Chapter : 3
MEASURES OF CENTRAL TENDENCY
The frequency distribution distribution summarizes the given mass of data, but for practical purposes there is usually a need need for furth further er conden condensat sation ion,, partic particula ularly rly when when we want want to compar comparee two or more more differ different ent distri distribut bution ions. s. We may even even reduce educe the entire entire distr distribu ibutio tion n to one number number which which represe epresents nts the distribution. distribution. We calculate ‘Measures ‘Measures of central tendency’ for this purpose. These measures measures summarize the given given mass mass of data data in much much more more concis concisee fashio fashion n than than a frequ frequenc encyy distri distribut bution ion.. Frequ Frequenc encyy distribution has too many details while an average reduces the large number of observations to one figure. The term ‘averages’ is used very often e.g., average Indian, average marks, or average size, etc., Sometimes it means ‘typical or usual’ like average Indian. It may also refer to the result of a specific process of calculation like average marks of students. Aver Average age is used used to reduce educe two or more more aggre aggregat gates es to a common common denomina denominator tor,, in order order to make make comparisons. It can be used to compare the totals for time time periods of different different lengths, e.g., if we have the figures of production production for time periods of different lengths, e.g., if we have the figures of production production for the months of January and February, February, in 1985 the production for the month of January is 4000 units while for the month of February February it is 3640 units. We cannot compare compare the two figures, figures, 4000 and 3640 units. units. The reason is, January has 31 days while February February has 28 days. Here we find the average daily production production 4000 129.03 by dividing the total total by the number of days. The average daily production production of January January is 31 3640 130 units i.e., There is no significant units while the average daily production of February is 28 difference difference between the production rate for the two months. Though the total production production in February is less, the daily production rate is almost the same. The number of deaths due to traffic accidents in two different periods should not be compared directly. The number should be compared with the total population and deaths per thousand should be calculated. The number of accidents is affected by the number of vehicles on the road and therefore we can also compare compare the number of accidents per 100 vehicles. Aver Average agess are are also also used used as a measur measuree of typical typical size. size. It gives gives one figure figure that that is typical typical of all all the observati observations ons that are essentially essentially different. different. If the items items are are scatter scattered, ed, the measur measuree will not be very satisfactory while while for homogeneous data the average will be a good representative representative of the data. But it is necessary to have this kind of summary statement for many statistical data. There are five averages which are conceptually different and each of them is from some point of view of a ‘central’ value of the distribution. The averages are also referred referred to as ‘measures ‘measures of central tendency’ because they are used to describe a magnitude near the centre of a distribution about which the values cluster. If we have the distribution of marks of students, very few students will get marks like 4, 5, 8, …. and similarly there there will be a small number of students getting above 80. Most of the students will have marks between 40 and 60 and the average will be somewhere within these limits, that is, average is a central figure. Averages are also known as ‘measures of location’. Each of these averages has its own advantages and disadvantages. disadvantages. But there are are certain characteristics, which make the average a good representative representative of the given data. DESIDERATA FOR SATISFACTORY AVERAGE
1. An average average shoul should d be rigidl rigidlyy defined; defined; otherw otherwise ise its value value will be affecte affected d by the bias of the person who calculates it. It cannot be a good representative representative if it is not a fixed value.
1
2. It shoul should d be base based d on all all the the obse observ rvat atio ions ns.. observations are left out of calculation.
It sill sill not be a good good repr repres esen enta tati tive ve if some some
3. It should should be easy easy to calculat calculatee and easy to unders understan tand. d. If the calcul calculati ations ons requ requir ires es tediou tediouss mathematical process, process, it will not be understood by many and it use will be limited. 4. It should should be be capable of further further algebraic algebraic treatment. treatment. This makes the the average more useful. 5. It should should not be affect affected ed much by sampl sampling ing fluctua fluctuation tions. s. If two indep independen endentt samples samples are are taken from the same population, the average should not differ significantly. It should also be remembered that the average should be expressed in the same unit as the series given. i.e., If we have the heights of 50 children in cms and the average is 130, it should be written as 130 cms. If the income of 100 families are given in (’00 Rs.) and the average is 30, the average should be expressed as 30(100 30(100 Rs.) or Rs.3000/-. Now we consider the types of averages. ARITHMETIC MEAN:
Arithmetic mean is defined as the sum of all the observations in the distribution divided by the number of ....., xn , its arithmetic mean is defined as observations, i.e., i.e., if a variable variable x takes the values values x1 , x2 , x3 , .. n
x
x1
x2
x3
xn
xi xi i 1
n
i.e.,
n
x
n
x
i
n
....., xn with corresponding frequencies f1 , f 2 , f3 , .. ......., f n , then If a variable x tames values x1 , x2 , x3 , .. their arithmetic mean is defined as n
x
f1 x1 f 2 x2 f 3 x3 f n xn f f2 3f nf 1
f x f x f f i
i 1 n
i
i
i 1
i
i
i
Short-cut Method :
x A
f d f i
i
c
i
Where A is the assumed mean chosen from xi values,
d i
xi
A c
and c is the length of the class-
interval. Exercise :
1.
Find Find the the arith arithmet metic ic mean mean for for the the follo followin wing g sets sets of obse observa rvatio tions: ns: a) 125, 132, 132, 127, 127, 139, 139, 140, 140, 142, 142, 137, 137, 122, 122, 120 120 and 130 b) 13.1, 15.2, 11.9, 10.2, 12.5, 14.3, 11.2, 10.8 10.8 c) 1357, 1357, 1454, 1454, 1389, 1389, 1405, 1405, 1485 1485 d) 53, 31, 31, 35, 35, -25, -25, 100, 100, 60, -16, -16, 13, 13, -3, -3, 95 [ Answers: a) 131.4
b) 12.4
c) 1418
d) 34.3 ]
2
2.
Calc Calcul ulat atee the the me mean an for for the the foll follow owin ing g dat data: a: xi: f i:
12 5
14 10
16 15
18 12
20 8
22 3
[ Answer: 16.6415 ] 3.
Calc Calcul ulat atee the the mean mean for for the the fol follo lowi wing ng dis distr trib ibut utio ion: n: Size of Shoe: No. of pairs:
6
7
32
8
40
52
9
10 32
11 25
9 12
10 9
11 4
40
[ Answer: 8.3394 ~ 8 ] 4.
Calc Calcul ulat atee the the me mean an for for the the foll follow owin ing g dat data: a: xi: f i:
5 11
6 15
7 20
8 16
[ Answer: 7.5287 ] 5.
The following data represents frequency distribution distribution of weights of children, find its arithmetic mean .
Wt. in Kgs. No. of Children
11
12 11
7
13 15
14 13
15
16
9
4
[ 13.305 kgs ] 6.
The following following data data represent representss distribution distribution of marks (out of 10) for a class of students. arithmetic mean.
Marks: No. of Students:
0 2
1 4
2
3
5
7
4 11
5 15
6 13
7 10
8 7
9 3
Find the
10 1
[ Answer: 5.06 ] 7.
Find Find the the arithm arithmeti eticc mean mean for the follow following ing distri distribut bution ion:: Marks: No. of students
8.
0-10 10-20 6 11 [ Answer: 22.91 ]
20-30 15
30-40 8
40-50 3
Calculat Calculatee the the arithm arithmetic etic mean for the followin following g data data givin giving g daily daily wages wages of of worker workers. s. Wages in Rs.: No. of workers:
20-40 7
40-60 12
60-80 16
80-100 13
100-120 13
120-140 4
44-55 5
Ans: 31.54
[ Answer: 77.69 ] 9.
Find Find the the mea mean n for for the the fol follo lowi wing ng data data:: Age in years Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60
No. of persons 15 33 54 80 97 100
[ Answer: 27.1 ] 10.
Find Find the the arith arithmet metic ic mean mean for the follow following ing:: Daily wages: No. of persons:
5-15 3
15-25 8
25-35 13
35-45 10
3
11.
Calcu Calculat latee the the arithm arithmeti eticc mean mean for the follow following ing data data represe epresenti nting ng monthl monthlyy salary salary of a group group of employees. Salary in Rs.: No. of persons:
700-800 32
800-900 43
900-1100 55
110-1500 22
1500-1600 18
[ Answer: Rs.1012.06 ] 12. 12.
