CENTRAL TENDENCY (1 st Semester) Presented By Dr. Porinita Dutta Department of Statistics

Size: px

Start display at page:

Download "CENTRAL TENDENCY (1 st Semester) Presented By Dr. Porinita Dutta Department of Statistics"

Jonah Hicks
5 years ago
Views:

1 CENTRAL TENDENCY (1 st Semester) Presented By Dr. Porinita Dutta Department of Statistics

2 OUTLINES Descriptive Statistics Introduction of central tendency Classification Characteristics Different measures of central tendency Uses of different measures

3 Descriptive Statistics

4 Introduction Central tendency (or measures of central tendency) A central tendency or a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. In statistics, a central tendency (or measure of central tendency) is a central value for a probability distribution.

5 Generally, Measures means Central Tendency means Methods average value of a statistical series The combined term measures of central tendency means the method of finding out the central value or average value of a statistical series.

6 Measures of central tendency also known as averages describes the tendency of individual items to cluster around the central in a frequency distribution. Sometimes it is also known as measure of location. The most important objective of measures of central tendency is to determine a single representative value for the entire series.

7 Classification There are two major classification of central tendency or averages. They are: (a) Mathematical or computational (i)arithmetic mean or mean, (ii)geometric mean, (iii)harmonic mean. (b) Positional (iv) Median, (v)mode The above five measures are the most common use measures of central tendency.

8 Characteristics The characteristics to be satisfied by an ideal measure of central tendency are : It should be rigidly defined. It should be easy to understand and easy to calculate. It should be based on all the observations of the series. It should be suitable for further mathematical treatment. It should be affected least by fluctuations of sampling. It should not be affected much by extreme values.

9 Different measures of central tendency Arithmetic Mean or Mean Most commonly used measure of central tendency. Average of all observations. The sum of all the observations divided by the number of observations. Best measure of central tendency.

10 Measure of central tendency-- Mean Its formula is The arithmetic mean x of n observations is given by x = n i=1 n x i, i = 1, 2, 3,, n (for ungrouped data) For grouped data, the arithmetic mean is given by x = n i=1 N f i x i n, i = 1,2, 3,, n ; N = i<1 f i where f i is the frequency of the variable x i.

11 Measure of central tendency-- Mean Ungrouped data x = A + d n d = x A A = Assumed mean Grouped data x = A + fd N d = x A Continuous data with equal class intervals x = A + fd N h d x A = h h = class width

12 Measure of central tendency-- Mean Example1: Find the arithmetic mean of 5, 8, 12, 15,20 Solution: Arithmetic mean(x ) = n x = 5:8: :20 5 = 60 5 =12

13 Measure of central tendency-- Mean Example 2 Find the arithmetic mean from the following frequency distribution. Wages(x) : No.of Person (f): Solution: x = fx f = = 52.07

14 Properties of Arithmetic Mean 1) The algebraic sum of deviations of a set of values from their arithmetic mean is zero. Symbolically, (a) x i x =0 for ungrouped data. (b) f i x i x =0 for grouped data. 2) The sum of the squares of deviations about mean is minimum. 3) If n 1 and n 2 are the number of observations and x 1 and x 2 be their respective means of the two series then mean x of the combined series is x = n 1 x 1 : n 2 x 2 n 1 :n 2

15 Merits Demerits (i) It is rigidly defined. (ii) It is easy to calculate and easy to understand. (iii) It is based on all the observations of the series. (i) It is affected very much by extreme values. (ii) A.M cannot be obtained if a single observation is missing. (iv) It is suitable for further mathematical treatment. (v) It is not much affected by fluctuations of sampling. (iii) It cannot be determined by inspection nor it can be located graphically.

16 Harmonic Mean Harmonic mean (H.M) of a number of observations, none of which is zero, is the reciprocal of the arithmetic mean of the reciprocals of the given values. Thus, the harmonic mean(h), of n observations x i, i = 1, 2, 3,, n is given by: H = 1 n 1 n i<1 1 xi

17 Geometric Mean Geometric Mean of a set of n numbers is the n th root of their product. If one value of a series is zero, the GM is zero and if one or more values are negative, the GM is meaningless. If x i, i = 1, 2, 3,.., n are the values of a variable x then GM = n x 1 x 2 x n = x 1 x 2 x 3 1 n

18 Relationship between Arithmetic Mean(AM), Geometric Mean(GM) and Harmonic Mean(HM) (a) AM GM HM (b) AM HM (GM) 2

19 Measures of Central tendency-median Median is that value of a variable which divides the data into two equal halves so that same proportion of values lie above and below the median value. Median is a positional average.

20 Median After arranging the data in ascending or descending order the middle most item is termed as median. In case of odd number of items there is one middle term but in case of even number of items there are two middle terms and median is the AM of two values.

21 (i) When number of items n is odd- Median = value of n:1 2 (ii) When n is even- Median = value of n 2 th item th item:value of n 2 :1 th item 2

22 Median for Continuous Frequency Distribution For computing median in case of continuous frequency distribution we shall have to determine the median class. Median class can be defined as that class for which the cumulative frequency is just greater than N 2.

