Measures of Central Tendency

Concept of Central Tendency

• A measure of central tendency is a typical value around which other figures congregate

- Simpson & KalfaOR

An average is a single value which is used to represent all of the values in the series.

Measures of Central Tendency

Mean (mathematical average)

Median (positional average)

Mode (positional average)

Arithmetic Mean Geometric mean Harmonic mean

Simple Arithmetic Mean

Weighted Arithmetic Mean

Mean of Composite Group

Basics

• Mean Average

• Median Mid positional value

• Mode Most frequently occurring value

5

Arithmetic Mean Ungrouped (Raw) Data

sObservation ofNumber

sObservation of Sumx

n

xi

EXAMPLETable 4.1 : Equity Holdings of 20 Indian Billionaires

( Rs. in Millions)2717 2796 3098 3144 3527

3534 3862 4186 4310 4506

4745 4784 4923 5034 5071

5424 5561 6505 6707 6874

Example

For the above data, the A.M. is  2717 + 2796 +…… 4645+….. + 5424 + ….+ 6874 = --------------------------------------------------------------------------

20 

= Rs. 4565.4 Millions

x

Arithmetic Mean Grouped Data

• N= = Total frequency

• Here, xi is the mid value of the class interval.

i

i

ff ixx

n

iif

1

example

• Calculate arithmetic mean from the following frequency distribution of marks at a test in statistics.

MarksMarks No.of No.of studentsstudents

2525 22

3030 33

3535 44

4040 88

4545 99

5050 44

5555 33

6060 22

• The details of the monthly salary of 100 employees of a firm are given below:

Monthly salary (in Rs.) No. of employees1000 181500 26

2000 312500 163000 55000 4

• In grouped data, the middle value of each group is the representative of the group bz when the data are grouped, the exact frequency with which each value of the variable occurs in the distribution is unknown.

• We only know the limits within which a certain number of frequencies occur.

• So, we make an assumption that the frequencies within each class are distributed uniformly over the range of class interval.

Example

• A company manufactures polythene bags. The bags are evaluated on the basis of their strength by buyers. The strength depends on their bursting pressures. The following data relates to the bursting pressure recorded in a sample of 90 bags. Find the average bursting pressure.

example

Bursting pressure

( 1 )

No. of bags

( fi ) ( 2 )

Mid Value of Class Interval

( xi ) ( 3 )

Fixi ( 4 )Col.(4) = Col.(2) x Col.

(3)

5-10 10 7.5 75

10-15 15 12.5 187.5

15-20 20 17.5 350

20-25 25 22.5 562.5

25-30 20 27.5 550

    

Sum fi =90   fixi =1725

values of fi and fixi , in formula

= 1725/90

= 19.17

i

i

ff ixx

EXAMPLE (short cut method)

• Calculate the mean of the following distribution of monthly wages of workers in a factory :

Monthly Monthly wages(in wages(in Rs.)Rs.)

No. of No. of workersworkers

100-120100-120 1010

120-140120-140 2020

140-160140-160 3030

160-180160-180 1515

180-200180-200 55

• The following frequency distribution represents the time taken in seconds to serve customers at a fast food take away. Calculate the mean time taken by to serve customers

Time taken (in seconds)

frequencies

40-60 660-80 1280-100 15100-120 12120-140 10140-160 5

Weighted Arithmetic Mean

• It takes into account the importance of each value to the overall data with the help of the weights.

• Frequency i.e. the no. of occurrence indicates the relative importance of a particular data in a group of observations.

• Used in case the relative importance of each observation differs or when rates, percentages or ratios are being averaged.

• The weighted AM of the n observations:

• AM is considered to be the best measure of central tendency as its computation is based on each and every observation.

wi

wi ixx

Example

• 5 students of a B.Sc. (Hons) course are marked by using the following weighing scheme :– Mid-term = 20%– Project = 10%– Attendance = 10%– Final Exam = 60%Calculate the average marks in the

examination.Marks of the students in various

components are:

StudStudent ent

midtmidtermerm

ProjeProjectct

AttenAttendncednce

FinFinalal

11 6565 7070 8080 8080

22 4848 5858 5454 6060

33 5858 6363 6565 5050

44 5858 7070 5454 6060

55 6060 6565 7070 7070

• A professor is interested in ranking the following five students in the order of merit on the basis of data given below:

• Attendance average will count for 20% of a student’s grade; the homework 25%; assignment 35%; midterm examination 10% and final examination 10%. What would be the students ranking.

Student

Attendance

Homework

Assignment

Midterm

final

A 85 89 94 87 90B 78 84 88 91 92C 94 88 93 86 89D 82 79 88 84 93E 95 90 92 82 88

Mean of composite group

• If two groups contain respectively, n1 and n2

observations with mean X1 and X2, then the combined mean (X) of the combined group of n1+n2 observations is given by :

21

221112

nn

XnXnX

Example

• There are two branches of a company employing 100 and 80 employees respectively. If arithmetic means of the monthly salaries paid by two branches are Rs. 4570 and Rs. 6750 respectively, find the A.M. of the salaries of the employees of the company as a whole.

• A factory has 3 shifts :- Morning, evening and night shift. The morning shift has 200 workers, the evening shift has 150 workers and night shift has 100 workers. The mean wage of the morning shift workers is Rs. 200, the evening shift workers is Rs. 180 and the overall mean of the workers is Rs. 160. Find the mean wage of the night shift workers.

Properties of A.M.

