[PDF] Measures of Central Tendency According to Simpson and Kafka

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1

UNIT-1

FREQUENCY DISTRIBUTION

Structure:

1.0 Introduction

1.1 Objectives

1.2 Measures of Central Tendency

1.2.1 Arithmetic mean

1.2.2 Median

1.2.3 Mode

1.2.4 Empirical relation among mode, median and mode

1.2.5 Geometric mean

1.2.6 Harmonic mean

1.3 Partition values

1.3.1 Quartiles

1.3.2 Deciles

1.3.3 Percentiles

1.4 Measures of dispersion

1.4.1 Range

1.4.2 Semi-interquartile range

1.4.3 Mean deviation

1.4.4 Standard deviation

1.2.5 Geometric mean

1.5 Absolute and relative measure of dispersion

1.6 Moments

1.7 ȕȖ

1.8 Skewness

1.9 Kurtosis

1.10 Let us sum up

1.11 Check your progress : The key.

2

1.0 INTRODUCTION

According to Simpson and Kafka a measure of central tendency is typical range of the data that is used to represent all the values in the series. Since an average is somewhere within the range of data, it is sometimes called a measure of

1.1 OBJECTIVES

The main aim of this unit is to study the frequency distribution. After going through this unit you should be able to : describe measures of central tendency ; calculate mean, mode, median, G.M., H.M. ; find out partition values like quartiles, deciles, percentiles etc; know about measures of dispersion like range, semi-inter-quartile range, mean deviation, standard deviation;

DQGFRHIficients, skewness, kurtosis.

1.2 MEASUIRES OF CENTRAL TENDENCY

The following are the five measures of average or central tendency that are in common use : (i) Arithmetic average or arithmetic mean or simple mean (ii) Median (iii) Mode (iv) Geometric mean (v) Harmonic mean Arithmetic mean, Geometric mean and Harmonic means are usually called Mathematical averages while Mode and Median are called Positional averages. 3

1.2.1 ARITHMETIC MEAN

To find the arithmetic mean, add the values of all terms and them divide sum by the number of terms, the quotient is the arithmetic mean. There are three methods to find the mean : (i) Direct method: In individual series of observations x1, x xn the arithmetic mean is obtained by following formula.

1 2 3 4 1...............nnx x x x x xAMn

(ii) Short-cut method: This method is used to make the calculations simpler. Let A be any assumed mean (or any assumed number), d the deviation of the arithmetic mean, then we have .fdMAN d=(x-A)) (iii)Step deviation method: If in a frequency table the class intervals have equal width, say i than it is convenient to use the following formula. fuM A in where u=(x-A)/ i ,and i is length of the interval, A is the assumed mean. Example 1. Compute the arithmetic mean of the following by direct and short -cut methods both:

Class 20-30 30-40 40-50 50-60 60-70

Freqyebcy 8 26 30 20 16

Solution.

Class Mid Value

x f fx d= x-A

A = 45

f d

20-30 25 8 200 -20 -160

30-40 35 26 910 -10 -260

40-50 45 30 1350 0 0

50-60 55 20 1100 10 200

6070 65 16 1040 20 320

Total N = 100

By direct method

By short cut method.

Let assumed mean A= 45.

4 Example 2 Compute the mean of the following frequency distribution using step deviation method. :

Class 0-11 11-22 22-33 33-44 44-55 55-66

Frequency 9 17 28 26 15 8

Solution.

Class Mid-Value f d=x-A

(A=38.5) u = (x-A)/i i=11 fu

0-11 5.5 9 -33 -3 -27

11-22 16.5 17 -22 -2 -34

22-33 27.5 28 -11 -1 -28

33-44 38.5 26 0 0 0

44-55 49.5 15 11 1 15

55-66 60.5 8 22 2 16

Total N = 103 -58

Let the assumed mean A= 38.5, then

M = A + -58)/103

= 38.5 - 638/103 = 38.5 - 6.194 = 32.306

PROPERTIES OF ARITHMETIC MEAN

Property 1 The algebraic sum of the deviations of all the variates from their arithmetic mean is zero. Proof . Let X1, X Xn be the values of the variates and their corresponding frequencies be f1, f2n respectively.

Let xi

Xi = Xi 0L quotesdbs_dbs5.pdfusesText_9

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