Page 1 Measures of Central Tendency 1. INTRODUCTION In the previous chapter, you have read about the tabular and graphic representation of the data. In this chapter, you will study the measures of central tendency which is a numerical method to explain the data in brief. You can see examples of summarising a large set of data in daytoday life, like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm, etc. Baiju is a farmer. He grows food grains in his land in a village called Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are interested in knowing the economic condition of small farmers of Balapur. You want to compare the economic CHAPTER Studying this chapter should enable you to: • understand the need for summarising a set of data by one single number; • recognise and distinguish between the different types of averages; • learn to compute different types of averages; • draw meaningful conclusions from a set of data; • develop an understanding of which type of average would be the most useful in a particular situation. 202021 Page 2 Measures of Central Tendency 1. INTRODUCTION In the previous chapter, you have read about the tabular and graphic representation of the data. In this chapter, you will study the measures of central tendency which is a numerical method to explain the data in brief. You can see examples of summarising a large set of data in daytoday life, like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm, etc. Baiju is a farmer. He grows food grains in his land in a village called Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are interested in knowing the economic condition of small farmers of Balapur. You want to compare the economic CHAPTER Studying this chapter should enable you to: • understand the need for summarising a set of data by one single number; • recognise and distinguish between the different types of averages; • learn to compute different types of averages; • draw meaningful conclusions from a set of data; • develop an understanding of which type of average would be the most useful in a particular situation. 202021 MEASURES OF CENTRAL TENDENCY 59 condition of Baiju in Balapur village. For this, you may have to evaluate the size of his land holding, by comparing with the size of land holdings of other farmers of Balapur. You may like to see if the land owned by Baiju is – 1. above average in ordinary sense (see the Arithmetic Mean) 2. above the size of what half the farmers own (see the Median) 3. above what most of the farmers own (see the Mode) In order to evaluate Baiju’s relative economic condition, you will have to summarise the whole set of data of land holdings of the farmers of Balapur. This can be done by the use of central tendency, which summarises the data in a single value in such a way that this single value can represent the entire data. The measuring of central tendency is a way of summarising the data in the form of a typical or representative value. There are several statistical measures of central tendency or “averages”. The three most commonly used averages are: • Arithmetic Mean • Median • Mode You should note that there are two more types of averages i.e. Geometric Mean and Harmonic Mean, which are suitable in certain situations. However, the present discussion will be limited to the three types of averages mentioned above. 2. ARITHMETIC MEAN Suppose the monthly income (in Rs) of six families is given as: 1600, 1500, 1400, 1525, 1625, 1630. The mean family income is obtained by adding up the incomes and dividing by the number of families. = = Rs 1,547 It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X . In general, if there are N observations as X 1 , X 2 , X 3 , ..., X N , then the Arithmetic Mean is given by X + X + X +...+ X N 1 2 3 X= N The right hand side can be written as 1 X N N i i ? = . Here, i is an index which takes successive values 1, 2, 3,...N. For convenience, this will be written in simpler form without the index i. Thus X X = N ? , where, SX = sum of all observations and N = total number of observations. 202021 Page 3 Measures of Central Tendency 1. INTRODUCTION In the previous chapter, you have read about the tabular and graphic representation of the data. In this chapter, you will study the measures of central tendency which is a numerical method to explain the data in brief. You can see examples of summarising a large set of data in daytoday life, like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm, etc. Baiju is a farmer. He grows food grains in his land in a village called Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are interested in knowing the economic condition of small farmers of Balapur. You want to compare the economic CHAPTER Studying this chapter should enable you to: • understand the need for summarising a set of data by one single number; • recognise and distinguish between the different types of averages; • learn to compute different types of averages; • draw meaningful conclusions from a set of data; • develop an understanding of which type of average would be the most useful in a particular situation. 