This presentation tells about Standard Deviation and its most of the properties.

1. Properties Of
Standard Deviation
BY: SAHIL JINDAL

2. Standard Deviation
Standard deviation is a widely used measurement of variability or diversity
used in statistics and probability theory. It shows how much variation or
"dispersion" there is from the "average" (mean, or expected/budgeted value).

3. Calculation
To calculate SD, you first need to find out variance:
 Variance (S2) = average squared deviation of values from mean
 Standard deviation (S) = square root of the variance

4. Example
Q. A hen lays eight eggs. Each egg was
weighed and recorded as follows:
60 g, 56 g, 61 g, 68 g, 51 g, 53 g, 69 g, 54 g.
A.
 First, calculate the mean (59).
 Then find the standard deviation.
Weight of eggs, in grams
Using the information from the above table, we can see that
The sum of each observation minus the mean, squared equals 320
Weight (x) (x - mean) (x -
mean)^2
60 1 1
56 -3 9
61 2 4
68 9 81
51 -8 64
53 -6 36
69 10 100
54 -5 25
472 320

5. Types
Low Standard
Deviation indicates
that the data points
tend to be very
close to the mean.
High Standard
Deviation indicates
that the data is
spread out over a
large range of
values.

6. Properties
 Standard deviation is only used to measure
spread or dispersion around the mean of a
data set.
 Standard deviation is never negative.
 Standard deviation is sensitive to outliers. A
single outlier can raise the standard deviation
and in turn, distort the picture of spread.

7. Cont..
 For data with approximately the same mean,
the greater the spread, the greater the
standard deviation.
 If all values of a data set are the same, the
standard deviation is zero (because each value
is equal to the mean).

8. Impact on ‘б’ of Change in Origin & Scale
 If a constant is added to all the observations then their standard deviation will
remain constant, i.e., Standard Deviation is not affected by addition of a constant
to all observations. For ex.
 If S.D. of 1, 2, 3 is 0.82 and 10 added is to these, then Standard Deviation of 11, 12, 13 will
also be 0.82.
 If a constant is subtracted to all the observations then their standard deviation will
remain constant, i.e., Standard Deviation is not affected by subtraction of a
constant to all observations. For ex.
 If S.D. of 11, 12, 13 is 0.82 and 10 subtracted is to these, then Standard Deviation of 1, 2, 3
will also be 0.82.

9. Cont.
 It is minimum when it is calculated through mean.
 If all the observations are multiplied by a constant ‘c’ then their standard deviation
will also increase ‘c’ times, i.e., Standard Deviation is affected by multiplication of all
observations by a constant. For ex.
 If S.D. of 1, 2, 3 is 0.82, then Standard Deviation of 2, 4, 6 will be 1.64.
 If all the observations are divided by a constant ‘c’ then their standard deviation will
also decrease ‘c’ times, i.e., Standard Deviation is affected by division of all
observations by a constant. For ex.
If S.D. of 2, 4, 6 is 1.64, then Standard Deviation of 1, 2, 3 will be 0.82.