23 The value of median now can be obtained by the formula. Median=L + N 2 ;F f h Where. L= Lower limit of median class. N= f= Total frequency F= Cumulative frequency of the class preceding the median class. f= frequency of median class. h= length of the median class

24 Measures of Central Tendency- Mode Mode is the value which occurs most frequently in a set of observations. In case of discrete frequency distribution, mode is the value of variable corresponding to maximum frequency. If there is only one mode, the distribution is called unimodal and if there are two modes, the distribution is bimodal.

25 Ungrouped Data Mode cannot be easily determined from individual series unless it is converted to a discrete (or continuous) frequency distribution.

26 Mode for Continuous Frequency Distribution Mode= L + f 1;f 0 2f 1 ;f 0 ;f 2 h where L= Lower limit of the modal class. f 1 = frequency of the modal class. f 0 = frequency of the class preceeding the modal class. f 2 = frequency of the class succeeding the modal class. h = width of the modal class.

27 Relationship among Mean, Median and Mode For a symmetrical distribution the mean, median and mode are identical. But if the distribution is moderately asymmetrical, there is an emperical relationship between them. The relationship is Mean-Mode=3(Mean-Median)

28 Uses of Different Measures of Central Tendency AM: It is the most important, widely used and best measure of central tendency. In the determination of average income, average price, average cost of production, average sales etc. i.e., those phenomenon which are capable of direct quantitative measurements, AM is the most appropriate measure. Median: Median is most useful average in case of open end classes frequency distributions. To find the average of qualitative data median is the most suitable one.

29 Uses of Different Measures of Central Tendency Mode: Mode is mostly used in business and commerce. GM: It is the most useful when smaller items are to be given importance. They are used in the construction of Index Numbers. HM: HM is specially useful in averaging rates and ratios where time is variable and distance is constant.

30 Thank You

31 Nonparametric Test (3 rd semester) Presented by Dr. Porinita Dutta Department of Statistics

32 CONTENTS 1 2 Introduction Assumptions for NP Methods 3 Uses of NP Methods 4 5 Advantages and Disadvantages Basic steps involved in a nonparametric test of hypothesis

33 INTRODUCTION Parametric statistical methods are based on some assumption about the population from which the sample has been drawn. Nonparametric test is a test as that does not depend on the particular form of the frequency function of the parent population from which the samples are drawn.

34 The term nonparametric (NP) is used in the sense that no hypothesis is established about the parameters of the population from which the samples are drawn. Whereas the term distribution free is used in the sense that no assumption is made about the form of the population density function.

35 ASSUMPTIONS FOR NP TECHNIQUES (i) The sample at hand is a random sample drawn from population whose median is unknown. a (ii) All the observations in the sample are independent. (iii) The variable of interest is continuous. (iv) The sampled population is symmetric. (v) The observations are measured at least on ordinal scale. (vi) Lower order moments exist.

36 USES OF NP TECHNIQUES (i) When the hypothesis does not involve a parameter of the probability function of the population. (ii)when the sample sizes so small that the sampling distribution of the statistics do not approximate the normal distribution and when no assumption can be made about the set of the population distribution from which the sample is drawn. (iii) The observations are not as accurate as required for a parametric inference. Also when the measurements are on the nominal or ordinal scale.

37 (iv) One wants to avoid complicated analysis of data. (v) One is interested in quick results.

38 ADVANTAGES AND DISADVANTAGES ADVANTAGES NP methods are very simple and easy to calculate. No assumption is made regarding the form of the population. Parametric techniques cannot apply to the data which are measure the nominal scale. Since the Socio-economic data are not normally distributed. So, NP test are used in Sociology, educational statistics and psychometry. NP test can be apply to the data which are given in ranks. DISADVANTAGES All NP methods are not as simple as they are claimed to be. NP methods can be used only if the measurements are in nominal and ordinal scale. No NP methods exist for testing interactions in Analysis of Variance. NP test are designed to test statistical hypothesis only and not for estimating the parameters.

39 BASIC STEPS INVOLVED IN NP TEST OF HYPOTHESIS Various basic steps involved in a test are: (i) First of all look for the assumptions necessary for the validity of a test procedure. (ii) The sample data required should be collected. (iii) The null and alternative hypotheses should be established. (iv) The test statistic should be decided. (v) The decision criteria should be fixed to decide about the rejection or acceptance of null and alternative hypotheses. (vi) The interpretation to the conclusions drawn should be given.

40 THANK YOU

Author : Dr. Pushpinder Kaur. Educational Statistics: Mean Median and Mode

Author : Dr. Pushpinder Kaur. Educational Statistics: Mean Median and Mode B.ED. PART- II ACADEMIC SESSION : 2017-2018 PAPER XVIII Assessment for Learning Lesson No. 8 Author : Dr. Pushpinder Kaur Educational Statistics: Mean Median and Mode MEAN : The mean is the average value