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# MEASURES OF DISPERSION

## If everything were the same, we would have no need

of statistics. We often need to measure the extent to
which scores in a dataset differ from each other.
Such a measure is called dispersion.

Range (R)
Mean Deviation (MD)
Variance
Standard Deviation (SD)
Example

## Groups Scores Mean

46 48 50 52 54 50
Section
E
30 40 50 60 70 50
Section
F
Objectives of Dispersion

## To know the average variation of

different values from the average of
a series

## To compare between two or more

series expressed in different units

## To know whether Central Tendency

truly represent the series or not
Range

## The range is the difference between the

highest and lowest values of a dataset.

## For the dataset {4, 6, 9, 3, 7}

the lowest value is 3, highest is 9
so the range is 9-3=6.
Mean Deviation

## It is the mean of the absolute deviations of a

set of data about the mean. For sample size N,
It is defined by
Example
x x x^ l x x^ l
92 2 2
75 -15 15
95 5 5
90 0 0
98 8 8
Total 30
MD = 30 / 5 = 6
Variance
Example
x x - 49.2 (x - 49.2 )2
44 -5.20 0027.04
50 0.80 0000.64
38 11.2 0125.44
96 46.8 2190.24
42 -7.20 0051.84
47 -2.20 0004.84
40 -9.20 0084.64
39 -10.2 0104.04
46 -3.20 0010.24
50 0.80 0000.64
Total 2600.40
2 = 2600 . 40 / 10 = 260 . 40
Standard Deviation
Coefficient of Variation
Why Coefficient of Variation

## The coefficient of variation (CV) is used to compare

different sets of data having different of measurement.
The wages of workers may be in dollars and the
consumption of meat in their families may be in
kilograms. The standard deviation of wages in dollars
cannot be compared with the standard deviation of
amounts of meat in kilograms. Both the standard deviations
need to be converted into coefficient of variation for
comparison. Suppose the value of CV for wages is 10% and
the value of CV for kilograms of meat is 25%. This means
that the wages of workers are consistent.
Example
A company has 2 sections with 40 and 65 employees .Their
average weekly wages are \$450 and \$350. The SDs are
7 and 9. (i) Which section has a larger wage bill? (ii)
Which section has larger variability in wages?

## Wage bill for section A = 40 x 450 = 18000

Wage bill for section B = 65 x 350 = 22750
Section B is larger in wage bill.

## CV for Section A = 7/450 x 100 =1.56 %

CV for Section B = 9/350 x 100 = 2.57%
There is greater variability in the wages of section B.
Skewness
Kurtosis
Correlation

## It is a statistical technique that can show whether and

how strongly pairs of variables are related.

## The average weight of people 5'5'' is less than the

average weight of people 5'6'', and their average
weight is less than that of people 5'7'', etc. Correlation
can tell us just how much of the variation in peoples'
weights is related to their heights.
Types of correlation

## Positive Negative Zero

correlation correlation correlation
Positive correlation

## As the values of a variable increase, the

values of the other variable also increase.

## As the values of a variable decrease, the

values of the other also decrease.

## Relation between price and supply

Negative correlation

## As the values of a variable increase, the values

of the other variable decrease.

## As the values of a variable decrease, the values

of the other variable increase.

Zero correlation

## Here, change in one variable has

no effect on the other variable.

## Relation between height and

exam grades
Correlation coefficient
Interpretation of correlation coefficient

r = 0 indicates no relation
r = + 1 indicates a perfect positive relation
r = - 1 indicates a perfect negative relation
Values of r between 0 and 0.3 (0 and - 0.3)
indicate a weak positive (negative) relation
Values of r between .3 and .7 (- .3 and - .7)
indicate a moderate positive (negative)
relation
Values of r between 0.7 and 1 (- 0.7 and -1)
indicate a strong positive (negative) relation
Example
x y xy x2 y2
0 1 0 0 1
1 2 2 1 4
2 3 6 4 9
3 4 12 9 16
4 5 20 16 25
5 6 30 25 36
Total 15 21 70 55 91
x = 15, y = 21, x y = 70, x2 = 55, y2 = 91, n = 6
Putting these values in the equation of r, we get: r = 1
Regression
Interpretation

## a gives expected amount of

change in Y for X=0

## b gives expected amount of

change in Y for 1 unit change in X
Example
x y xy x2 y2
0 1 0 0 1
1 2 2 1 4
2 3 6 4 9
3 4 12 9 16
4 5 20 16 25
5 6 30 25 36
Total 15 21 70 55 91
x = 15, y = 21, x y = 70, x2 = 55, y2 = 91, n = 6
Using these values, we get: a = 1 , b = 1
Correlation vs. Regression

Correlation Regression
It cannot predict It can predict

+
Probability
Example

## 5 cards are chosen at random from a

deck of 52 playing cards. What is the
probability of choosing 5 aces?

P (5 aces) = 0 / 52 = 0
Experiment

## An action where the result

is uncertain is called an
experiment.

Tossing a coin
Sample Space

## All possible outcomes of an

experiment is called sample space.

## A die is rolled, the sample space :

S = {1, 2, 3, 4, 5, 6}
Event

A single result of an
experiment is called an event.

## Getting a Tail when tossing a

coin
Example
A die is rolled, find the probability that
an even number is obtained.

## Sample Space , S = {1, 2, 3, 4, 5, 6}

The event "an even number is obtained"
E = {2, 4, 6}
P (E) = n (E) / n(S) = 3 / 6
Example

## A teacher chooses a student at

random from a class of 30 girls.
What is the probability that the
student chosen is a girl?

P (girl) = 30 / 30 = 1
Example
In a lottery, there are 10 prizes and 25
blanks. A lottery is drawn at random.
Find the probability of getting a prize?

P (getting a prize)
= 10 / ( 10 + 25 )
= 10 / 35
Example
At a car park there are 60 cars, 30 vans and 10 Lorries. If
every vehicle is equally likely to leave, find the probability of:
a)Van leaving first b) Lorry leaving first.

## a) Let S be the sample space and A be the event of a

van leaving first. So,
n (S) = 100 , n (A) = 30 : P ( A ) = 30 / 100

## b) Let B be the event of a lorry leaving first. So,

n (S) = 100 , n (B) = 10, : P ( B ) = 10 / 100
Example
In a box, there are 8 black, 7 blue and 6 green
balls. One ball is picked up randomly. Find the
probability that ball is neither black nor green?

## Total number of balls = (8 + 7 + 6) = 21

Let E= event that the ball drawn is neither black
nor green = event that the ball drawn is blue.
P ( E ) = 7 / 21
Example
Two coins are tossed, find the probability that
two heads are obtained.
Each coin has 2 possible outcomes:
H (heads)and T (Tails)

Sample space, S = {(H, T), (H, H), (T, H), (T, T)}

## The event "two heads are obtained", E = {(H, H)}

P (E) = 1 / 4
Example
Two dice are rolled; find the probability that the sum
of the values is equal to 1.
The sample space,
S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }
Let E be the event "sum equal to 1".
P (E) = 0 / 36 = 0
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