2. Measures of Central Tendency (Averages of First Order) · • Average discovers uniformity in variability. • The tendency of the variables to accumulate at the center of the

J. K. SHAH CLASSES Measures of Central Tendency

2. Measures of Central Tendency (Averages of First Order)

MEASURES OF CENTRAL TENDENCY

Introduction:

• Central tendency is defined as the tendency of the data to concentrate towards the

central or middle most region of the distribution.

• In other words, Central Tendency indicates average.

• Any average is a representative value of the entire distribution value

• Average discovers uniformity in variability.

• The tendency of the variables to accumulate at the center of the distribution (data) is

known as measures of central tendency.

• Measures are popularly also known as averages.

• It helps in discovering the uniformity in variability.

Average

Mathematical Avg. Positional Avg.

A. M G. M H. M Median Mode

The criteria for Ideal Measures of Central Tendency

1. It should be simple to understand Mean, Median & Mode are easy to compute.

2. It should be based on all the observations.

3. It should be rigidly defined (except Mode).

4. It should not be affected by extreme values (except Median & Mode).

5. It should have sampling stability or it should not be affected by sampling fluctuations.

(A.M, G.M, H.M. not affected).

6. It should be capable of further algebraic treatment.

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ARITHMETIC MEAN

• It is the best measure of central tendency and most commonly used measure

• The only drawback of this measure is that it gets highly affected by presence of

extreme values in the distribution.

• Calculation of AM

1. For Simple series: A.M. = x

xn

=∑

2. 1 2 3 1 2 3

For simple frequency distribution :

, , ,........., be a series, occuring with frequency , , ,....., respectively, thenn nLet x x x x f f f f

1 1 2 2

1 2

......A.M. = ; N = Total Frequency

......

n n

n

fx fx f x f x f xx

N f f f f

+ + += = =

+ + +

∑ ∑∑

3. For Grouped Frequency Distribution:

a) Direct Method

1 1 2 2

1 2

......A.M. =

......

n n

n

fx fx f x f x f xx

N f f f f

+ + += = =

+ + +

∑ ∑∑

Where, x = mid - values or class marks

b) Method of Assumed Mean using Step Deviation (By changing of origin and scale)

if

fdAxMA ⋅

+==∑∑

.

• i

axd

−=

Where,

X = mid-values or original values if it is a discreet series

a = Assumed Mean i.e., a value arbitrarily chosen from mid-values or any other

values

I = class width or any arbitrary value

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Properties

1. If all values of the variable are constant, then AM is constant.

2. x

xn

=∑ ; Thus, x nx=∑ i.e. (no. of observations) x (average).

3. Sum of deviations of values from their arithmetic mean is always zero.

4. When the values of x are equi-distant, then AM = First value + Last value

2

5. If the frequencies of variable increases or decreases by the same proportion, the

value of AM will remain unaltered.

6. Weighted AM of first “n” natural numbers, when the values are equal to their

corresponding weights, will be given by 2 1

3

nx

+=

7. Sum of squares of deviation is minimum when the deviation is taken from AM.

8. AM is dependent on the change of origin and scale.If Y = a ± bx, then,

Y a bx= ±

9. Formula for calculating Combined Mean is given by: 1 1 2 2

1 2

c

n x n xx

n n

+=

+

Where,

1x = mean of the first group

2x = mean of the second group

1n = number of samples in the first group

2n = number of samples in the second group

GEOMETRIC MEAN (GM)

1. thn1 2 3 1 2 3 , , ,........., be a simple series, then G.M. = . . ....... (n root of the product)n nLet x x x x x x x x

2. 1 2 3 1 2 3 , , ,........., be a series, occuring with frequency , , ,....., respectively, thenn nLet x x x x f f f f

31 2N1 2 3G.M. = . . ...... nf ff f

nx x x x

3. ( . )nGM =Product of the observation

4. It is capable of further algebraic treatment.

5. It is not affected by sampling fluctuations.

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6. It is not affected by extreme values.

7. GM cannot be calculated if any variable assumes value 0 or negative value.

8. GM is particularly useful in cases where we have to find out average rates or ratios of

quantities which are changing at a cumulative rate, i.e., the change is related to the

immediate preceding data. For example, average rate of depreciation by WDV

method or average rate of growth of population.

9. GM is extensively used in the construction of index numbers.

10. GM is the most difficult average to calculate and understand because it involves the

knowledge of logarithms.

