BUSINESS STATISTICS

1. Prof. Rajkumar Teotia
Institute of Advanced Management and Research (IAMR)
Address: 9th Km Stone, NH-58, Delhi-Meerut Road, Duhai,Ghaziabad (U.P)
- 201206
Ph:0120-2675904/905 Mob:9999052997 Fax: 0120-2679145
e mail: rajkumarteotia@iamrindia.com

3. Median is the central value of the variable that divide the series
into two equal parts in such a way that half of the items lie
above the value and the remaining half lie below this value.
Median is defined as the value of the middle item (or the mean
of the values of the two middle items) when the data are
arranged in an ascending or descending order of magnitude.

4. Thus, in an ungrouped frequency distribution if the n values are
arranged in ascending or descending order of magnitude, the
median is the middle value if n is odd.
When n is even, the median is the mean of the two middle
values.

5. Example
Suppose we have the following series:
15, 19,21,7, 10,33,25,18 and 5
Solution
We have to first arrange it in either ascending or descending
order.
These figures are arranged in an ascending order as follows:
5,7,10,15,18,19,21,25,33
Now as the series consists of odd number of items, to find out the
value of the middle item, we use the formula
Where n is the number of items

6. In this case, n is 9, as such
N + 1 = 9 + 1 = 5
2 2
That is, the size of the 5th item is the median.
So median is 18

7. Example
Suppose we have the following series:
15, 19,21,7, 10,33,25,18, 5 and 23
Solution
We have to first arrange it in either ascending or descending
order. These figures are arranged in an ascending order as
follows:
5, 7, 10, 15, 18, 19, 21,23,25,33
Now as the series consists of even number of items, to find out
the value of the middle item, we use the formula
Where n is the number of items

8. In this case, n is 10, as such
N + 1 = 10 + 1 = 5.5
2 2
That is, the size of the 5.5th item is the median. We have to take
the average of the values of 5th and 6th item. This means an
average of 18 and 19, which gives the median as 18.5.

9. In the case of a grouped series, the median is calculated with
the help of the following formula:
Median = L + N - P.c.f x i
Where,
2
f
L = Lower limit of median class
P.c.f = Previous commutative frequency of median class
f = frequency of median class.
i = Size of the median class.
N = total no of observation or the total of the frequency.

10. Example – From the following data, calculate median.
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of students 10 20 30 50 40 30
Solution-
Step I- First we will find out the commutative frequency
Marks(x) No. of students (f) Commutative
frequency
C.f
0-10
10-20
20-30
30-40
40-50
50-60
10
20
30
50
40
30
10
30
60
110
150
180
N = 180

11. Step II - Size of N item = size of 180 item = 90th item
2 2
Step III-Commutative
frequency which includes 90th = 110 Class
corresponding to 110 = 30 – 40 (is the median class)

12. Marks(x) No. of students
(f)
Commutative
frequency
C.f
0-10
10-20
20-30
30-40
40-50
50-60
10
20
30
50
40
30
10
30
60
110
150
180
N = 180
PCF
Median
Class
f
L

13. Step iv – now we will apply the following formula
Median = L + N - P.c.f x i
Median = 30 + 90 – 60 x 10
50
Median = 36
2
f

14. Example: from the following data calculate median
Marks 45 55 25 35 5 15
No. of students 40 30 30 50 10 20

15. Solution-
Step I- First we will find out the commutative frequency
Marks in
ascending
order (x)
No. of students (f) Commutative
frequency
C.f
5
15
25
35
45
55
10
20
30
50
40
30
10
30
60
110
150
180
N = 180

16. Step II - Size of N + 1 item = size of 181 item = 90.5th item
2 2
Step III- Commutative frequency which includes 90.5th = 110
Median = size of item corresponding to 110 = 35

17.  Unlike the arithmetic mean, the median can be computed
from open-ended distributions. This is because it is located
in the median class-interval, which would not be an open-ended
class
 As it is not influenced by the extreme values, it is preferred
in case of a distribution having extreme values.
 In case of the qualitative data where the items are not
counted or measured but are scored or ranked, it is the most
appropriate measure of central tendency.

