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Inclusive Series

An inclusive series includes the upper limit and the lower limit in the class interval.For example in the class interval 20-29 the frequencies(number of students) for 20 and 29 are included. In such series the upper limit of class interval does not repeat itself as a lower limit of the next class interval. Thus there is a gap between the upper limit of a class interval and the lower limit of the next class interval. For example 20-29, 30-39, 40-49, 50-59 etc represents an inclusive series. Thus all the items ranging between 20-29 are included in that class interval. Likewise, all items ranging between 30-39 would be included in that class interval. Illustration:

Pocket Expenses No of Students
(x) (f)
20-29 10
30-39 8
40-49 6
50-59 4
60-69 2

This is an inclusive series of the pocket expenses of the students of a class.Inclusive series are used when there is some definite difference between the values of various items in the population. In the above table if a student has 29.5 or 39.5 pocket expenses these can be expressed only if the inclusive series is converted into an exclusive series.To see how to convert an inclusive series into an exclusive series click here.

MEAN- It is the sum total of all the observations divided by the number of observations
There are three ways of calculating the mean of an lnclusive series.

1.Direct Method
2.Short Cut Method
3.Step Deviation Method

Let us take the inclusive series given below to find the mean of -

Pocket Expenses No of Students
(x) (f)
20-29 10
30-39 8
40-49 6
50-59 4
60-69 2

Direct Method

Mean = m*f/ f

Let L1=lower limit
L2=upper limit
them mid value m=(L1+L2)/2

Pocket Expenses Mid Value Frequency m*f
(x) (m) (f)  
20-29 24.5 10 245
30-39 34.5 8 276
40-49 44.5 6 267
50-59 54.5 4 218
60-69 64.5 2 129
    f=30 m*f=1135

Mean = m*f/ f = 1135/30 = 37.83

Thus average pocket expenses are $37.83

Short Cut Method

Mean = A + f*d / f

L1=lower limit of the class interval
L2=Upper limit of the class interval
A = The mid value of the middle class interval(in this case the 40-49 interval).
so A=44.5
d = (m-A) The deviation of the mid values of the respective class intervals from A.
C = Class interval which is 10 in this case.

Pocket Expenses Mid Value Frequency Deviation f*d
  m=(L1+L2)/2 (f) d=(m-A)  
20-29 24.5 10 -20 -200
30-39 34.5 8 -10 -80
40-49 44.5 6 0 0
50-59 54.5 4 +10 +40
60-69 64.5 2 +20 +40
    f=30   f*d= -200

Mean = 44.5 + (-200)/30
= 37.83

Step Deviation Method

Mean = A + (f*d'/f) * C

Where d'= d/C
and d=(m-A)
C=10 the class interval

A=44.5

Pocket Expenses Mid Value Frequency Deviation Step Deviation f*d'
  m=(L1+L2)/2 (f) d=(m-A) d'=d/C  
20-29 24.5 10 -20 -2 -20
30-39 34.5 8 -10 -2 -8
40-49 44.5 6 0 0 0
50-59 54.5 4 +10 +1 +4
60-69 64.5 2 +20 +2 +4
    f=30     f*d'=-20

Mean = 44.5 + (-20/30) * 10
= 37.83

Therefore the average pocket expensise are $37.83

MEDIAN - It is the middle most value of the given data

We will have to first convert the inclusive series into an exclusive series.To see how to do this click here.
C.M - Cumulative Frequency.C.M of a class interval is the cumulation(sum) of all the frequencies till that class interval.

Marks Conversion No. of students Cumulative Frequency
(x)   (f) (C.M)
410-419 409.5-419.5 14 14
420-429 419.5-429.5 20 34
430-439 429.5-439.5 42 76
440-449 439.5-449.5 54 130
450-459 449.5-459.5 45 175
460-469 459.5-469.5 18 193
470-479 469.5-479.5 7 200

Median = L + [ (N/2 - C.M) / f ]*C

Over here N is the C.M of the last class interval.First find the the N/2 term.
The N/2 term is 200/2=100 th term.
This lies in the 130th cumulative frequency and the corresponding median class is 439.5-449.5

L=the lower limit of the class interval.It is 439.5
N/2 = 100
C.M = 76
f = 54
C = 10

Median = 439.5 + [ (100 - 76)/54]*10
=439.5 + 4.44
= 443.94

To calculate the median graphically click here

MODE- It is the observation that has appeared the maximum number of times.

We will have to first convert the inclusive series into an exclusive series for calculating the mode.To see how to do this click here.

 

Marks Conversions No. of students
(frequency)		  
(x)   (f)  
10-19 9.5-19.5 10  
20-29 19.5-29.5 12  
30-39 29.5-39.5 18 f0
40-49 39.5-49.5 30 f1
50-59 49.5-59.5 16 f2
60-69 59.5-69.5 6  
70-79 69.5-79.5 8  

A glance at the above table reveals that 39.5-49.5 is the Modal class interval.The actual value of mode is given by:

z(mode)= L1 + [(f1-f0) / (2f1-f0 -f2)] * i
Where:
L1=lower limit of the class interval
L2=Upper limit of the class interval
f0 = The frequency of the class interval previous to the modal class interval.
f1 = The frequency of the modal class interval
f2 = The frequency of the class interval coming after the modal class interval.
i = The intervals of the classes which 10 in this case.

Over here
L1=39.5
f0=18
f1=30
f2=16
i=10

z(mode)= 39.5 + [(30-18) / (2*30-18 -16)] * 10
= 44.12
Therefore the mode of the above data is 44.12

To see how to calculate mode graphically using a histogram, click here.