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.
What is Measure of Central Tendency?
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.
Why is mean used as a Measure of
Central Tendency?
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
For
the Frequency Distribution
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 xbar 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
Sample Standard Deviation
Why are there two
different methods of Standard Deviation S.D.?
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
What is Variance?
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:
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
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.
Here, N = 10
The mean of the given Data set is (10 + 20 + 45 + 45 + 24 + 8 + 19 + 45 + 23 +
35)/10 = 27.4
X_{i} (Observations) 
(x_{i } µ) 
(x_{i } 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.∑(x_{i
} µ) = 0 
3.∑(x_{i
} 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.
Given no. of people belonging to
different age groups. Find the S.D. for grouped data?
Age group 
Midvalue 
Frequency 
d = x  A/h 
fd 
fd^{2} 
2030 
25 
3 
3 
9 
27 
3040 
35 
61 
2 
122 
244 
4050 
45 
132 
1 
132 
132 
5060 
55 
153 
0 
0 
0 
6070 
65 
140 
1 
140 
140 
7080 
75 
51 
2 
102 
204 
8090 
85 
2 
3 
6 
18 
Total 

542 

15 
765 
Here, we take d = (x  55)/10 now,
Mean = A + h* ∑fd/N where N= ∑f
Mean = 54.72 years.
{Variance is independent of change
of origin but not of scale}
→ σ = 11.88 years
5. Relationship of Standard Deviation with other Statistical
Measures
Relationship
between Standard Deviation and Variance
 Standard Deviation is positive
square root of Variance.
Relationship
between Standard Deviation and Precision
 Standard Deviation and
precision are inversely related to each other.
6. Implications of Standard Deviation
Degree of variability of given
observations. To understand it in a better way, consider three sets of observations
given below:
1. 
14 
2 
1 
3 
5 
Mean = 5
Standard Deviation = 5.24404
Variance = 27.5
2. 
5 
4 
6 
4 
6 
Mean = 5
Standard Deviation = 1
Variance = 1
3. 
5 
5 
5 
5 
5 
Mean = 5
Standard Deviation = 0
In
these three observations, we noticed that the mean is 5 in all the three cases,
whereas the Standard Deviation in 3^{rd} observation is 0 since there
is no variation in values. In second observation, the Standard Deviation is
very less since the values are closely related to each other and in first case,
the Standard Deviation is very high due to very high degree of variation in the
values.
7. Important Properties of Standard Deviation
 Standard Deviation is always
positive.
 Standard Deviation of a
constant is 0 i.e. σ (c) = 0 where c is a constant.
 σ (X + c) = σ (X)
 σ (c * X) = c* σ (X)
 The above two properties show
that Standard Deviation is independent of change of origin but not of
change of scale.
 σ (X + Y) = √(variance(X) +
variance(Y) + 2*covariance(X * Y)
 Standard Deviation is high when
difference between values is large.
 Standard Deviation is nothing
but sum of distance of values from mean divided by total number of
observations.
8. Standard Deviation and Standard Normal Distribution
Figure of normal distribution with mean µ
If
X is a random variable which is distributed normally such that X~N (µ, σ^{2})
which has mean µ and σ S.D. then, the length within 2σ limits covers 68.26 of
the area of the distribution. The area within 4 σ limits i.e. from µ  2*σ to µ
+ 2*σ covers 95.46 area (% probability), the area between µ3 *σ and µ+3 *σ is
99.7773 and rest of the area falls after µ±3σ.
9. Real Life Examples where Standard Deviation is used
 In finance, Standard Deviation
is applied to the annual rate of return of an investment to measure
the investment's volatility.
 Standard Deviation is used in
models based on real life to check the variation in policies in real life
situation in a much easier way.
 Standard Deviation is used to
check hypothesis testing.
 Standard Deviation is used to
find out confidence limits which is used to find out if there is a
significant deviation in quality of the products and the limit within
which defects are permissible/tolerable limits and limits after which
measures are to be taken.
 Standard Deviation is used to
find correlation coefficient between two random variables.
 Standard Deviation is widely
used in Normal Distribution.
 Standard Deviation is used to check pvalues, ttest.
How to calculate Standard Deviation
using excel or Scientific Calculators?
Excel and Scientific Calculators
both have an option to calculate the population as well as the sample S.D.
Calculating
of S.D. Using Excel
The command to find S.D. in excel is
STDEVP (values) where values are chosen from a column.
Calculating
of S.D. Using Casio Calcuator Fx 115Es
 Click on the mode button.
 Press 3 for Stats.
 1 for S.D. of single variable.
 Enter your data
 AC
 Shift +1 for Stats
 Select σ_{x} option.
10. Precaution that should be taken while using Standard
Deviation
 It must be ascertained that which formula should be used for calculating Standard Deviation.
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