202021 MEASURES OF CENTRAL TENDENCY 59 condition of Baiju in Balapur village. For this, you may have to evaluate the size of his land holding, by comparing with the size of land holdings of other farmers of Balapur. You may like to see if the land owned by Baiju is – 1. above average in ordinary sense (see the Arithmetic Mean) 2. above the size of what half the farmers own (see the Median) 3. above what most of the farmers own (see the Mode) In order to evaluate Baiju’s relative economic condition, you will have to summarise the whole set of data of land holdings of the farmers of Balapur. This can be done by the use of central tendency, which summarises the data in a single value in such a way that this single value can represent the entire data. The measuring of central tendency is a way of summarising the data in the form of a typical or representative value. There are several statistical measures of central tendency or “averages”. The three most commonly used averages are: • Arithmetic Mean • Median • Mode You should note that there are two more types of averages i.e. Geometric Mean and Harmonic Mean, which are suitable in certain situations. However, the present discussion will be limited to the three types of averages mentioned above. 2. ARITHMETIC MEAN Suppose the monthly income (in Rs) of six families is given as: 1600, 1500, 1400, 1525, 1625, 1630. The mean family income is obtained by adding up the incomes and dividing by the number of families. = = Rs 1,547 It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X . In general, if there are N observations as X 1 , X 2 , X 3 , ..., X N , then the Arithmetic Mean is given by X + X + X +...+ X N 1 2 3 X= N The right hand side can be written as 1 X N N i i ? = . Here, i is an index which takes successive values 1, 2, 3,...N. For convenience, this will be written in simpler form without the index i. Thus X X = N ? , where, SX = sum of all observations and N = total number of observations. 202021 60 STATISTICS FOR ECONOMICS How Arithmetic Mean is Calculated The calculation of arithmetic mean can be studied under two broad categories: 1. Arithmetic Mean for Ungrouped Data. 2. Arithmetic Mean for Grouped Data. Arithmetic Mean for Series of Ungrouped Data Direct Method Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations. Example 1 Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58. X X N = S = + + + + = 40 50 55 78 58 5 56 2 . The average mark of students in the economics test is 56.2. Assumed Mean Method If the number of observations in the data is more and/or figures are large, it is difficult to compute arithmetic mean by direct method. The computation can be made easier by using assumed mean method. In order to save time in calculating mean from a data set containing a large number of observations as well as large numerical figures, you can use assumed mean method. Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observation. You can, then, take the summation of these deviations and divide it by the number of observations in the data. The actual arithmetic mean is estimated by taking the sum of the assumed mean and the ratio of sum of deviations to number of observations. Symbolically, (HEIGHT IN INCHES) 202021 Page 4 Measures of Central Tendency 1. INTRODUCTION In the previous chapter, you have read about the tabular and graphic representation of the data. In this chapter, you will study the measures of central tendency which is a numerical method to explain the data in brief. You can see examples of summarising a large set of data in daytoday life, like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm, etc. Baiju is a farmer. He grows food grains in his land in a village called Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are interested in knowing the economic condition of small farmers of Balapur. You want to compare the economic CHAPTER Studying this chapter should enable you to: • understand the need for summarising a set of data by one single number; • recognise and distinguish between the different types of averages; • learn to compute different types of averages; • draw meaningful conclusions from a set of data; • develop an understanding of which type of average would be the most useful in a particular situation. 202021 MEASURES OF CENTRAL TENDENCY 59 condition of Baiju in Balapur village. For this, you may have to evaluate the size of his land holding, by comparing with the size of land holdings of other farmers of Balapur. You may like to see if the land owned by Baiju is – 1. above average in ordinary sense (see the Arithmetic Mean) 2. above the size of what half the farmers own (see the Median) 3. above what most of the farmers own (see the Mode) In order to evaluate Baiju’s relative economic condition, you will have to summarise the whole set of data of land holdings of the farmers of Balapur. This can be done by the use of central tendency, which summarises the data in a single value in such a way that this single value can represent the entire data. The measuring of central tendency is a way of summarising the data in the form of a typical or representative value. There are several statistical measures of central tendency or “averages”. The three most commonly used averages are: • Arithmetic Mean • Median • Mode You should note that there are two more types of averages i.e. Geometric Mean and Harmonic Mean, which are suitable in certain situations. However, the present discussion will be limited to the three types of averages mentioned above. 2. ARITHMETIC MEAN Suppose the monthly income (in Rs) of six families is given as: 1600, 1500, 1400, 1525, 1625, 1630. The mean family income is obtained by adding up the incomes and dividing by the number of families. = = Rs 1,547 It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X . In general, if there are N observations as X 1 , X 2 , X 3 , ..., X N , then the Arithmetic Mean is given by X + X + X +...