This document discusses measures of dispersion, which describe how widely or narrowly values are spread around the mean or median. It defines dispersion and explains that measures can be either absolute, measuring the amount of variation, or relative, measuring the degree of variation. Common measures discussed include range, standard deviation, and standard error. Standard deviation is presented as the most important absolute measure, calculated as the square root of the mean of squared deviations from the mean. Standard error measures variation between sample means. Examples are provided to demonstrate calculating different dispersion measures.

1. Measures of Dispersion

2. Mean , median mode cannot reveal
the manner in which all the values of
variables are scattered about the
statistical average
• Measure of central tendency are less useful in
statistical analysis as compared to measure of
dispersion of values around the central tendency.
• Two series of data may have the same mean or
median, the values in one series may be widely
dispersed as compared to values in other.

3. Problem

4. • Both arrays comprise of 19 numbers in a
ascending order.
• The arithmetic mean of the first and second
arrays are the same 95/19 = 5
• The mean and mode are 5 for both the arrays
• It is necessary to know the scatter of
dispersion of values from their mean and
median.

6. Dispersion

7. The variation of values of
a variable from their
statistical average is
called dispersion.
Dispersion

8. A measure of statistical dispersion
is a real number that is zero if all
the data are identical and increases
as the data becomes more diverse.
It cannot be less than zero.

9. Two type of dispersion
To know the amount of variation
[ absolute measure ]
The degree of variation
[ relative measure ]

10. To measure the amount of variation
• Range
• Quartile deviation/ semi interquartile range
• Mean deviation
• Standard deviation
• Standard error

11. To measure the degree of variation
• The coefficient of range
• The coefficient of quartile deviation
• Coefficient of mean deviation
• Coefficient of variation.

12. Range
• It is the difference between the largest and
the smallest values of variable.
• The range is expressed in the same unit as the
original data.
• The coefficient of range is expressed as a
number because the units cancel during
division.

14. Problem I
The Hb level in mg/dl of six patients are
9
5
11
8
13
7
Calculate range and coefficient of range

15. Solution for Problem

16. Problem II
Weight in
kg
10-20 20-30 30-40 40-50 50-60 60-70
Number
of person
6 10 16 14 8 4
CALCULATE RANGE AND COEFFICIENT OF RANGE

18. Advantage and limitation
•It is simple to understand
•It is easy to calculate
•It is not a reliable measure
of measurement.

19. Standard deviation
• It is also called root mean square
deviation.
• It is the positive square root of the
arithmetic mean of the square
deviations of individual observations
from their arithmetic mean.

20. • It can be applied to probability
distribution random variable
population and data set.
• It is usually denoted by Greek letter
• It is formulated by sir Francis Galton [
1822-1911]
Standard deviation

21. Standard deviation
• If any data points are close to mean then the
standard deviation is zero .
• It save as a measure of uncertainty.
• It is the most important absolute measure of
dispersion.
• The square of the standard deviation is
variance
• Variance = [SD]2

22. Problem III calculate the standard
deviation for the following one
Weight in Kg [x]
43
65
42
58
67
55
62
58
43
57

23. Direct method
Weight in Kg [x] x2
43 1849
65 4225
42 1764
58 3364
67 4489
55 3025
62 3844
58 3364
43 1849
57 3249
550 31022

24. Solution for problem III

25. Age x 45 55 65 75 85
F number of
people
6 12 17 12 3
Problem IV-Calculate S.D for
ungrouped frequency

26. Calculate S.D for ungrouped frequency
Age x 45 55 65 75 85
F number of
people
6 12 17 12 3
fx 270 660 1105 900 255
fx2 12150 36300 71825 67500 21675
fx 3190
fx2 209450

27. Solution for problem IV

28. Problem V -Calculate S.D for grouped
frequency
Age Number F
35-40 3
40-45 8
45-50 13
50-55 10
55-60 7
60-65 4
U=x-a/c X is mid value
A = mean= 49.5 C=5

29. Calculate SD for ungrouped frequency
Age Number F x U=x-a/c fu fu2
35-40 3 37 -2 -6 12
40-45 8 42 -1 -8 8
45-50 13 47 0 0 0
50-55 10 52 1 10 10
55-60 7 57 2 14 28
60-65 4 62 3 12 36
45 22 94
The formula for calculation of SD for grouped or continous are same

30. Solution for problem V

31. Advantage of standard deviation
• It is most widely used dispersion which
provides an average distance for each element
from the mean.
• It is based on all the values of variables.
• It is suitable for algebraic treatment.
• It is less affected by sampling fluctuation
• It is expressed in the same unit as the unit of
measurement.

32. Disadvantage
• It is difficult to understand.
• The process of calculation is complicated.
• It cannot be used to compare the variability of
2 or more sets of distribution given in
different unit of measurement.

33. Standard error
• It is also called standard deviation of the mean.
• It is a measure of variation of arithmetic mean in
a set of observation or measurement.
• It is smaller in value compared to standard
deviation
• While standard deviation is a measure of inter-
individual variability within a sample, the
standard error measures the inter-sample
variability.