11. Logarithm of GM of “n” observations is equal to the AM of the logarithm of these “n”

observations.

12. GM is based on all observations

13. If all the observations assumed by a variable constant, say K, then the GM of the

observations is also K

14. GM of the product of two variables is the product of their GM’s i.e., if z = xy, then GM

of z = (GM of x) ⋅ (GM of y)

15. GM of the ratio of two variables is the ratio of GM’s of two variables i.e., if z = x/y then

GM of GM of x

GM of yz =

16. Combined GM: G12 = [ 1n

1G 2 1 2

1

n

2G ]n n+

HARMONIC MEAN (HM)

1. Let 1 2 3

1 2 3

, , ,........., be a simple series, then H.M. = 1 1 1 1

.......n

n

nx x x x

x x x x+ + + +

2. Let 1 2 3 1 2 3 , , ,........., be a series, occuring with frequency , , ,....., respectively, thenn nx x x x f f f f

31 2

1 2 3

H.M. =

....... n

n

N

f ff f

x x x x+ + + +

3. HM cannot be calculated if any variable assumes value 0, as inverse of 0 is

undefined.

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4. HM has a very restricted use, and they are usually used for calculating average

speed, average rates of quantities, etc.

5. It is based on all the values.

6. It is capable of further algebraic treatment.

7. It is not affected by extreme values, sampling fluctuations.

8. If y = ax is affected by the change of scale then

HM(y) = a HM (x)

GM(y) = a GM (x)

9. If all the observations are constant, HM is constant

10. Combined H.M: 12H =

2

2

1

1

21

H

n

H

n

nn

+

+

Rule for using AM and HM

When the average to be calculated is of the form a/b, where a and b are different quantities

then

i. Use HM when ‘a’ is constant

ii. Use AM when ‘b’ is constant

For eg,

Avg. speed = ? Distance = same (given)

Use H. M

Avg. speed = ? Time = same (given)

Use A. M

Relation between AM, GM & HM

1. If the values are equal,

AM = GM = HM.

2. If the values are distinct,

AM > GM > HM.

3. H.M. x A.M. G

H.M. x A.MG2

=

=

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MEDIAN:

1. Median is defined as the positional average and is regarded as the second best

average after arithmetic mean.

2. Median is suitable when there is a wide range of variation in data or distribution

pattern is to be studied at a varying level.

3. Median is suitable for qualitative data.

4. Median is suitable for distributions with open ends.

5. Median can be located graphically using Cumulative Frequency Polygon or Ogives.

6. The absolute sum of deviations is minimum when the deviations are taken from

Median, and this property of Median is known as “Minimal Property”.

7. Median is dependent on change of Origin & Scale.

Y = a ± bx

Then,

Me (Y) = a ± b × Me(x)

Calculation

For Simple Series

Median = value corresponding to (n + 1)/2th term in the distribution

Note 1: Arrange the data in the ascending or descending order

Note 2: If the value of (n+1)/2th term is a fraction then the average of the values within

which it is lying is the median.

Note 3: If n is odd median = simply the middle most value and if n is even median =

average of 2 mid values

For Simple Frequency Distribution:

Median = value corresponding to the (N+1)/2th Term in the ‘less than’ type Cumulative

Frequency column where,

N = Total Frequency

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For Grouped Frequency Distribution:

12Median = .

m

NF

l if

−

+

l1 = Lower boundary of the median class i.e., the class where Cumulative Frequency N/2

falls

N = Total frequency

F = Cumulative frequency of the pre-median class.

fm = Frequency of the median class

i = Width of the median class

MODE

1. Mode is that value of the distribution which occurs with highest frequency.

2. Mode is a crude method of finding out average and it provides only a Bird’s Eye view

of the distribution.

3. It is the most unstable average and the quickest method of finding out the average

where we need to find out the most common value of the distribution

4. It is not affected by extreme values but it is affected by sampling fluctuations

5. Multimodal distributions are ill-defined

6. Mode is dependent on the change of origin and scale

7. If y = a ± bx then, Mo(y) = a± b Mo(x)

8. Mode can be located graphically using Histogram or Area Diagram or Frequency

Diagram.

9. Mode does not take into account all of the observations.

10. When the classes are of unequal width, we consider frequency densities instead of

class frequency to locate mode, where frequency density = Class Frequency / Width

of the Class

Calculation of Mode for Simple Series:

1. For simple series, there is no mode as all values occur with frequency = 1, i.e., same

frequency.