+ X N 1 2 3 X= N The right hand side can be written as 1 X N N i i ? = . Here, i is an index which takes successive values 1, 2, 3,...N. For convenience, this will be written in simpler form without the index i. Thus X X = N ? , where, SX = sum of all observations and N = total number of observations. 202021 60 STATISTICS FOR ECONOMICS How Arithmetic Mean is Calculated The calculation of arithmetic mean can be studied under two broad categories: 1. Arithmetic Mean for Ungrouped Data. 2. Arithmetic Mean for Grouped Data. Arithmetic Mean for Series of Ungrouped Data Direct Method Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations. Example 1 Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58. X X N = S = + + + + = 40 50 55 78 58 5 56 2 . The average mark of students in the economics test is 56.2. Assumed Mean Method If the number of observations in the data is more and/or figures are large, it is difficult to compute arithmetic mean by direct method. The computation can be made easier by using assumed mean method. In order to save time in calculating mean from a data set containing a large number of observations as well as large numerical figures, you can use assumed mean method. Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observation. You can, then, take the summation of these deviations and divide it by the number of observations in the data. The actual arithmetic mean is estimated by taking the sum of the assumed mean and the ratio of sum of deviations to number of observations. Symbolically, (HEIGHT IN INCHES) 202021 MEASURES OF CENTRAL TENDENCY 61 Let, A = assumed mean X = individual observations N = total numbers of observa tions d = deviation of assumed mean from individual observation, i.e. d = X – A Then sum of all deviations is taken as Sd=S (XA) Then find Then add A and Sd N to get X Therefore, You should remember that any value, whether existing in the data or not, can be taken as assumed mean. However, in order to simplify the calculation, centrally located value in the data can be selected as assumed mean. Example 2 The following data shows the weekly income of 10 families. Family A B C D E F G H I J Weekly Income (in Rs) 850 700 100 750 5000 80 420 2500 400 360 Compute mean family income. TABLE 5.1 Computation of Arithmetic Mean by Assumed Mean Method Families Income d = X – 850 d' (X) = (X – 850)/10 A 850 0 0 B 700 –150 –15 C 100 –750 –75 D 750 –100 –10 E 5000 +4150 +415 F 80 –770 –77 G 420 –430 –43 H 2500 +1650 +165 I 400 –450 –45 J 360 –490 –49 11160 +2660 +266 Arithmetic Mean using assumed mean method X A d N Rs = + = + = S 850 2 660 10 1116 ( , )/ , . Thus, the average weekly income of a family by both methods is Rs 1,116. You can check this by using the direct method. Step Deviation Method The calculations can be further simplified by dividing all the deviations taken from assumed mean by the common factor ‘c’. The objective is to avoid large numerical figures, i.e., if d = X – A is very large, then find d'. This can be done as follows: d' = d c X A c =  . The formula is given below: X A d N c = + ' × S where d' = (X – A)/c, c = common factor, N = number of observations, A= Assumed mean. Thus, you can calculate the arithmetic mean in the example 2, by the step deviation method, X = 850 + (266/10) × 10 = Rs 1,116. 202021 Page 5 Measures of Central Tendency 1. INTRODUCTION In the previous chapter, you have read about the tabular and graphic representation of the data. In this chapter, you will study the measures of central tendency which is a numerical method to explain the data in brief. You can see examples of summarising a large set of data in daytoday life, like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm, etc. Baiju is a farmer. He grows food grains in his land in a village called Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are interested in knowing the economic condition of small farmers of Balapur. You want to compare the economic CHAPTER Studying this chapter should enable you to: • understand the need for summarising a set of data by one single number; • recognise and distinguish between the different types of averages; • learn to compute different types of averages; • draw meaningful conclusions from a set of data; • develop an understanding of which type of average would be the most useful in a particular situation. 202021 MEASURES OF CENTRAL TENDENCY 59 condition of Baiju in Balapur village. For this, you may have to evaluate the size of his land holding, by comparing with the size of land holdings of other farmers of Balapur. You may like to see if the land owned by Baiju is – 1. above average in ordinary sense (see the Arithmetic Mean) 2. above the size of what half the farmers own (see the Median) 3. above what most of the farmers own (see the Mode) In order to evaluate Baiju’s relative economic condition, you will have to summarise the whole set of data of land holdings of the farmers of Balapur. This can be done by the use of central tendency, which summarises the data in a single value in such a way that this single value can represent the entire data. The measuring of central tendency is a way of summarising the data in the form of a typical or representative value. There are several statistical measures of central tendency or “averages”. The three most commonly used averages are: • Arithmetic Mean • Median • Mode You should note that there are two more types of averages i.e. Geometric Mean and Harmonic Mean, which are suitable in certain situations. However, the present discussion will be limited to the three types of averages mentioned above. 