60

55

153

0

0

0

60-70

65

140

1

140

140

70-80

75

51

2

102

204

80-90

85

2

3

6

18

Total

 

542

 

-15

765


Solution: