BACHELOR'S DEGREE PROGRAMME
TermEnd Examination, June, 2011
ELECTIVE COURSE: COMMERCE
ECO7: ELEMENTS OF STATISTICS
Time: 2 hours Maximum Marks: 50
Note: There are three sections and all are compulsory.
SECTION  A
1. Fill in the blanks with appropriate word(s) given in brackets. 5x1=5
(a) Two dimensional diagram indicates the area. (Length/area).
(b) Percentages are units of analysis. (Collection/analysis).
(c) Statistics does not study individuals. (individuals/ aggregates).
(d) Mode can be determined with the help of a histogram. (Median/Mode).
(e) The median is also the 50th percentile. (median/mean).
2. State whether the statements given below are True or False. 5x1=5
(a) Statistics are either enumerated or estimated, but reasonable standard of accuracy must be maintained. True
(b) When figures are estimated/approximated to the next significant number, they give rise to compensating errors.F
(c) A frequency distribution with one variable is called univariate frequency distribution. True
(d) The sum of absolute deviations (i.e. deviations ignoring signs) from the median is the least. True
(e) Standard deviation is the positive square root of the variance. True
SECTION  B
Attempt any two of the following. 8+7=15
3. (a) Prepare a frequency distribution by inclusive method starting with 0  4:
6

20

37

32

17

22

18

12

16

40

3

30

38

18

20

5

6

36

30

32

2

36

6

31

15

18

40

32

26

38

Solution:
Class Interval

Tally Marks

Frequency

0 – 4
5 – 9
10 – 14
15 – 19
20 – 24
25 – 29
30 – 34
35 – 39
40 – 44

II
IIII
I
III
I
IIIII
IIII
II

2
4
1
6
3
1
6
5
2

N = 30

(b) Distinguish between a histogram and historigram.
Ans: Histogram: A histogram in another kind of graph that uses bars in its display. This type of graph is used with quantitative data. Ranges of values, called classes, are listed at the bottom, and the classes with greater frequencies have taller bars.
Historigram: A time series is an arrangement of statistical data in a chronological order. The graph of time series with time on xaxis and dependent variable on yaxis is called historigram. If the actual time series are graphed, the historigram is called absolute historigram. The graph obtained by plotting the index numbers of the given values is called index historigram.
4. (a) Estimate the value of arithmetic mean using the empirical relationship if mode is 30.6 and median is 28.4. 5+10=15
Solution: Empirical relationship between Mean, Median & Mode:
=> 3 Median = 2 Mean + Mode
=> 3 x 28.4 = 2 Mean + 30.6
=> 85.2 – 30.6 = 2 Mean
=> 54.6 = 2 Mean
=> Mean
(b) Calculate the value of mode for the following distribution:
X,midpoint :

30

50

70

90

110

130

150

170

190

f:

6

9

11

14

20

15

10

8

7

Solution: Calculation of Mode
X (Mid point)

F

C.I.
 
30
50
70
90
110
130
150
170
190

6
9
11
14 = f0
20 = f1
15 = f2
10
8
7

20 – 40
40 – 60
60 – 80
80 – 100
100 – 120
120 – 140
140 – 160
160 – 180
180 – 200

Modal Class = ( 100 120 )
Now,
Mode

5. (a) Co  efficient of variations of two distributions are 60% and 80%, their standard deviations are 20 and 16 respectively. What are their arithmetic means? 5+10=15
Solution: Coefficient of variation
.1st distribution (C.V)
.2nd distribution (C.V)
Now,
C.V. of 1st distribution
Again, C.V. of 2nd distribution
(b) Calculate Bowley's Co  efficient of skewness from the following data:
Sales No. of
(Rs. In Lakhs)

Companies

Below 50
50 – 60
60 – 80
80 100

8
12
20
25

Solution:
Sales

F

C.F.

40 – 50
50 – 60
60 – 80
80 – 100

8
12
20
25

8
20
40
65

Solution:
Again,
Again,
Again,
Bowley’s coefficient of Skewness
SECTION  C
6. Distinguish between any two of the following. 5+5=10
(a) Census and Sample Investigation.
Ans: Difference between Sample Investigation and Census Investigation:
Sample Investigation: It is a Investigation under which only a part taken out of the universe is investigated. It is not essential to investigate every individual item of the Universe.
Census Investigation and complete enumeration: Under Census Investigation detail information regarding every individual person or item of a given universe is collected.
Difference between Census and Sample Investigation:
The following are the differences between Census and Sample method of investigation:
(a) Under Census method, each and every individual item is investigated whereas under sample Investigation only a part of universe is investigated.
(b) There is no chance of sampling error in census Investigation whereas sampling error cannot be avoided under sample Investigation.
(c) Large number of enumerators is required in census whereas less number of enumerators is required in sample Investigation.
(d) Census Investigation is more time consuming and costly as compared to sample Investigation.
(e) Census Investigation is an old method and it less systematic than the sample Investigation.
(b) Exclusive and Inclusive class Intervals.
Ans: Difference between Exclusive and Inclusive method
Exclusive method: Under this method, the upper class limit is excluded but the lower class limit of a class is included in the interval. Thus an observation that is exactly equal to the upper class limit, according to the method, would not be included in that class but would be included in the next class. On the other hand, if it were equal to the lower class limit then it would be included in that class. For Example, In the class 800900 the upper class limit 800 is included and the lower class limit 900 is excluded.
Inclusive method: The Inclusive Method does not exclude the upper class limit in a class interval. It includes the upper class in a class. Thus both class limits are parts of the class interval. For Example, In the class 800–899, both upper class limit and the lower class limit 800 and 899 are included.
(c) Biased and Unbiased Errors.
Ans: Statistical errors are of two types: biased and unbiased.
Biased errors: These errors arise from any bias is selection, estimation, etc. For example, if in place of simple random sampling, deliberate sampling has been used in a particular case some bias is introduced in the results and hence such errors are called biased sampling errors.
Unbiased errors: These errors arrive due to chance differences between the members of population that are included in the sample and those not included. An error in statistics is the difference between value of a statistic and that of the corresponding parameter.
(d) 'Less than' and 'More than' ogives.
Ans: Less than Method: In the less than method we start with upper limits of class and go on adding the frequencies. When these frequencies are plotted we get a rising curve.
More than Method: Here, we start with lower limit and go on subtracting the frequencies of each class. When these frequencies are plotted a decreasing curve will be obtained.
The basic difference between the less than and more than type is that in less than ogives, frequencies are added starting from the upper limit of the 1st class interval of the frequency distribution. On the other hand, in case of more than ogives, frequencies are added starting from the lower limit of 1st class interval of the frequency distribution. Accordingly, while in case of less than Ogive the cumulative total tends to increase, in case of more than Ogive, the cumulative total tends to decrease.