2. For simple frequency distribution Mode can be calculated by mere inspection. The

variable occurring with the highest frequency is the mode of the distribution. Now a

distribution can be uni-modal or bi-modal, but not multi-modal.

o If only one value occurs with the highest frequency, then there is only one mode.

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o If two variable occurs with the same highest frequency, then there are two

modes.

o If all variables occurs with same frequency, then there is no mode.

o If more than one variable occurs with same highest frequency, then also there is

no mode.

Calculation of Mode for Grouped Frequency Distribution:

1

1

1 2

Mode = .2

m

m

f fl i

f f f

−+

− −

L1 = Lower boundary of the modal class i.e., the class with highest frequency.

fm = Frequency of the modal class

f1 = Frequency of the pre-modal class

f2 = Frequency of the post-modal class

i = Class width

Concept of Symmetrical & Asymmetrical Distribution:

1. When in a distribution all the measures of central tendencies are equal, the

distribution is said to be symmetrical.

2. For symmetrical distribution; Mean = Median = Mode.

3. Any deviation from this symmetry makes the distribution asymmetrical or skewed.

4. For moderately skewed distribution: Mean – Mode = 3(Mean – Median)

OTHER PARTITION VALUES (FRACTILES)

Partition values means which divides distribution in equal parts.

• Quartiles

o There are 3 quartiles (Q1, Q2, Q3), which divides the distribution in 4 equal parts

representing 25%, 50% and 75% of the data respectively.

o Q2 is nothing but the median of the data.

o For symmetrical data, Q2 is simple average of the extreme quartiles Q1 (lower

quartile) and Q3 (upper quartile).

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• Deciles

o There are 9 deciles (D1, D2, ……, D9), which divides the distribution in 10 equal

parts representing 10%, 20% ……. 90% of the data respectively.

o D5 is nothing but the median of the data.

• Percentiles

o There are 99 percentiles (P1, P2, ……, P99), which divides the distribution in 100

equal parts representing 1%, 2% ……. 99% of the data respectively.

o P50 is nothing but the median of the data

• All partition values are dependent on the change of Origin and Scale.

• All partition values can be calculated graphically through Cumulative Frequency

Polygon or ogives.

Calculation of Partition Values

Type of Series Quartiles Deciles Percentiles

Simple Series

1

4

1,2,3

i

nQ i

i

+ =

=

1

10

1,2,3,.....,9

i

nD i

i

+ =

=

1

100

1,2,3,.....,99

i

nP i

i

+ =

=

Simple Frequency Dist

correspo-

1-nding to CF; i

4

iQ value

N

=

+

correspo-

1-nding to CF; i

10

iD value

N

=

+

correspo-

1-nding to CF; i

100

iP value

N

=

+

Group Frequency Dist 1

4+i

q

iNf

Q l if

−

=

110+i

d

iNf

D l if

−

=

1100+i

p

iNf

P l if

−

=

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Arithmetic Mean

1. Find the arithmetic mean of 14, 16, 19, 25, 21.

a) 18

b) 19

c) 20

d) 21

2. Find the arithmetic mean of the numbers 3, 5, 7, 9, … 47.

a) 24

b) 25

c) 27

d) 31

3. Mean of 0, 3, 5, 6, 7, 9, 12, 0, 2 is:

a) 4.89

b) 5.7

c) 5.6

d) 6.5

4. Find the arithmetic mean of the natural numbers from 1, 2, 3, …, n.

a) n

b) (n + 1)

c) (n + 1)/2

d) None of the above

5. A variable x takes all integral values from 1 to 10 with equal frequencies. Find its arithmetic mean.

a) 5

b) 6

c) 5.5

d) cannot be determined

6. If the AM of 3, 5, x, 12, 17 be 9, find the value of x.

a) 5

b) 6

c) 7

d) 8

7. The AM of the following: -2, 3, 0, 5, 4 is:

a) 1

b) 2

c) 3

d) Cannot be determined

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8. The algebraic sum of deviations of 8, 1, 6 from their arithmetic mean is:

a) 1

b) 2

c) – 1

d) 0

9. The arithmetic mean of 8, 1, 6 with weights 3, 2, 5 respectively is:

a) 5

b) 5.6

c) 6

d) 4.6

10. If there are 3 observations 15, 20, 25 then the sum of deviation A.M. is

a) 0

b) 5

c) – 5


11. If 7, 4, 9 be the AM of three separate groups containing 6, 2, 5 observations

respectively, then the AM of all the 13 observations is given by:

a) 7.29

b) 7.31

c) 7.00

d) 6.89

12. If a variable assumes the values 1, 2, 3, 4, 5 with frequencies 1, 2, 3, 4, 5

respectively, find its arithmetic mean.