2. ARITHMETIC MEAN Suppose the monthly income (in Rs) of six families is given as: 1600, 1500, 1400, 1525, 1625, 1630. The mean family income is obtained by adding up the incomes and dividing by the number of families. = = Rs 1,547 It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X . In general, if there are N observations as X 1 , X 2 , X 3 , ..., X N , then the Arithmetic Mean is given by X + X + X +...+ X N 1 2 3 X= N The right hand side can be written as 1 X N N i i ? = . Here, i is an index which takes successive values 1, 2, 3,...N. For convenience, this will be written in simpler form without the index i. Thus X X = N ? , where, SX = sum of all observations and N = total number of observations. 202021 60 STATISTICS FOR ECONOMICS How Arithmetic Mean is Calculated The calculation of arithmetic mean can be studied under two broad categories: 1. Arithmetic Mean for Ungrouped Data. 2. Arithmetic Mean for Grouped Data. Arithmetic Mean for Series of Ungrouped Data Direct Method Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations. Example 1 Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58. X X N = S = + + + + = 40 50 55 78 58 5 56 2 . The average mark of students in the economics test is 56.2. Assumed Mean Method If the number of observations in the data is more and/or figures are large, it is difficult to compute arithmetic mean by direct method. The computation can be made easier by using assumed mean method. In order to save time in calculating mean from a data set containing a large number of observations as well as large numerical figures, you can use assumed mean method. Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observation. You can, then, take the summation of these deviations and divide it by the number of observations in the data. The actual arithmetic mean is estimated by taking the sum of the assumed mean and the ratio of sum of deviations to number of observations. Symbolically, (HEIGHT IN INCHES) 202021 MEASURES OF CENTRAL TENDENCY 61 Let, A = assumed mean X = individual observations N = total numbers of observa tions d = deviation of assumed mean from individual observation, i.e. d = X – A Then sum of all deviations is taken as Sd=S (XA) Then find Then add A and Sd N to get X Therefore, You should remember that any value, whether existing in the data or not, can be taken as assumed mean. However, in order to simplify the calculation, centrally located value in the data can be selected as assumed mean. Example 2 The following data shows the weekly income of 10 families. Family A B C D E F G H I J Weekly Income (in Rs) 850 700 100 750 5000 80 420 2500 400 360 Compute mean family income. TABLE 5.1 Computation of Arithmetic Mean by Assumed Mean Method Families Income d = X – 850 d' (X) = (X – 850)/10 A 850 0 0 B 700 –150 –15 C 100 –750 –75 D 750 –100 –10 E 5000 +4150 +415 F 80 –770 –77 G 420 –430 –43 H 2500 +1650 +165 I 400 –450 –45 J 360 –490 –49 11160 +2660 +266 Arithmetic Mean using assumed mean method X A d N Rs = + = + = S 850 2 660 10 1116 ( , )/ , . Thus, the average weekly income of a family by both methods is Rs 1,116. You can check this by using the direct method. Step Deviation Method The calculations can be further simplified by dividing all the deviations taken from assumed mean by the common factor ‘c’. The objective is to avoid large numerical figures, i.e., if d = X – A is very large, then find d'. This can be done as follows: d' = d c X A c =  . The formula is given below: X A d N c = + ' × S where d' = (X – A)/c, c = common factor, N = number of observations, A= Assumed mean. Thus, you can calculate the arithmetic mean in the example 2, by the step deviation method, X = 850 + (266/10) × 10 = Rs 1,116. 202021 62 STATISTICS FOR ECONOMICS Calculation of arithmetic mean for Grouped data Discrete Series Direct Method In case of discrete series, frequency against each observation is multiplied by the value of the observation. The values, so obtained, are summed up and divided by the total number of frequencies. Symbolically, X fX f = S S Where, S fX = sum of the product of variables and frequencies. S f = sum of frequencies. Example 3 Plots in a housing colony come in only three sizes: 100 sq. metre, 200 sq. meters and 300 sq. metre and the number of plots are respectively 200 50 and 10. TABLE 5.2 Computation of Arithmetic Mean by Direct Method Plot size in No. of d' = X–200 Sq. metre X Plots (f) f X 100 fd' 100 200 20000 –1 –200 200 50 10000 0 0 300 10 3000 +1 10 260 33000 0 –190 Arithmetic mean using direct method, X= X N 33000 260 126.92 Sq.metre ? = = Therefore, the mean plot size in the housing colony is 126.92 Sq. metre. Assumed Mean Method As in case of individual series the calculations can be simplified by using assumed mean method, as described earlier, with a simple modification. Since frequency (f) of each item is given here, we multiply each deviation (d) by the frequency to get fd. Then we get S fd. The next step is to get the total of all frequencies i.e. S f. Then find out S fd/S f. Finally, the arithmetic mean is calculated by X A fd f = + S S using assumed mean method. Step Deviation Method In this case, the deviations are divided by the common factor ‘c’ which simplifies the calculation. Here we estimate d' = d c X A c =  in order to reduce the size of numerical figures for easier calculation. Then get fd' and S fd'. The formula for arithmetic mean using step deviation method is given as, X A fd f c = + ' × S S Activity • Find the mean plot size for the data given in example 3, by using step deviation and assumed mean methods. 202021Read More
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