Summary Sheet

Session Number :

Subject Expert :

6 Nagesh P.Nagesh P.

Department of Management Studies

S.J. College of Engineering

Mysore – 570 006.

Measures of Central TendencyMode

It is the value which occurs with the maximum frequency. Thus, it

corresponds to the values of variable which occurs most frequently. The

model class of a frequency distribution is the class with highest frequency. It

is denoted by ‘z’. Mode is the value of variable which is repeated the greatest

number of times in the series.

Ex: If we say modal marks obtained by students in class test is 42, it mean

that the largest number of student have secured 42 marks.

If each observations occurs the same number of times, we can

say that there is ‘no mode’. If two observations occur the same

number of times, we can say that it is a ‘Bi-modal’. If there are 3

or more observations occurs the same number of times we say

that ‘multi-modal’ case. When there is a single observation

occurs mot number of times, we can say it is ‘uni-modal’ case.

For a grouped data mode can be computed by following

equations with usual notations.

Mode = 21m

1m

fff2

)ff(h

where, fm = max frequency (modal class frequency)

f1 = frequency preceding to modal class.

f2 = frequency succeeding to modal class

h = class width.

or Mode = 21

2

ff

hf

Ex: 1. Find the modal for following data. Marks

(CI) No. of

students (f)

1 – 10 3

11 – 20 16

21 – 30 26

31 – 40 31 Max. frequency

41 – 50 16

51 – 60 8

f=N=100

We shall identify the modal class i.e. 31-40.

Where, fm = 31, f1 = 26, f2 = 16, h = 10,

2

3130 and 30.5

Mode (z) = 21m

1m

fff2

)ff(h

= 162631x2

26)-(3110 30.5

= 33.

Or Mode = 21

2

ff

hf

=

)1626(

16x105.30

= 34.30

It can be noted that there exists slightly different

mode values in the second method.

Partition values

Median divides in to two equal parts.

Just as one point divides as series in to two equal parts (halves),

3 points divides in to four points (Quartiles) 9 points divides in

to 10 points (deciles) and 99 divide in to 100 parts (percentage).

The partitioned values are useful to know the exact composition

of series.

Measures of quartiles

Measure Individual and Discrete senses

Continuous series

Q1

item4

1Nth

item4

nth

Q2

item4

1N2th

itemn4

2 th

Q3 item1N4

3 th itemn4

3 th

Ex-1: From the following marks find Q1, Median and Q3 marks

23, 48, 34, 68, 15, 36, 24, 54, 65, 75, 92, 10, 70, 61, 20, 47, 83, 19, 77

Let us arrange the data in array form.

Sl. No.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

x 10 15 19 20 23 24 34 36 47 48 54 61 65 68 70 75 77 83 92 Q1 Q2 Q3

a. Q1 = item1n4

1 th = 1194

1 = 20x

4

1 = 5th item Q1=23

b. Q2 = item1n42 th = 20x

4

2 = 10th item Q2 = 48

c. Q3 = item1n4

3 th = 20x4

3 = 15th item Q3 = 70

Q1 = item1n4

1 th = 3214

1, Q1 = 80.25th item

Just above 80.25, the cf is 100. Against 100 cf, value is 5.

Q1 = 5, Q2 = item1n2

1 th , Q2 = 321x2

1, 160.5th item

Just above 160.5, the cf is 230. Against 230 cf value is 6.

Q2 = 6, Q3 = item1n4

3 th , Q3 = x4

3321 = 240.75th item

Just above 240.75, the cf is 260. Against 260 cf value is 6.5. Q3 = 6.5

Ex - 2: Locate the median and quartile from the following data. Size of shoes X

4 4.5 5 5.5 6 6.5 7 7.5 8

Freq. f 20 36 44 50 80 30 30 16 14 N=f=320

Cf 20 56 100 Q1

150 230 Q2

260 Q3

290 306 320

Ex - 3: Compute the quartiles from the following data.

First locate Q1 for ¼ N = 25 = 30, h = 10, f = 12, c = 20

(Q1) =

CN4

1

f

h = 2025

12

1030 = 34.16

a. Locate Q2 (Median) - Q2 corresponds to N/2 = 50

Q2 =

C

2

N

f

h = 3250

28

1040 = 46.42

Q3 corresponds to ¾ N = 75,

Q3 =

CN4

3

f

h = 6075

20

1050 = 57.5

Deciles

The deciles divide the arrayed set of variates into ten portions of

equal frequency and they are some times used to characterize

the data for some specific purpose. In this process, we get nine

decile values. The fifth decile is nothing but a median value.

We can calculate other deciles by following the procedure which

is used in computing the quartiles.

Formula to compute deciles.

,CN10

1

f

hD

1

CN

20

2

f

hD

2 on,so&

Percentiles

Percentile value divides the distribution into 100 parts of

equal frequency. In this process, we get ninety-nine percentile

values. The 25th, 50th and 75th percentiles are nothing but

quartile first, median and third quartile values respectively.

Formula to compute. Percentile is given below.

P25 = ,CN100

25

f

h

P26 =

CN100

26

f

h & so, on

Ex: Find the decile 7 and 60th percentile for the given

data

CI f Cf 10-20 1 1 20-30 3 4 30-40 11 15 40-50 21 36

50-60 43 79 P60

60-70 32 111 D70

70-80 9 120 f=N= 120

Calculate the missing frequency from the data if the median is 50.

CI f cf

10-20 2 2

20-30 8 10

30-40 6 16

40-50 ? f1 16+f1

50-60 15 31+f1

median class

60-70 10 41+f1

f = 41 + f1

16 + f1 = ½ (41 + f1)

= 2 (16 + f1) = 41 + f1

= 32 + 2f1 = 41 + f1

f1 = 9

Find out the missing values of the variate for the following data

with mean is 31.87.