18. The values of a variate that divide the series or the distribution into four
equal parts are known as quartiles. Since three points are required to
divide the data into four equal parts, we have three quartiles Q1, Q2, Q3.
 The first quartile (Q1):- it is known as a lower quartile, is the value of a
variate below which there are 25% of the observation and above which
there are 75% of the observations.
 The second quartile (Q2):- it is known as a middle quartile or median, is
the value of a variate which divides the distribution into two equal parts.
It means there are 50% of the observations above it and 50% of the
observations below it.
 The Third quartile (Q3):- it is known as an upper quartile, is the value
of a variate below which there are 75% of the observations and above
which there are 25% of the observations.

19. Q1 = size of N + 1 th item
4
Q2 = size of 2( N + 1) th item
4
Q3 = size of 3( N + 1) th item
4

20. Example: - from the following data calculate first and third quartile.
Marks 5 15 25 35 45 55
No. of students 10 20 30 50 40 30
Solution:-
Step I: - Calculation of commutative Frequencies
Marks No. of students (f) Commutative
frequency
C.f
5
15
25
35
45
55
10
20
30
50
40
30
10
30
60
110
150
180
N = 180

21. Step II: -
Q1 = size of N + 1 th item
4
= Size of 180 + 1 th item = 181 = 45.25 th item
4 4
= size of 45.25 th item = 25, So Q1 = 25
Step III: -
Q3 = size of 3(N + 1) th item
4
Q3 = 135.7 th item
Q3 = 45

22. Computation of Quartiles for grouped data:-
Q1 = L + N - P.c.f x i
4
f
Q2 = L + N - P.c.f x i
2
f
Q3 = L + 3N - P.c.f x i
4
f

23. DECILES
The deciles of a variate that divide the series or the distribution
into ten equal parts are called deciles. Each part contains 10% of
the total observations. Since nine points are required to divide the
data into 10 equal parts, we have nine deciles that are D1 to D9
Computation of Deciles for ungrouped data and
discrete series(after arranging the size of item in
ascending or descending order):-
DJ = size of J (N + 1) th item
10
Where,
J = 1, 2, 3, 4, 5, 6, 7, 8, 9,

24. Example: - from the following data calculate first and second Deciles.
Marks 5 15 25 35 45 55
No. of students 10 20 30 50 40 30
Solution:-
Step I: - Calculation of commutative Frequencies
Marks No. of students (f) Commutative frequency
C.f
5
15
25
35
45
55
10
20
30
50
40
30
10
30
60
110
150
180
N = 180

25. Step II: -
D1 = size of N + 1 th item
10
= Size of 180 + 1 th item = 181 = 18.1 th item
10 10
= size of 18.1 th item = 25, So D1 = 15
Step III: -
D2 = size of 2(N + 1) th item
10
D2 = 36.2 th item
D2 = 25

26. DJ = L + J N - P.c.f x i
Where,
10
f
J = 1, 2, 3, 4, 5, 6, 7, 8, 9,

27. The value of a variate which divides a given series or
distribution into 100 equal parts are known as percentiles.
Each percentile contains 1% of the total number of
observations. Since ninety nine points are required to divide
the data into 100 equal parts, we have 99 percentiles, P1 to P100

28. PJ = size of J (N + 1) th item
100
Where, J = 1 to 100
Computation of Percentiles for grouped data:-
PJ = L + J N - P.c.f x i
100
f
Where, J = 1 to 100

29. Example: - from the following data calculate Q1, D8 and P10.
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of students 10 20 30 50 40 30
Solution:-
Step I: -Calculation of commutative Frequencies
Marks No. of students
(f)
Commutative
frequency
C.f
0-10
10-20
20-30
30-40
40-50
50-60
10
20
30
50
40
30
10
30
60
110
150
180
N = 180
P10
Q1
D8

30. Step II: - Calculation of Q1
Q1 = L + N - P.c.f x i
N th item = 180 = 45th item
4 4
Q1 = 20 + 45 – 30 x 10 = 25
30
4
f

31. Step III: - Calculation of D8
D8 = L + 8N - P.c.f x i
8 N th item = 8 x 180 = 144th item
10 10
D8 = 40 + 144 – 110 x 10 = 48.5
40
10
f