Find Find the the mea mean n for for the the foll follow owin ing g data data.. Class Interval: Frequency:
20-30 9
30-50 14
50-70 20
70-90 12
90-100 5
5-10 10-15 2 8 [ Answer: Rs.27.97 ]
15-25 12
25-35 15
35-45 11
[ Answer: 57 ] 13. 13.
Calc Calcul ulat atee the the arit arithm hmet etic ic me mean an:: Sales in ‘000 Rs.: No. of shops:
14.
The follo following wing data repr represent esentss yield yield per acre acre (in kgs.) kgs.) for for a number number of of farms. farms. Find the the arithm arithmetic etic mean. Yield pe per ac acre:
700-750
No. of farms: 15.
750-800
800-850
32 43 55 [ Answer: 825.8 kgs. ]
850-900
900-950
22
17
9501000 18
The followin following g is the the distri distributi bution on of heights heights in cms of of 50 stude students. nts. Find the mean. mean. Height in cms: No. of students:
16. 16.
45-50 5
140-145 145-150 7 10 [ Answer: 152.4 cms ]
150-155 15
155-160 13
160-165 5
Find Find the the ari arith thme meti ticc mea mean: n: 148152 3
Height in cms: No. of persons:
152156 5
156160 9
160164 15
164168 10
168172 6
172176 2
[ Answer: 162 cms ] 17.
The follow following ing data data repres represents ents the the distribut distribution ion of balance balance amount amountss in bank account accountss at the end of March 2002. 2002. Find the average balance balance amount. Amount in Rs.: No. of accounts:
500599 25
600699 42
700799 55
800899 70
900999 62
10001099 50
11001199 35
12001299 11
[ Answer: Rs.877.21 ] 18.
Find Find the the aver average age tax for the follow following ing data. data. Tax in Rs.: No. of Employees:
100399 12
400699 20
700999 25
1000-1299 35
1300-1599 15
1600-1899 8
[ Answer: Rs.966.89 ] 19.
Find Find the the arith arithmet metic ic mean mean for for the the foll followi owing ng data. data. No. of units produced: No. of factories:
50-99 4
100-149 9
150-199 11
200-249 15
250-299 12
300-349 8
350-399 2
4
[ Answer: 218.76 ] 20.
The follo following wing data data repr represent esentss salary salary of emplo employees yees in an office. office. Find the average average salary salary.. Salary in Rs.: No. of Employees. 900 – 1000 4 1000 – 1200 11 1200 – 1400 19 1400 – 1600 22 1600 – 1800 18 1800 – 1900 9 1900 – 2000 3 [ Answer: Rs.1473.26 ]
21.
If the the mean for the the followin following g data is Rs.56/ Rs.56/-, -, find the missin missing g frequen frequency cy.. Wages in Rs.: No. of persons:
30-40 10
40-50 20
50-60 40
60-70 …
70-80 8
80-90 6
[ Answer: 16 ] 22.
If the averag averagee marks of stude students nts are are 26.75, 26.75, find the the number number of student studentss belong belonging ing to the class class interval 10 – 20. Marks: No. of students:
23.
0 - 10 3 [ Answer: 7 ]
10-20 …
20-30 15
30-40 10
40-50 5
If the average average wages wages of workers workers are are Rs.73. Rs.73.25, 25, find find the number number of workers workers with with wage wage between between Rs.80 Rs.80 and Rs.100. Wages in Rs.: No. of persons:
20-40 10
40-60 18
60-80 22
80-100 …
100-120 11
120-140 5
[ Answer: 14 ] 24.
If the the mean value for the the followin following g data data is 33, 33, find the missin missing g frequen frequency cy.. Marks: No. of students:
0-10 5
10-20 10
20-30 25
30-40 30
40-50 …
50-60 10
[ Answer: 20 ] 25.