a) 4.5

b) 4

c) 5

d) 3.67

13. The weighted AM of 3, 5 and 10 with weights x, y and z respectively is known to be 8.

What is the weighted AM if the weights are changed to 2x, 2y and 2z respectively?

a) 7

b) 8

c) 4

d) 16

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14. AM of first N odd natural numbers is:

a) N

b) N + 1

c) (N + 1) / 2

d) None of these

15. The weighted mean of first n natural numbers, whose weights are proportional to their

corresponding number is:

a) (n + 1)/2

b) (n + 1) (2n + 1) / 6

c) n(n + 1)

d) 2 1

3

n +

16. If mean of n observations 12, 22, 32, …., n2 is 46

11

n, then the value of n is:

a) 11

b) 12

c) 22


17. Find the A.M. of the following distribution:

Weights 100 110 120 130 140

No. of Men 15 20 25 30 10

a) 110

b) 100

c) 120

d) 130

18. The average weight of the following frequency distribution is 117 lbs. Find the value

of X.

X : 100 110 120 x+25 140

F : 1 4 2 2 1

a) 110

b) 100

c) 120

d) 105

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19. If there are two groups containing 30 and 20 observation and having 50 and 60 as

arithmetic means, then the combined arithmetic mean is:

a) 52

b) 54

c) 55

d) 56

20. The average monthly salary of a group of 20 workers in a small factory is `̀̀̀ 2,020. There are 5 skilled workers, 8 semiskilled workers and the remaining unskilled

workers. The average salary of skilled workers is `̀̀̀ 2,500 and that of the semiskilled

workers is `̀̀̀ 2,000 what is the average salary of the unskilled workers in that factory?

(a) 1600 (b) 1700 (c) 1800 (d) None of these

21. The mean salary for a group of 40 female workers is `̀̀̀ 5200 per month and that for a

group of 60 male workers is `̀̀̀ 6800 per month. What is the combined salary?

a) `̀̀̀ 6160

b) `̀̀̀ 6260

c) `̀̀̀ 6560


22. The mean annual salary paid to all employees of a company was `̀̀̀ 5,000. The mean

annual salaries paid to male and female employees were `̀̀̀ 5,200 and `̀̀̀ 4,200 respectively. Determine the percentage of males and females employed by the company.

a) 60%, 40%

b) 80%, 20%

c) 50%, 50%

d) 40%, 60%

23. The algebraic sum of the deviations of 25 observations measured from 45 is – 55. Find the arithmetic mean of the observations.

a) 42.8

b) 50.8

c) 52.8

d) 50.9

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24. If the mean of numbers 27 + x, 31 + x, 89 + x, 107 + x, 156 + x is 82, then the mean of

130 + x, 126 + x, 68 + x, 50 + x, 1 + x is:

a) 79

b) 157

c) 75


25. If y = 5x - 20 and AM of x is 30, then find the AM of y.

a) 130

b) 140

c) 30


26. If the relationship between two variables u and v are given by 2u + v + 7 = 0, and if

AM of u is 10, then the AM of v is:

a) 17

b) – 17

c) 27

d) – 27

27. The mean of a set of numbers is K. If each number is increased by A, then the mean

of the new set is:

a) K

b) K + A

c) KA

d) None of these

e)

28. The arithmetic mean of a variable y is 10. Find the mean of the variable 5y – 10.

a) 50

b) 40

c) 25

d) 30

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29. The mean of a set of observation is x . If each observation is divided by α , 0α ≠ and

then is increased by 10, then the mean of the new set is :

(a)x

α (b)

10x

α

+ (c)

10x α

α

+ (d) 10ax +

30. The sum of deviation of a set of values x1, x2, ……., xn measured from 50 is -10 and

the sum of deviation of values from 46 is 70. The values of n and the mean are :

(a) 20,49.5 (b) 20, 49 (c) 21, 39.5 (d) none of these

31. The average weight of students in a class of 35 students is 40 kg. If the weight of the

teacher be included, the average rises by (1/2) kg; the weight of the teacher is :