32. Step IV: - Calculation of P10
P10 = L + 10N - P.c.f x i
10 N th item = 10 x 180 = 18th item
100 100
P10 = 10 + 18 – 10 x 10 = 14
20
100
f

33. Mode is often said to be that value in a series which occurs most
frequently or which has the greatest frequency. But it is not
exactly true for every frequency distribution. Rather it is that
value around which the items tend to concentrate most heavily.
Calculation of mode in case of ungrouped data
Example- Find the mode of the following series: 8, 9, 11, 15, 16,
12, 15, 3, 7, 15
Solution- There are ten observations in the series wherein the
figure 15 occurs maximum number of times three. The
mode is therefore 15.

34. In the case of grouped data, mode is determined by the following formula:
MO = L + Δ1 x i
Δ1 + Δ2
Where,
MO = Mode.
L = Lower limit of the modal class.
Δ1 = The difference between the frequency of the modal class and the frequency of the pre modal class.
Δ2 = The difference between the frequency of the modal class and the frequency of the post-modal class.
i = The size of the modal class

35. Example- calculate the modal sales of the 100 companies from the following data
Sales in Rs(lakhs) 58-60 60-62 62-64 64-66 66-68 68-70 70-72
No. of companies 12 18 25 30 10 3 2
Solution-
Sales in
Rs(lakhs)
Since the maximum frequency is 30 is in the class 64-66, therefore 64-66 is the
modal class
No. of companies
58-60 12
60-62 18
62-64 25
64-66 30
66-68 10
68-70 3
70-72 2
Modal class

36. Mode is determined by the following formula
MO = L + Δ1 x i
MO = 64 + 5 x 2
5 + 20
MO = 64.4
Δ1 + Δ2

37. Example: -
from the following data, calculate Mode:
Marks 5 10 11 12 13 14 15 16 18 20
No. of students 4 6 5 10 20 22 24 6 2 1

38. Solution:-
First we will do grouping of the above data with the help of
grouping table. A grouping table must have six columns.
Marks Column 1 Column 2 Column 3 Column 4 Column 5 Column 6
5 4 x x x
10 6 10 15 x
11 5 11 21
12 10 15 35
13 20 30
52
14 22
42
66
15
24
46
52
16 6 30 32
18 2 8 9 x
20 1 3 x x x

39. ANALYSIS TABLE
Column No. Marks
5 10 11 12 13 14 15 16 18 20
1 1
2 1 1
3 1 1
4 1 1 1
5 1 1 1
6 1 1 1
TOTAL 1 3 5 4 1
The highest total in the analysis table is five. The item
corresponding to it is 14. Therefore, the mode is 14

40. Mode can be also determined indirectly by applying the
following formula:
Merits of Mode
Mode = 3 median - 2 mean
 In many cases it can be found by inspection.
 It is not affected by extreme values.
 It can be calculated for distributions with open end classes.
 It can be located graphically.
 It can be used for qualitative data.

41. Demerits of Mode
 It is not based on all values.
 It is not capable of further mathematical treatment.
 It is much affected by sampling fluctuations.

42. Having discussed mean, median and mode, we now turn to the
relationship amongst these three measures of central tendency. We shall
discuss the relationship assuming that there is a unimodal frequency
distribution.
1-Symmetrical Distribution
When a distribution is symmetrical, the mean, median and mode are the
same as is shown below in the following figure.
Mean = median = mode

43. 2-Asymmetrical Distribution
Asymmetrical distributions are of following type
 Positively skewed
 Negatively skewed
 ‘L’ shaped positively skewed
 ‘J’ shaped negatively skewed
Positively skewed
Mean ˃ Median ˃ Mode

44. Negatively skewed
Mean ˂ Median ˂ Mode

45. ‘L’ Shaped Positively skewed
Mean ˂ Mode
Mean ˂ Median

46. ‘J’ Shaped Negatively Skewed
Mean ˃ Mode
Mean ˃ Median

47. Moderately Skewed /Asymmetrical Distribution
Under Peak of curve Divides area in
halves
Centre of Gravity
Mode Median Mean