Find the missi missing ng frequen frequencies cies if if the mean mean is 21.9 and and the total total of frequen frequencies cies is is 75. Class Interval: Frequencies:
0-5 2
5-10 5
10-15 7
15-20 …
20-25 …
25-30 16
30-35 8
35-40 3
[ Answer: 13 and 21 ]
COMBINED ARITHMETIC MEAN
If n1 and n2 are the number of observations of two groups with means x1 and x2 then their combined arithmetic mean, denoted by x12 is given by x12
n1 x1 n2 x2 n1 n2
5
This can be extended to three groups also as x 123
n1 x 1
n1
n2 x 2
n2
n3 x 3
n3
PROBLEMS :
1. The average average marks of a group group of 100 student studentss in Accountancy Accountancy are are 60 and for another another group group of 50 students, the average marks are are 90. Find the average marks of the combined group group of 150 students. [ 70 marks ]
2. The average daily wages for 90 workers in a factory is Rs.59/-, the average wages for 50 male worker workerss out out of them is Rs.63/-. Rs.63/-. Find Find the average average wages wages for the remaini emaining ng female female workers workers.. [ Rs.54/- ] 3. The average average marks marks of a class class of studen students ts are are 76. The averag averagee marks of of boys and and girls girls are are 69 and 83 respectively. respectively. If there there are 100 boys in in the class find the number number of girls girls in the the class. [ No. of girls girls = 100 ]
4. The mean daily wages of a group of employees are Rs.180/- The mean daily wages of men and women are are Rs.186/- and Rs.175/Rs.175/- respectively respectively.. Find the the ratio of men and women women in in the group. [ 5 : 6] 5. If the average average marks in a certain certain test test of boys and girls girls in a class are are 80 and 85 respecti respectively vely and if the the average marks for the entire class class are 83.75, find the percentage percentage of boys in the class. [ 25% ] 6. The mean mean weight weight of a group group of 70 70 workers workers is 60 kgs. kgs. The second second group group consis consists ts of 80 worker workerss with average weight 57 kgs and there are 50 workers in the third group group with average weight 62 kgs. Find the average weight weight of the combined combined group group of 200 workers. [ 59.3 kgs ] 7. The mean marks of 100 boys in a class class are are 45. The mean marks of the entire entire class of 150 students are are 50. Find the mean marks of the remaining group of girls. [ 60 ]
8. There are are three groups groups in a class of 100 students. The first contains 25 students students with average pocket money Rs.62/-, the second group consists consists of 50 students with average pocket money Rs.55/-. Find the average pocket money of the students from the third group if the average for the entire class is Rs.58/-. [ Rs.60/Rs.60/- ] 9. The average average monthl monthlyy salary salary of employees employees of of a firm is Rs.520 Rs.5200/-. 0/-. The average average salari salaries es of gents gents and ladies from from the firm are are Rs.6000/- and Rs.4800/Rs.4800/- . Find the percentage percentage of gents and ladies ladies in the firm. [ 1 : 2 ]
10. A garment factory makes both men’s men’s and women’s women’s shirts. The average profit of the factory is 8% of sales. Average profit on men’s men’s shirt is 10%. Women’s omen’s shirts form 60% of the total sales. What is the average profit on sales of women’s women’s shirts? [ 6.67% ] 11. There are men, men, women and children children working in a factory. factory. The total number of workers is 500. The average average daily wages wages of 250 male workers workers is Rs.100/-. Rs.100/-. The average average daily wages wages of 150 women workers is Rs.80/-. What is the average daily wages of children children working working in that factory, factory, given that the average daily daily wages of all all the 500 workers workers taken together together is Rs.82/Rs.82/- [ Rs.40/- ] 12. The sum of the deviations deviations of a certain number of observations observations measured measured from 4 is 72 and and the sum of the deviations deviations of the same observation observationss from 7 is -3. Find the number number of observations observations and their mean. [ 25 & 6.88 ]
13. The mean of a certain number of observations observations is 40. If two more observations observations with values 50 and 64 are added to the data, the the mean rises to 42. 42. Find the the number number of items in the original data. [ 15 ]
6
14. The mean weight of 98 students as calculated from from a frequency distribution distribution is found to be 50 kgs. It is later discovered that the frequency of the class interval 30 – 40 was wrongly taken as 8 instead of 10. Calculate the the correct correct arithmetic arithmetic mean. [ 49.7 kgs ] 15. The mean monthly monthly salaries salaries paid to all 77 employees employees in a company was Rs.78/Rs.78/-.. The mean monthl monthlyy salaries of 32 of them was Rs.75/- and that of the other 25 was Rs.82/-. What was the mean salary of the remaining? remaining? [ Rs.77.80 ] MEDIAN
Definition:
....., xn are n observations arranged either in ascending order or in descending order and if If x1 , x2 , x3 , .. i.
n is ODD, then there will be only one middle term and the value of the middle term is the median.
x i.e., Median = n 1 .
ii.