(a) 40.5 kg (b) 50 kg (c) 41 kg (d) 58 kg

Geometric Mean

32. Find the Geometric mean of : 8, 4, 2.

a) 2

b) 4

c) 8


33. Calculate the GM of 108, 930, 498, 237

34. Find the GM of the following: 3, -2, 4, 0, 5.

a) 2

b) 3

c) 1

d) cannot be determined

35. If GM of x is 10 and GM of y is 15, then the GM of xy is:

a) 150

b) Log 10 x Log 15

c) Log 150

d) None of these

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36. The GM of the numbers 6, 62, 63, …., 6n is:

a) 6n/2

b) 62/n

c) 6(n+1)/2

d) None of these

37. The GM of the first n terms of the GP a, ar, ar2, ar3,...... is:

a) arn

b) arn/2

c) arn-1

d) ar(n-1)/2

38. The interest paid on each of three different sums of money yielding 3%, 4%, and 5%

simple interest p.a. respectively is the same. What is the average yield percent on the

total sum invested.

a) 3.83%

b) 4.83%

c) 2.83%

d) None of these

Harmonic Mean

39. Harmonic mean of 3, 4, 5 is:

a) 2

b) 3.33

c) 3.85


40. The HM of - 3, 2, 4, 0, -5 is:

a) 1

b) 2

c) 3


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41. What is the HM of 1, ½, 1/3, ¼, …, 1/n?

a) n

b) 2n

c) 2 / (n + 1)

d) n (n + 1) / 2

42. Find the Harmonic mean of the following numbers: 1 1 1

1, , ,......,3 5 2 1n −

a) 1

1n +

b) 1

1n −

c) 2

n

d) 1

n

43. Rani bought 6 rupees worth of orange from five markets at 15p, 20p, 25p, 30p, and 50p per orange respectively. What is the average price of an orange?

a) 22p

b) 28p

c) 24p

d) 25p

44. What would be the average price, if she purchased 20 oranges from each market? (take data from the previous question)

a) 28p

b) 24p

c) 25p

d) none of the above

45. A motor car covered a distance of 50 miles four times. The first time at 50 mph, the second at 20 mph, the third at 40 mph and the forth at 25 mph. Calculate the average speed.

a) 29

b) 33.75

c) 29.63


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46. A travelled some distance by cycle at a speed of 15 km per hour. On return journey he

travelled the same distance at a speed of 10 km per hour. What was his average

speed per hour during the entire journey?

a) 12.5 km per hour

b) 13 km per hour

c) 12 km per hr

d) 15 km per hour

47. An aeroplane flies from A to B at the rate of 500 km/hour and comes back from B to A

at the rate of 700 km/hour. The average speed of the aeroplane is:

a) 600 km per hour

b) 583.33 km per hour

c) 550 km per hour


Combined Properties of AM, GM and HM 48. The AM of two observations is 36 and their GM is 24. Then the HM is:

a) 14

b) 15

c) 16

d) 18

49. The arithmetic mean of two observations is 25 and their geometric mean is 15. The

two observations are:

a) 25 and 25

b) 45 and 5

c) 35 and 15

d) 30 and 20

50. If the A.M. of some observations be 25 and their H M is 9, find their G. M.

a) 15

b) 9

c) 18

d) 24

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51. If A.M. = 25; H. M. = 9, then GM = ?

a) 15

b) 20

c) 21

d) 19

Median

52. Find the median of the following numbers: 2, 5, 8, 4, 9, 6, 7.

a) 9

b) 6

c) 8


53. Find the median of the following numbers: 5, 8, 6, 9, 11, 4.

a) 6 b) 7 c) 8 d) None of the above

54. Calculate median for the following data :

No. of students 6 4 16 7 8 2

Marks 20 9 25 50 40 80

a) 20

b) 25

c) 35

d) 28

55. From a batch of 13 students who had appeared for an examination 4 students have

failed. The marks of the successful candidates were 41, 57, 38, 61, 36, 35, 71, 50, 40.

Calculate the median marks.

a) 40

b) 50

c) 38


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56. You are given below a certain statistical distribution. Calculate the most suitable average.

