measures of dispersion - najah videos...2019/02/18 · •central tendency measures do not reveal...
TRANSCRIPT
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?