2
n is EVEN, then there will be two middle terms, the average of the values of the two middle terms is the median. i.e., Median =
x n x n1 2
2
.
2
In the case of a frequency distribution, distribution, it is calculated as Median = l1
l2 l1 n f
2 pcf
l 1 is the lower limit of the C.I. of the median class
Where
l 2 is the upper limit of the C.I. of the median class n
is the total of frequencies f is the frequency at the median class
pcf is the previous cumulative frequency Note:
i. ii.
iii.
The very first cumulative frequency value for which
n 2
becomes less than determines the median
class. To calculat calculatee the median median the class intervals intervals must be continu continuous. ous. If they they are are not continuou continuouss then then we have to make them continuous by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. The value value of median median thus thus calculate calculated d must must be be a value in the the C.I. C.I. of the median median class. class.
PROBLEMS :
1.
Find Find the the med media ian n of of the the foll follow owin ing g set setss of of obs obser erva vati tion ons: s: i. 43, 43, 30, 30, 44, 67, 67, 35, 40, 40, 59 [ 43 ] ii. ii.
2.
16, 16, 19, 19, 27, 27, 10, 10, 5, 7, 12, 12, 15
[ 13.5 13.5 ]
Calc Calcul ulat atee the the me medi dian an of the the fol follo lowi wing ng dist distri ribu buti tion on::
7
xi: f i:
15 7
20 15
25 19
30 23
35 20
40 15
45 8
50 5
[ Answer: 30 ] 3.
Find Find the the med media ian n for for the the foll follow owin ing g data data rep reprresen esenti ting ng hei heigh ghts ts of of 45 stud studen ents ts.. Ht. in cms: No. of students:
4.
166-170 12
170-174 15
174-178 6
178-182 2
Calc Calcul ulat atee the the me medi dian an wage wage for for the the foll follow owin ing g dat data: a: Wages in Rs.: No. of workers:
5.
158-162 162-166 3 7 [ Answer: 170.13 cms ]
Less than 35 24 [ Rs.41.16 ]
35 – 40 62
40 – 45 99
45 – 50 18
Over 50 15
35 – 50 18
50 – 65 32
65 – 80 18
Above 80 12
For For the the fol follow lowing ing dat data a, fi find the the med media ian n ag age. Age in years: No. of persons:
Below 35 20 [ 55.625 years ]
6.
Find Find the the med media ian n and and the the two two qua quart rtil iles es for for the the foll follow owin ing g dat data. a. Rainfall in cms: No. of Years:
20-25 2
25-30 5
30-35 8
35-40 12
40-45 10
45-50 7
50-55 6
55-59 6
60-64 2
240260 23
260300 24
[ Median = 39.17, Q1 = 33.44 and Q3 = 45.36 ] 7.
For the follow following ing distri distribut bution ion of weight weightss of of 60 60 stud student ents, s, find find the the three three quarti quartiles les.. Weights in Kgs: No. of students:
30-34 3
35-39 5
40-44 12
45-49 18
50-54 14
[ Median = 47.28, Q1 = 42.42 and Q3 = 52 ] 8.
For the follow following ing distri distribut bution ion of weight weightss of of 60 60 stud student ents, s, find find the the three three quarti quartiles les.. 100140180200220140 180 200 220 240 No. of Salesmen: 14 45 52 80 32 [ Median = 206, Q1 = 183.27 and Q3 = 227.19 ] Commission in Rs.
9.
The The medi median an mar marks ks of of 100 100 stud studen ents ts in in Acc Accou ount ntan ancy cy ar are 56. 56. It was was lat later er fou found nd tha thatt mark markss of one one student student were wrongl wronglyy consider considered ed as 76 instead of 67. What would would be the correct correct median? median? [ Median is unaltered ]
10.
In a batc batch h of 25 25 studen students, ts, 10 10 stude students nts failed failed in a test test,, by obta obtaini ining ng less less than than 35 mark marks. s. Those Those who who passed the test got 40, 45, 57, 60, 49, 52, 75, 72, 80, 87, 55, 58, 65, and 60 marks. What was the median of the marks of all the 25 students? students? [ 45 ]
11.