Value Frequency

Less than 100 40

100 – 200 89

200 – 300 148

300 – 400 64

400 and above 39

a) Median = 240

b) Median = 241.22

c) Mean = 241


57. Two variables x and y are given by y = 2x – 3. If the median of x is 20, what is the

median of y?

a) 20

b) 37

c) 40

d) 35

58. Two variables x and y are given by y = 2x – 11. If the median of x be 30, find the ratio

of the median of y to that of x.

a) 4 : 3

b) 49 : 13

c) 49 : 30

d) 41 : 30

Mode

59. The mode of the following observations is: 4, 3, 2, 5, 3, 4, 5, 3, 7, 3, 2, 6

a) 2

b) 3

c) 4

d) 6

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60. Find the Mean, Median and Mode of the following numbers: 7, 4, 3, 5, 6, 3, 3, 2, 4, 3, 4, 3, 3, 4, 4, 3, 2, 2, 4, 3, 5, 4, 3, 4, 3, 4, 3, 1, 2, 3.

a) 3.33, 3, 3

b) 3, 3, 3

c) 3.47, 3, 3

d) 3.47, 3.47, 3

61. Find the mode of the following: 0, 3, 5, 6, 7, 9, 12, 0, 2.

a) 2 or 10

b) 2

c) 2

d) 0

62. Find the mode from the following distribution:

Marks Number of Candidates

1 – 5 7

6 – 10 10

11 – 15 16

16 – 20 32

21 – 25 24

26 – 30 18

31 – 35 10

36 – 40 5

41 – 45 1

a) 18

b) 18.83

c) 17.83


63. From the following cumulative distribution of marks of 22 students, the arithmetic mean, median and mode of the distribution would be:

Marks Number of students

below 10 3

below 20 8

below 30 17

below 40 20

below 50 22

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a) 23, 23, 24

b) 23.18, 23.33, 24

c) 23.33, 23.18, 24.83

d) 23.81, 23.33, 24.33

64. If x and y are related by x – y – 10 = 0 and mode of x is known to be 23, then the mode of y is:

a) 3

b) 13

c) 20

d) 23

Combined Properties of Measures of Central Tendency

65. If arithmetic mean is 26.8, median is 27.9, then what is the value of mode?

a) 29

b) 30.1

c) 31.1

d) 29.9

66. If the Mean and Mode of a certain set of numbers be 60.4 and 50.2 respectively, find approximately the value of the Median.

a) 55

b) 56

c) 57

d) 58

67. If in a moderately asymmetrical distribution mode and mean of the data are 6λλλλ and 9λλλλ respectively, then median is :

(a) 8 λλλλ (b) 7 λλλλ (c) 6 λλλλ (d) 5 λλλλ

Fractiles – Other Partition Values

68. Find Quartiles, D9, P10 of the following data: 96, 180, 98, 75, 190, 80, 102, 100, 94, 75, 185, 160.

69. Find Q1, D5, P80 of the following distribution:

Weight in lbs 115 120 130 140 145

No. of men 8 20 35 24 12

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70. If for a symmetrical distribution Q1 = 20 and Q3 = 30, find the median.

a) 20

b) 25

c) 30

d) 15

71. Find the quartiles of the values: 35, 30, 48, 40, 25, 36, 45

a) 30, 35, 41

b) 30, 36, 45

c) 29, 36, 45


72. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12, is:

a) 11

b) 13

c) 11.50

d) 10.70

73. Calculate the first and third quartiles for the following data:

Class Limits Frequency

10 – 19 5

20 – 29 9

30 – 39 14

40 – 49 20

50 – 59 25

60 – 69 15

70 – 79 8

80 – 89 4

a) 37.36, 60.83

b) 37, 61

c) 37.63, 60.38


74. The value of 45th and 57th percentiles for the following data on marks obtained by 100 students is:

Marks Number of Students

20 – 25 10

25 – 30 20

30 – 35 20

35 – 40 15

40 – 45 15

45 – 50 20

: 308 :


a) 33.75, 37.33

b) 35.73, 37.35

c) 33.75, 43.33

d) 33.75, 34.33

Errors and Missing Frequency

75. The mean of 20 observations is 85; but it was later found that two of the observations were wrongly read as 75 and 70 instead of 57 and 60. Find the actual mean.

a) 85

b) 83

c) 83.6

d) 80

76. The AM, Median and Mode of a group of 50 observations were found to be `̀̀̀ 47, `̀̀̀ 46

and `̀̀̀ 45 respectively. It was discovered later that one of the observations was wrongly recorded as 67 instead of 76. Find the correct values of AM, Median and Mode.

