Revision

Measures of Dispersion

Outcomes

• By the end of this lecture, the student will be able to Know definition, uses and types of statistics.

Measures of Dispersion

These are methods which used for measuring variability (or homogenicity) of observations.

a) The Range:

It is defined as the highest observation the lowest observation. It is a simple measure, easily and quickly obtained. Sometimes we cannot differentiate between the amount of variation among different groups if they have equal largest and smallest observations.

This results from the fact that this methods neglects all intermediate o

This results from the fact that this methods neglects all intermediate observations.

e.g.

1st group: 9 7 5 3 1 range = 9-1 = 8

2nd group: 9 3 4 3 1 range = 9-1 = 8

It is defined as the average of the absolute deviation of each observation from the arithmetic meanN.B. Absolute deviation means difference between

two quantities and this difference is given a +ve sign always. It is denoted by

Mean absolute Deviation =

b) The mean absolute Deviation:

n

xxi

The range is a good measure of dispersion but it does not have good mathematical properties. Ex :

The mean Absolute deviation= 12/5 = 2.4

1 5 4

3 5 2

5 5 0

7 5 2

9 5 4

xix

xxi

12 25

xxn i

• It equals the mean of the squared deviations of observations from their arithmetic mean.

S2

We use (n-1) instead of n as a correction for small values.

So, S2 =

c) The variance: (S2)

n

xxi

2

1

2

n

xxi

Mathematically this equation equals to:

This formula is better and is easier in computation.

It is the on commonly used.

1

2

2

2

nnx

xs

ii

It is defined as the positive Square route or the variance. It should always be defined in the same unit as the original variables.

I. For ungrouped data:

d) The Standard Deviation: (S)

1

2

2

nnx

xs

ii

Ex:

2

2

26

8

4

5

6

ii

ii

xx

xx

150

9 3

64

16

25

36

92173

734

2135150155

26150

12

2

2

2

..

.

.

S

s

nnx

xs

ii

II. Computation of the standard deviation from grouped data:

a. Using the long method:

Steps Determine the mid point for each interval . Find the product for each interval and the sum of these products . Find the product for each interval by multiplying by the corresponding value and then find the

sum of these product

jj xf

jx

jj xf

2

jj xf jx

jj xf2

jj xf

Find the variance S2 from the formula:

Standard Deviation

2s S

1

2

2

2

j

j

jj

jj

f

f

xfxf

S

Ex :

Age in years

Frequencyfj

Mid pointxj fj xj fj xj

2

10- 3 12.5 37.5 468.75

15- 7 17.5 122.5 2143.75

20- 6 22.5 135 3037.5

25-29 4 27.5 110

Total (Σ) 20 405 8675

years

.S

24.9 S

405-8675

S

5

924

12020

12

2

2

2

S

f

f

xfxf

Sj

j

jj

jj

AssignmentTitle Student Name

The Standard Deviation احمد احمد محمد امليوسف اسعد اميره

مرشدي صالح اميرهالموجود عبد انجي

References

• Biostatistical analysis: Jerrold H. Zar