In a group group of 25 chi childr ldren en the the median median heig height ht is 164 164 cms cms and the the heigh heights ts of the the talle tallest st and and short shortest est boy in the group are 170 cms and 154 cms respectively. respectively. To this group 4 children are are added with the height heightss 152, 152, 150, 174, 174, and 171 cms. cms. Find the median median height height of the new new group group of of 29 children. children. [ 164 cms. ]
12.
If the the median median heig height ht for for the follo followi wing ng dist distrib ributi ution on is 162. 162.5 5 cms, cms, find the the missi missing ng frequ frequenc encyy. Hei Heigh ghtt in cms: cms:
150 150 – 155 155
155 155 – 160 160
160 160 – 165 165
165 165 – 170 170
170 170 – 175 175
175 175 – 180 180
8
No. of students:
3
6
8
…
3
1
[ Answer: 5 ] 13.
If the median marks are 43.25, find the missing frequency: Age in years: 10-19 20-29 30-39 40-49 No. of Persons: 2 5 10 8
50-59 …
60-69 5
70-79 3
[ Answer: 7 ] 14.
If the the median median for for the the follow following ing dis distri tribut bution ion is is Rs.26. Rs.26.25, 25, find find the the missi missing ng frequ frequenc encyy. Wages in Rs.: 12.5 – 17.5 17.5 – 22.5 22.5 – 27.5 27.5 – 32.5 32.5 – 37.5 37.5 – 42.5 42.5 – 47.5 47.5 – 52.5 52.5 – 57.5 Total
15.
No. of persons 2 22 10 … 3 4 6 1 1 63
[ 16 ]
If the the median median mark markss in Hist History ory for for a group group of stude students nts are are 27, 27, find find the the number number of of studen students ts getti getting ng marks between 30 and 40. Marks: No. of Students:
0 – 10 5
10 – 20 5
20 – 30 10
30 – 40 …
40 – 50 3
[ Answer: 11 ] 16.
The follow following ing data data repr represe esents nts the weekly weekly wages wages in Rs. Rs. of a group group of worker workers. s. If the the medi median an is is Rs.114, find the missing frequency. 60-75 3
Weekly wages in Rs. No. of workers:
75-90 3
90-105 6
105-120 5
120-135 …
135-150 6
[ Answer : 7 ] 17.
If the the first first and and the thir third d quart quartile iless for the the follo followin wing g distri distribut bution ion ar are given given to be 23.1 23.125 25 and and 43.5 43.5 respectively, find the missing frequencies. Weekl eeklyy wage wagess in Rs. Rs. No. of workers:
0 – 10 5
10 – 20 …
20 – 30 20
30 – 40 30
40 – 50 …
50 – 60 10
[ Answer: 15 & 25 ] 18.
Find the missing missing frequen frequencies cies given given that that the first first quart quartile ile is 320 320 and and the third third quart quartile ile is 550. Weekly wages in Rs. No. of workers:
100-200 7
200-300
300-400
400-500
500-600
600-700
10
…
20
16
…
[ Answer: 15 & 12 ] 19.
If the the median median of of the foll follow owing ing dist distrib ributi ution on is 146 146 and and the tota totall of the the frequ frequenc encies ies is is 229, 229, find find the missing frequencies. C.I.: Frequency
110-120 12
120-130 …
130-140 34
140-150 65
150-160 46
160-170 …
170-180 18
9
: [ Answer: 30 & 25 ] 20. 20.
