Chap4 measure of dispersion and skewness
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This document discusses measures of dispersion and skewness including variance, standard deviation, coefficient of variation, skewness, and box plots. It defines variance and standard deviation for ungrouped and grouped data. Coefficient of variation expresses standard deviation as a percentage of the mean. Skewness can be positive (skewed to the right), negative (skewed to the left), or symmetrical. Pearson's coefficient of skewness measures the degree of skewness. A box plot uses quartiles to visualize the spread of a distribution and indicate its skewness.
Chap4 measure of dispersion and skewness
- 1. Jamri AB Chapter 4 : Measure of Dispersion and Skewness (EPjj) 1 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) 2 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) 3 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) 4 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 ______________________________________________________________________