• If a constant amount is added or subtracted from each value in the series, mean is also added or subtracted by the same constant amount. E.g. Consider the values 3,5,9,15,16

A.M. = 9.6If 2 is added to each value, then A.M. = 11.6 = 9.6 + 2.Thus, mean is also added by 2.

• Sum of the deviations of a set of observations say x1, x2, , xn from their mean is equal to zero.

A.M. is dependent on both change in origin and scale. The sum of the squares of the deviations of a set of

observation from any number say A is least when A is X.

Merits and demerits of Arithmetic Mean

Advantages Disadvantages

(i) Easy to understand and calculate(ii) Makes use of full data(iii) Based upon all the

observations.

 (i ) Unduly influenced by extreme values (ii) Cannot be calculated from the data with open-end class.e.g. below 10 or above 90

(iii) It cannot be obtained if a single observation is missing.

(iv) It cannot be used if we are dealing with qualitative characteristics which cannot be measured quantitatively; intelligence, honesty, beauty etc

Harmonic Mean The harmonic mean (H.M.) is defined as the reciprocal

of the arithmetic mean of the reciprocals of the observations. 

For example, if x1 and x2 are two observations, then the arithmetic means of their reciprocals viz 1/x1 and 1/ x2 is

  {(1 / x1) + (1 / x2)} / 2= (x2 + x1) / 2 x1 x2

The reciprocal of this arithmetic mean is 2 x1 x2 / (x2 + x1). This is called the harmonic mean.

 Thus the harmonic mean of two observations x1 and x2 is 2 x1 x2

-----------------

x1 + x2

• In general, for the set of n observations X1,X2……..Xn, HM is given by :

• And for the same set of observations with frequencies f1,f2……..fn, HM is given by:

ix

nHM

1

i

i

xfn

HM

• HM gives the largest weight to the smallest item and the smallest weight of the largest item

• If each observation is divided by a constant, K then HM is also divided by the same constant.

• If each observation is multiplied by a constant, K then HM is also multiplied by the same constant.

• It is used in averaging speed, price of articles.

• If time varies w.r.t. a fixed distance then HM determines the average speed.

• If distance varies w.r.t. a fixed time then AM determines the average speed.

• EXAMPLE : If a man moves along the sides of a square with speed v1, v2, v3, v4 km/hr, the average speed for the whole journey = 4

(1/v1)+(1/v2)+(1/v3)+(1/v4)

EXAMPLE

• In a certain factory a unit of work is completed by A in 4 min, by B in 5 min, by C in 6 min, by D in 10 min, and by E in 12 minutes.

– What is the average no. of units of work completed per minute?

Example

• The profit earned by 19 companies is given below:

calculate the HM of profit earned.

Profit Profit (lakhs)(lakhs)

No. of No. of companiescompanies

20-2520-25 44

25-3025-30 77

30-3530-35 44

35-4035-40 44

Geometric Mean

Neither mean, median or mode is the appropriate average in calculating the average % rate of change over time. For this G.M. is used.The Geometric Mean ( G. M.) of a series of observations with x1, x2,

x3, ……..,xn is defined as the nth root of the product of these

values . Mathematically G.M. = { ( x1 )( x2 )( x3 )…………….(xn ) }

(1/ n )

It may be noted that the G.M. cannot be defined if any value of x is zero as the whole product of various values becomes zero.

• When the no. of observation is three or more then to simplify the calculations logarithms are used.log G.M. = log X1 + log X2 + ……+ log Xn

N

G.M. = antilog (log X1 + log X2 + ……+ log Xn) N

For grouped data, G.M. = antilog (f1log X1 + f2log X2 + ……+ fnlog Xn)

N

Geometric mean

• GM is often used to calculate the rate of change of population growth.

• GM is also useful in averaging ratios, rates and percentages.

EXAMPLE

• A machinery is assumed to depreciate 44% in value in first year, 15% in second year and 10% in next three years, each percentage being calculated on diminishing value. What is the average % of depreciation for the entire period?

• Compared to the previous year the overhead expenses went up by 32% in 2002; they increased by 40% in the next year and by 50% in the following year. Calculate the average rate of increase in the overhead expenses over the three years.

Example• The annual rate of growth for a factory for 5 years is

7%,8%,4%,6%,10%respectively.What is the average rate of growth per annum for this period.

• The price of the commodity increased by 8% from 1993 to 1994,12%from 1994 to 1995 and 76% from 1995 to 1996.the average price increase from 1993 to 1996 is quoted as 28.64% and not 32%.Explain and verify the result.

37

Combined G.M. of Two Sets of Data

 If G1 & G2 are the Geometric means of two sets

of observations of sizes n1 and n2, then the combined Geometric mean, say G, of the combined series is given by :

n1 log G1 + n2 log G2

log G = ------------------------------- n1 + n2

Example

• The GM of two series of sizes 10 and 12 are 12.5 and 10 respectively. Find the combined GM of the 22 observations.

Combined G.M. of Two Sets of Data

10 log 12.5 + 12log 10 log G = ------------------------------- 10 + 12

22.9691= ------------ = 1.04405

22Therefore,  

G = antilog 1.04405 = xThus the combined average rate of growth for the period of 22 years is x%.

Relationship Among A.M. G.M. and H.M. The relationships among the magnitudes of the three types of Means calculated from the same data are as follows: (i) H.M. ≤ G.M. ≤ A.M.  i.e. the arithmetic mean is greater than or equal to the geometric which is greater than or equal to the harmonic mean. ( ii ) G.M. = i.e. geometric mean is the square root of the product of arithmetic mean and harmonic mean.

( iii) H.M. = ( G.M.) 2 / A .M.

... MHMA *