An inco incomp mple lete te dist distri ribu buti tion on is give given n bel below ow:: C.I.: Frequency:
10-20 13
20-30 30
30-40 …
40-50 65
50-60 …
60-70 25
70-80 18
You are given given that the median value is 46 and the total number of frequen frequencies cies is 230. Also calculate the mean of the completed data. [ Answer: 34 & 45 and the mean value = 45.83 ]
MODE
Mode is that value of the variable, which characterizes more more items than any other value. It is the value of greatest greatest frequency or more precisely precisely greatest frequency frequency density. Mode cannot be calculated unless the data are are converted converted in the form of a discret discretee or a continuous continuous distribut distribution. ion. In some distributi distributions ons it is difficult to get the exact value of mode as observations may concentrate concentrate around two or more more values. In such cases the distribution distribution s bimodal, trimodal or multimodal. Mode is a measure which should be used with caution, only when the person believes that it has relevance. relevance. The mode can occur at an extreme value, value, in which vase it will be a poor measure or central central tendency. Example: Find the mode of the following data: 21, 44, 31, 21, 57, 36, 21, 44, 45, 21 On observing the given data, data, we see that the value 21 occurs 4 times times which is the maximum. Hence mode = 21. In the case of frequency distribution, Mode is calculated using the following formula: Mode = l1
Where
d 1 d1
d 2
c
l 1 is the lower limit of the C.I. of the modal class l 2 is the upper limit of the C.I. of the modal class. d 1 is the difference between the frequency at the mode class & the previous one d 2 is the difference between the frequency at the mode class & the next one. c is the length or the width width of the C.I.
PROBLEMS :
1.
Calc Calcul ulat atee the the valu valuee of of mod modee fr from the the fol follo lowi wing ng data data:: Income in Rs. No. of persons:
200 – 400 16
400 – 600 34
600 – 800 60
800-1000 37
1000-1200 13
[ Answer: Rs.706.12 ] 2.
The follow following ing data data gives gives the consum consumpti ption on of electr electrici icity ty.. Calcu Calculat latee the value value of mode. mode.
10
No. of Units: No. of consumers:
0 -100 9
100-200 18
200-300 35
300-400 32
400-500 28
500-600 10
70-90 12
90-110 8
110-130 6
25-30 20
30-35 8
35-40 7
[ Answer: 285 units ] 3.
The The foll follow owin ing g are are the the mark markss in a test test.. Find Find the the mode mode.. Marks: No. of students:
10-30 4
30-50 10
50-70 14
[ Answer: 63.3 ] 4.
Calc Calcul ulat atee the the mod modal al wag wages es for for the the fol follo lowi wing ng dist distri ribu buti tion on:: Wages in Rs.: No. of Employees:
10-15 3
15-20 5
20-25 15
[ Answer: Rs. 26.47 ] 5.
If the the mod modee for for the the foll follow owin ing g dist distri ribu buti tion on is is 130, 130, fin find d the the miss missin ing g freq freque uenc ncyy. Class In Interval: Frequency:
60-75 3
75-90 3
90-105 6
105-120 …
120-135 7
135-150 6
[ Answer: 5 ] 6
If the the mod modee of the the fol follo lowi wing ng dat data a is 750 750 and and the the tot total al of of the the freq freque uenc ncie iess is 186 186,, find find the the mis missi sing ng frequencies. Life in in hr hrs.: No. of bulbs:
200-400 10
400-600 …
600-800 50
800-1000 45
1000-1200 30
1200-1400 …
1400-1600 5
[ Answer: 35 & 11 ] 7.
Prov Provee that that the the val value ue of of medi median an lie liess betw betwee een n mean mean and and mod modee usin using g the the foll follow owin ing g data data.. Age (below) No. of persons:
10 11
20 35
30 50
40 79
50 89
60 100
[ Answer: Mean = 28.6, Median = 30, Mode = 34.24 ] 8.
Find Find the the mea mean, n, me medi dian an and and mod modee for for the the fol follo lowi wing ng data data.. Class In Interval: Frequency:
60-75 3
75-90 3
90-105 6
105-120 5
120-135 7
135-150 6
[ Answer: Mean = 111.15, Median = 114, Mode = 130 ] 9.
Find Find the the mea mean, n, me medi dian an and and mod modee for for the the fol follo lowi wing ng data data.. Class Interval: Frequency:
10-30 4
30-50 10
50-70 14
70-90 12
90-110 8
110-130 6
[ Answer: Mean = 70.37, Median = 68.57, Mode = 63.33 ] 10. 10.
If the the me medi dian an and and mode of the the foll follow owin ing g distr distrib ibut utio ion n are 33.5 33.5 and 34 res respe pect ctiv ivel elyy, find the the missing frequencies. Wages in Rs.: No. of Workers:
0-10 4
10-20 16
20-30 …
30-40 …
40-50 …
50-60 6
60-70 4
Total 230
[ Answer: 60, 100 and 40 ] 11.
