MEASURES OF DISPERSION

INTRODUCTION:

• Let us see the following Example:

Section A

(Population

A)

Section B

(Population

B)

10 25

20 30

30 35

40 35

50 40

60 45

10 20

10

30

3020

40

40

50

50

60

60

AA AA A A

B B BBB B

• Central Tendency measures do not reveal the variability present in the data.

• Dispersion is the extent to which values in a distribution differ from the average of the distribution.

• Three Main measures are: 1- The Range2- The Variance3- The Standard Deviation.

• 1- The Range

• is the difference between the largest value and the smallest value.

• R = XL – XS

• In our Example:

• But the range is a poor measure of dispersion since it depends just on two values.

• 2- The Variance:

• The average of the squared differences from the Mean.

• In population, 𝜎2 = 𝑖=1𝑁 𝑥𝑖−𝜇

2

𝑁

In Sample, 𝑠2= 𝑖=1𝑛 𝑥𝑖− 𝑥 2

𝑛−1

Shortcut formula, 𝑠2=𝑛 𝑖=1

𝑛 𝑥𝑖2 −( 𝑖=1

𝑛 𝑥𝑖)2

𝑛(𝑛−1)

In Our Example: Section

A

(Popul

ation

A)

Section

B

(Popul

ation

B)

10 25

20 30

30 35

40 35

50 40

60 45

But the meaning of variance is not clear

• 3- The Standard Deviation:

• It is the square root of the variance

𝐼𝑛 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 , 𝜎 = 𝑖=1𝑁 𝑥𝑖 − 𝜇 2

𝑁

𝐼𝑛 𝑆𝑎𝑚𝑝𝑙𝑒 , 𝑠 = 𝑖=1𝑛 𝑥𝑖 − 𝑥 2

𝑛 − 1

• Example : Areas of sprayable surfaces with DDT from a sample of 15 houses are as follows (m2):

𝟏𝟎𝟏, 𝟏𝟎𝟓, 𝟏𝟏𝟎, 𝟏𝟏𝟒, 𝟏𝟏𝟓,𝟏𝟐𝟒, 𝟏𝟐𝟓, 𝟏𝟐𝟓, 𝟏𝟑𝟎, 𝟏𝟑𝟑,𝟏𝟑𝟓, 𝟏𝟑𝟔, 𝟏𝟑𝟕, 𝟏𝟒𝟎, 𝟏𝟒𝟓

Find the variance and standard deviation of the above distribution. 𝑥 = 1875

𝑥2 = 236877

Variance = 178.71 standard deviation = 13.37

Variance For Grouped Data

𝑠2 =𝑛 𝑥𝑚

2. 𝑓 − 𝑥𝑚. 𝑓2

𝑛(𝑛 − 1)

• Example: Consider the following grouped data on the amount of time (in hours) that 80 college students devoted to leisure activities during a typical school week:

Time ( Hours) Number Of Students

10 – 14 8

15 – 19 28

20 – 24 27

25 – 29 12

30 – 34 4

35 – 39 1

4- The Coefficient of Variation:

• The coefficient of variation is most useful in comparing the variability of several different samples, each with different means.

• In Population 𝐶𝑉 =𝜎

𝜇.100%

• In Sample 𝐶𝑉 =𝑠

𝑥. 100%

• If Sample A has a mean of $100 and a standard deviation of $10 and Sample B has a mean of 300 pounds and a standard deviation of 20 pounds, which sample has greater variation?