77. The arithmetic mean of the following data is 67.45. Find the missing frequencies.

Height Number of students

60 – 62 5

63 – 65 18

66 – 68 f1

69 – 71 f2

72 – 74 8

Total 100

a) 40, 29

b) 27, 42

c) 42, 27

d) 38, 31

78. The following table gives the distribution of 100 families according to expenditure. If

mode of the distribution is 24, find the missing frequencies x1 and x2.

Expenditure Number of Families

00 – 10 14

10 – 20 x1

20 – 30 27

30 – 40 x2

40 – 50 15

: 309 :


a) 20, 24

b) 21, 23

c) 22, 22

d) 23, 21

79. Find the missing frequencies, if the first quartiles of the following data is 21.5.

Class Frequency

10 – 15 24

15 – 20 f1

20 – 25 90

25 – 30 122

30 – 35 f2

35 – 40 56

40 – 45 20

45 – 50 33

Total 460

a) 60, 55

b) 61, 54

c) 64, 51

d) 54, 61

Theoretical Aspects

80. The average discovers:

a) uniformity in variability

b) variability in uniformity

c) either a) or b)

d) both a) and b)

81. The correction factor is applied in:

a) inclusive type of distribution

b) exclusive type of distribution

c) both type of distribution


: 310 :


82. The best measure of central tendency is:

a) arithmetic mean

b) quartile

c) median

d) mode

83. Which of the following statements is not CORRECT:

a) The sum of deviations from mean is zero.

b) The mean of the two samples can be combined

c) Mean has the least sampling variability

d) Choice of assumed mean affect the actual mean

84. Which of the following is the basic assumption made while computing the arithmetic

mean from a grouped frequency distribution?

a) All the classes have equal weights

b) All class interval are of same width

c) All the values of a class are equal to the mid value of that class

d) All of the above

85. The most commonly used measure of central tendency is:

a) geometric mean

b) median

c) arithmetic mean

d) mode

86. When weighted averages are used?

a) when the data are put in the form of grouped frequency distribution

b) when data are not classified properly

c) when all the observations are not of equal importance

d) either b) or c) above

87. The sum of deviations of individual observations is zero from:

a) mode

b) median

c) arithmetic mean

d) harmonic mean

: 311 :


88. Which average is affected most by extreme observations?

a) mode

b) median

c) geometric mean

d) arithmetic mean

89. Median is better suited for __________ interval series.

a) positional

b) continuous

c) discrete


90. In case of odd number of observations which of the following describes median?

a) the simple average of the two middle values

b) simply the middle value

c) the weighted average of the two middle most values

d) any of the two middle most value

91. In case of even number of observations which of the following best describes median?

a) simply the middle value

b) the difference of the two middle values

c) the simple average of the two middle values

d) multiplicative effect of the two middle most values

92. For dealing with qualitative data the best average is:

a) arithmetic mean

b) mode

c) median

d) harmonic mean

93. Median can be calculated graphically using which of the following graphs:

a) histogram

b) pie charts

c) ogives


: 312 :


94. The positional measure of central tendency is:

a) geometric mean

b) median

c) harmonic mean

d) arithmetic mean

95. Which of the following measures of central tendency can have more than one value for a given set of observations?

a) mean

b) median

c) mode

d) both mean and mode

96. Mode can be determined graphically using:

a) ogives

b) histogram

c) bar diagrams

d) all of the above

97. Which of the following is the most unstable average:

a) median

b) mode

c) geometric mean

d) harmonic mean

98. The mode of the distribution is the value that has the greatest of _________.

a) Values

b) frequencies

c) either a) or b) above

d) neither a) nor b) above

99. In a symmetrical distribution mean ______ median _______ mode.

a) greater than, greater than

b) equal to, less than

c) equal to, equal to

d) less than, less than

: 313 :


100. Which of the following measures of central tendency satisfy a linear relationship

between two variables?

a) mean

b) median

c) mode

d) all of the above

101. The geometrical mean is the ___root of the product of all the measurements.

a) 5th

b) nth

c) (n + 1)th

d) (n – 1)th

102. If the set of observations contains both negative and positive values, which of the following measure of central tendency cannot be considered?

a) mode

b) median

c) geometric mean

d) arithmetic mean

103. Which of the following measures is the most difficult to compute?

a) harmonic mean

b) quartiles

c) mode

d) geometric mean

104. The harmonic mean is the ________ of the arithmetic mean of the inverse of values.

a) reciprocal

b) average

c) median


105. Quartiles divides the set of observations in how many parts?

a) 2

b) 10

c) 100

d) 4

: 314 :