Give Given n that that the the me mean an of wage wagess is Rs.4 Rs.418 18.7 .75 5 and and the mode mode is Rs.3 Rs.362 62.5 .50, 0, find find the the miss missin ing g frequencies. frequencies. Hence calculate the median median of the completed data. data. Wages in Rs.:
100-200
200-300
300-400
400-500
500-600
600-700
11
No. of Workers:
5
12
…
…
14
11
[ Answer: 22, 16 and Median = 406.25 ] 12. Find Find the missing missing freque frequenci ncies es for the follo followin wing g data, data, given that the the modal modal marks marks are are 53.25 53.25 and median is 52.5. Find the arithmetic arithmetic mean of the completed completed data. Marks: No. of Students:
20-29 10
30-39 18
40-49 25
50-59 …
60-69 15
70-79 12
80-89 …
[ Answer: 40, 10 and 52.8076 ] 13.
Find Find the miss missing ing fr frequenc equencies ies for for the the follow following ing dat data a given given that that the the mode mode of the the distri distribut bution ion is is 44 and the median is 45.8 Age in years: No. of persons:
10-20 10
20-30 10
30-40 …
40-50 50
50-60 29
60-70 15
70-80 …
80-90 10
[ Answer: 36 and 10 ] 14. 14.
Find Find the the missi missing ng freq freque uenc ncie iess if the mode mode of the follo followi wing ng dist distri ribu buti tion on is give given n to be 95 and and arithmetic mean 96. Weekl eeklyy Expe Expend ndit itur ure: e: No. of Families:
50 – 70 …
70 – 90 60
90 – 110 70
110 – 130 130 …
130 130 – 150 150 10
[ Answer: 20 and 40 ] 15. 15.
The The follo followi wing ng data data give givess the dist distri ribu buti tion on of mar marks ks of som somee stude student nts. s. The The arith arithme meti ticc mean mean of marks is 78 and the mode mode is 75. Find the missing missing frequencies. frequencies. Marks: No. of students:
10-30
30-50
50-70
70-90
90-110
110-130
130-150
5
…
25
30
…
10
5
[ Answer: 10 and 15 ] 16.
The median median age of the the foll follow owing ing distr distribu ibutio tion n is 44 years. years. The The modal modal age is 43 43 year years. s. Two of the the frequencies however are missing. Find those frequencies frequencies given the following data. Age in years: No.of persons:
25-30 8
30-35 …
35-40 24
40-45 30
45-50 …
50-55 20
55-60 14
[ Answer: 10 and 26 ] 17.
Find Find the missin missing g frequ frequenc encies ies give given n that that the the median median and and mode mode of the the distr distribu ibutio tion n are are 1504 1504 and 1500 respectively. Life in hours: No. of bulbs:
950-1150 ..
13501550 100
1150-1350 43
15501750 …
17501950 23
1950-2150 13
[ Answer: 20 and 81 ] 18.
The firs firstt and the thir third d quarti quartiles les of of the the follow following ing data data are are give given n to be be 12.5 12.5 marks marks and and 25 mark markss respectively. Find the missing frequencie frequencies. s. Marks: Frequency:
0-5 4
5-10 8
10-15 …
15-20 19
20-25 …
25-30 10
30-35 5
35-40 …
Total 72
[ Answer: 12, 11 and 3 ]
12
19. 19.
Find Find the the missi missing ng freq freque uenc ncie iess given given that that the the mode mode is 4400 4400 hours hours and and arith arithme meti ticc mean is 4100 4100 hours. Life in hours: No. of bulbs:
1000-2000 100
2000-3000 …
3000-4000 200
4000-5000 …
5000-6000 150
6000-7000 50
7000-8000 50
[ Answer: 150 and 300 ] 20. 20.
If the arit arithm hmet etic ic mean mean for the follo followi wing ng freq freque uenc ncyy dist distri ribu buti tion on is 54 year years, s, find find the the miss missin ing g frequency frequency and also calculate its mode and median. Age in years: No. of persons:
0 – 20 4
20 – 40 5
40 – 60 …
60 – 80 11
80 – 100 5
[ Answer: 15, 54.29 years and 54.67 years ]
13
