QMT181 Uitm Dungun 2014

1. Jamri AB Chapter 4 : Measure of Dispersion and Skewness (EPjj)
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CHAPTER 4 : MEASURE OF DISPERSION AND SKEWNESS
4.1 VARIANCE (S2) AND STANDARD DEVIATION (S)
a) Ungrouped data
Variance, S2 =
b) Grouped data
Variance, S2 =
4.2 COEFFICIENT OF VARIATION (CV)
 Sometimes we may need to compare the variability of two different data sets that have different units of measurement, the CV is one such measure.
 CV expresses standard deviation as a percentage of the mean and is computed as follows:
CV =
Example 1:
The following table represents the length of life (in seconds) of 20 mosquitoes subjected to a new spray of ABC’s brand in a laboratory experiment.
Length of life (in seconds)
Number of mosquitoes
10 and less than 15
6
15 and less than 20
9
20 and less than 25
3
25 and less than 30
2
a) Calculate the mean and the standard deviation.
b) The mean and the standard deviation for the length of life of mosquitoes using XYZ’s brand are 17.1 seconds and 5.5 seconds respectively. By using a suitable calculation determine which distribution is more consistent.
Solution :
Length of life
Number of mosquitoes
10 and less than 15
6
15 and less than 20
9
20 and less than 25
3
25 and less than 30
2

2. Jamri AB Chapter 4 : Measure of Dispersion and Skewness (EPjj)
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Mean
Standard deviation
CV (ABC)
CV (XYZ)
4.3 SKEWNESS
Skewed to the right (Positively Skewed Distributions)
In this distribution, the long tail indicates the presence of extreme values at the positive end of the distribution. This pulls the mean to the right. In a positively skewed distribution, the mean is pulled in a positive direction. The frequency curve would be:
Symmetrical Bell-Shaped Distribution,
This type of distribution is known as ‘normal’ and an approximately normal distribution occurs when measuring quantities such as heights, examination marks and ext.

3. Jamri AB Chapter 4 : Measure of Dispersion and Skewness (EPjj)
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Skewed to the left (Negatively Skewed Distributions)
In a negatively skewed distribution, the mean is pulled in a negative direction. The frequency curve would be:
Mode < median & mean = skewed to the right
Mode > median & mean = skewed to the left
PEARSON’S COEFFICIENT OF SKEWNESS
 We need to know two things to describe the ‘skewness’ of a distribution.
 One is the direction of the skew, positive or negative, and the other is a measure of the degree of skewness, i.e. how far the distribution is pulled in one direction.
 One measure of skewness is given by PEARSON’S COEFFICIENT OF SKEWNESS.
 The formula for skewness is
mean – mode or 3(mean – median)
standard deviation standard deviation
 Generally skewness can take any value between 3 and –3.

4. Jamri AB Chapter 4 : Measure of Dispersion and Skewness (EPjj)
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4.4 BOX PLOT
The following data show the times (in seconds) between an order being placed and the food being received at a MacDonald’s drive-thru window.
166 83 158 101 200
70 255 129 121 135
i. Find the value of first quarter (Q 1 ), Median (Q 2 ) and third quarter (Q 3 )
ii. Construct a box plot for the data.
iii. Comment on the skewness.
Solution :
1
2
3
4
5
6
7
8
9
10
70
83
101
121
129
135
158
166
200
255
Lower quartile or first quartile , Q 1 => position = n + 1 =
4
Q 1 =
Q 2 (median) => position = n + 1 =
2
Q 2 (median) =
Upper quartile or third quartile Q 3 => position = 3(n + 1) =
4
Q 3 =
Construct a box plot
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