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Chapter 4 Chapter 4

Measures of Variation Measures of Variation

(Dispersion) (Dispersion)



We have the distribution of yield of two rice varieties from

5 plots each

Variety 1: 45 42 42 41 40

Variety 2: 54 48 42 33 30

The mean yield of both varieties is 42 kg



Objectives of measuring variation:



To describe dispersion (variability) in a data



To compare the spread in two or more distributions



To determine the reliability of an average

3.1 Introduction 3.1 Introduction

The Range and Relative range

Range = Max – Min  Range is the crudest absolute measures of variation  (^) It is widely used in the construction of quality control charts and description of daily temperature RR = Range/(Max + Min) Properties of range  It is affected by extreme values  It does not take into account all observations  (^) It is easy to calculate and simple to understand 1.3 Types of measures of variation 1.3 Types of measures of variation

The mean and median deviation  MD – is the average of the absolute deviations taken from a central value, generally the mean or median n x x f n x x MD x i i n i i        1 ( ) n x x f n x x MD x  (^) i  i i     ~ ~ ) ~ (

Variance and Standard Deviation Variance and Standard Deviation

2 2 1 2 1 2 2

 

( x )

N

x

N

N

i N i

N x

i

N

Population Variance

s

x x

n

x

x

n

n

s s

i n i n

i

n

2 2 1 2 1 2 2

^ 

 

Sample Variance

Variance – is average of the squared deviations from the mean Standard deviation - is the square root of the variance Example: The height of nine students was measured in inches and the data is presented below. Height(x): 69 66 67 69 64 63 65 68 72 Calculate the population variance and standard deviations. Variance = 7.11 inch2; S.D = 2.66 inch  (^) The variance and Standard deviation of a grouped data is calculated by using the Class Mark.

Coefficient of variation (CV)

 it is the corresponding relative measure of standard deviation  It is used to compare the variability of two or more different groups  Less coefficient of variation – is said to be less variable or more consistent or more uniform or more homogeneous. 100 % x s CV

Example: The students of Biology and Chemistry took Stat

273 course. The following information was recorded

CV (Biology) = 29.11%

CV (Chemistry) = 17.2%



Chemistry students are more homogenous.

Departments Mean S.D

Biology 79 23

Chemistry 64 11

Example: Two sections were given an exam in certain course. The average score was 72 with standard deviation of 6 for section 1 and 85 with standard deviation of 5 for section 2. Student A from section 1 scored 84 and student B from section 2 scored 90. Who performed better relative to his/her group? Z-score for student A = 2. Z-score for student B = 1.

 Skewness

 Measure of asymmetry of a frequency distribution

 Symmetric or unskewed



Skewed to right



Skewed to left

1.4 Skewness and Kurtosis 1.4 Skewness and Kurtosis

Skewed to right

Skewed to left

 Kurtosis

 Measure of flatness or peakedness of a frequency distribution

 Platykurtic (relatively flat)

 Mesokurtic (normal)



Leptokurtic (relatively peaked)

Platykurtic - flat distribution