106. Deciles divides the set of observations in how many parts?

a) 2

b) 10

c) 100

d) 4

107. How many deciles are more than 50 percentiles?

a) four

b) one

c) two

d) all of the above

108. Percentiles are the values dividing a given set of observations into:

a) two parts

b) ten parts

c) four parts

d) hundred parts

109. How many deciles lie below the first quartile?

a) 5

b) 25

c) 2

d) 4

For the statements given state the appropriate type of average to be used:

110. The distribution has open-end classes.

a) mode

b) median

c) either mode or median

d) arithmetic mean

111. Marks obtained in Statistics by the students in an examination.

a) mode

b) arithmetic mean

c) median

d) harmonic mean

: 315 :


112. The size of shoes sold in a shop.

a) median

b) quartiles

c) mode

d) geometric mean

113. A frequency distribution having open-end classes.

a) mode

b) arithmetic mean

c) harmonic mean

d) median

114. Different rates of depreciation charges on diminishing balance.

a) arithmetic mean

b) geometric mean

c) harmonic mean

d) all of the above

115. Speeds in several journeys when the distance covered is the same in each journey.

a) arithmetic mean

b) geometric mean

c) harmonic mean

d) all of the above

116. Speeds in several journeys when the time taken is the same in each journey.

a) arithmetic mean

b) harmonic mean

c) both a) and b) above

d) neither a) nor b) above

117. When it is desired that the sum of absolute deviations from the average should be least.

a) geometric mean

b) arithmetic mean

c) median

d) mode

: 316 :


118. The distribution has wide range of variations.

a) mode

b) arithmetic mean

c) harmonic mean

d) median

119. The quantities are in ratios.

a) harmonic mean

b) geometric mean

c) arithmetic mean

d) either a) or b) above

120. When variability has also to be calculated.

a) geometric mean

b) quartile

c) percentile

d) arithmetic mean

121. The values changing at a cumulative rate have to be found.

a) harmonic mean

b) geometric mean

c) arithmetic mean

d) either a) or c) above

Miscellaneous Problems

122. An auto ride in Delhi costs Re. one for the first kilometer and sixty paise for each

additional kilometre. The cost for each kilometre is incurred at the beginning of

kilometre so that the rider pays for a whole kilometre. The average cost for 2.75

kilometre in rupees is :

(a) 0.75

(b) 0.80

(c) 0.90

(d) none of these

: 317 :


123. A train travels first 300 kilometres at an average rate of 30 k.p.h. and further travels

the same distance at an average rate of 40k.p.h. what is the average speed over the

whole distance?

a) 34.28 b) 36.28 c) 44.28 d) 34

124. An automobile driver travels from plain to a hill station, 100km. distance at an average

speed of 30 km per hour. He then makes the return trip at average speed of 20 km

per hour. What is his average speed over the entire distance (200 km.)?

a) 24.009 b) 34 c) 44.009 d) 42.009

125. A man purchases black pencils worth `̀̀̀ 100 at the rate of `̀̀̀ 5 per pencil, red pencils

worth `̀̀̀ 100 at the rate of `̀̀̀ 10 per pencil and blue pencils worth `̀̀̀ 100 at the rate of

`̀̀̀ 20 per pencil. The average cost of one pencil is :

(a) `̀̀̀ 10 (b) `̀̀̀ (60/7) (c) `̀̀̀ (35/3) (d) none of these

126. In a group of 13 boys, the shortest boy measures 145 cms, and the tallest 160 cms.

4 more boys are added to this group. Their heights were (cms) 140, 142, 161, 161. If

the median of the first group of 13 boys was 152 cms, what is the median of the group

of 17 boys after addition of the 4 new boys?

(a) 152 (b) 150 (c) 149 (d) 148

: 318 :


THEORY ANSWERS:

80 a 91 c 102 c 113 d

81 a 92 c 103 d 114 b

82 a 93 c 104 A 115 c

83 d 94 b 105 D 116 a

84 c 95 c 106 c 117 c

85 c 96 b 107 a 118 d

86 c 97 b 108 d 119 b

87 c 98 b 109 c 120 b

88 d 99 c 110 b 121 b

89 a 100 d 111 b

90 b 101 b 112 c

: 319 :

Documents

2. Measures of Central Tendency (Averages of First Order) · • Average discovers uniformity in variability. • The tendency of the variables to accumulate at the center of the