BACHELOR'S DEGREE PROGRAMME
TermEnd Examination December, 2012
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) Median can be determined with the help of Ogive (Median /Mean).
(b) Primary data are called First hand data. (first hand / second hand)
(c) A bar signifies area (area / length)
(d)Coefficient of variation is always expressed in terms of percentage (variance / coefficient of variation).
(e) A discrete variable is the result of Counting. (measurement / counting).
2. State whether the statements given below are true or false. 5x1=5
(a) In a census investigation all the units of the population are investigated. True
(b) An exclusive frequency distribution has nonoverlapping class limits. False
(Note: Exclusive are overlapping and inclusive are nonoverlapping)
(c) Negative values cannot be presented using a simple bar diagram. False
(d) Mode of a set of data cannot be greater than arithmetic mean. False
(e) Standard deviation is always computed from mean. True
SECTION  B
Attempt any two of the following: 8+7
3. (a) Prepare a frequency distribution by exclusive method starting with 5  10:
6

21

5

20

33

10

10

30

45

7

19

37

17

20

19

28

11

20

30

26

20

43

15

5

37

36

27

26

12

19

Solution:
Frequency Distribution Table
Class Interval

Tally Marks

Frequency

5 – 10
10 – 15
15 – 20
20 – 25
25 – 30
30 – 35
35 – 40
40 – 45
45 – 50

I I I I
I I I I
I I I I
I I I I
I I I I
I I I
I I I
I
I

4
4
5
5
4
3
3
1
1

N = 30

(b) Explain the process of constructing a cumulative frequency graph.
Ans: Cumulative Frequency Graph (Ogive): A cumulative frequency graph, also known as an Ogive, is a curve showing the cumulative frequency for a given set of data. The cumulative frequency is plotted on the yaxis against the data which is on the xaxis for ungrouped data. When dealing with grouped data, the Ogive is formed by plotting the cumulative frequency against the upper boundary of the class. An Ogive is used to study the growth rate of data as it shows the accumulation of frequency and hence its growth rate.
Let’s see the steps used for its construction. We have to follow the given steps for the construction of Ogive curve.
Step 1: To construct the ogive curve first, we have to select two axis i.e. x and y axis.
Step 2: In the simple frequency curves the frequency is plotted alongside the class interval.
Step 3: In the ogive curve the cumulative frequency is plotted alongside the lower limit and upper limit. It depends on series which are cumulated.
For the construction of ogive curve two methods are defined which are shown below.
1. Less than method.
2. More than method.
If we plot the ogive curve with the help of less than method then we use following steps:
Step 1: We have to start from the upper limits of class intervals and then add class frequencies to get the cumulative frequency distribution.
Step 2: We have to take upper limit in the xaxis direction.
Step 3: After this we take cumulative frequencies along the Yaxis direction.
Step 4: Now put the points (pi, gi), where ‘pi’ is the upper limit of a class and ‘gi’ is corresponding cumulative frequency.
Step 5: At last join the points which are obtained in the graph to get the ogive.
If we plot the ogive curve with the help of more than method, then we have to use following steps:
Step 1: In this method, we start from the lower limits of class intervals and then the total frequency is subtracted from the frequency to get the cumulative frequency distribution.
Step 2: In second step, we take the lower class limits along to the X  axis direction in the graph.
Step 3: Now, we take the cumulative frequencies along Yaxis direction in the graph.
Step 4: Now, put the points (pi, gi), where ‘pi’ is the lower limit of a class and ‘gi’ is corresponding cumulative frequency.
Step 5: At last join the points which are obtained in the graph to get the ogive.
4. (a) Calculate the value of median for the following frequency distribution: 10+5
Class :

6062

6365

6668

6971

7274

Frequency :

17

39

48

34

12

Solution:
Class

Frequency

Class Boundaries

Cumulative Frequency

60 – 62
63 – 65
66 – 68
69 – 71
72 – 74

17
39
48
34
12

59.5 – 62.5
62.5 – 65.5
65.5 – 68.5
68.5 – 71.5
71.5 – 74.5

17
56
104
138
150

Median = Size of
Median Class = ( 65.5 – 68.5 )
Now,
M
(b) The mean weight of 150 students of a class is known to be 60 kg. The mean weight of the boys is 70 kg while the mean weight of girls is 55 kg. Estimate the number of boys and girls in the class.
Ans: Given Here,
Now,
Again,
5. (a) The following data relates to the goals scored by team A in a football season: 10+5
No. of Goals scored in a match :

0

1

2

3

4

No. of Match played :

1

9

7

5

3

For team B, the arrange number of goals scored per match was 2.5 with a standard deviation of 1.25 goals. Find which team is more consistent in its performance.
(b) For a given data, Q1=58, Md=59 and Q3=61. Find the coefficient of skewness.
Ans: Coefficient of Skewness
6. What is sampling? Explain briefly the various methods of sampling. 3, 12
Ans: Meaning of Sampling and Its Types
Sampling refers to the statistical process of selecting and studying the characteristics of a relatively small number of items from a relatively large population of such items,, to draw statistically valid inferences about the characteristics about the entire population.
There are two broad methods of sampling used by researchers, nonrandom (or judgment) sampling and random (or probability) sampling. In judgment sampling the researcher selects items to be drawn from the population based on his or her judgment about how well these items represent the whole population. The sample is thus based on someone’s knowledge about the population and the characteristics of individual items within it. The chance of an item being included in the sample is influenced by the characteristic of the item as judged by an expert selecting the item. A judgment sampling system is simple and less expensive to use. Also when there is very little known about the population under study a pilot study based on judgment sample is carried out to permit design of a more rigorous sampling system for a detailed study.
In random sampling, individual judgment plays no part in selection of sample. Each item in the sample stands equal chance of being included in the sample. In case of random sampling, the researcher is required to use specific statistical processes to ensure this equal probability of every item in the population. A random sampling system enables more reliable results of statistical analysis with measurable margins of errors and degree of confidence.
To improve the cost effectiveness of data collection and analysis, several variations of the random sampling are used by researchers. Some of the most common types of random sampling methods are
(1) simple random sampling,
(2) systematic sampling,
(3) stratified sampling, and
(4) cluster sampling.
Simple random sampling ensures that each possible sample has an equal probability of being selected, and each item in the entire population has an equal chance of being included in the sample.
In systematic sampling the items are selected from the population at a uniform interval defined in terms of time, order or space. For example an observation may be made every half an hour, or from a long queue of people every fourth person may be selected, or in a bunch of documents every tenth document may be selected.
In stratified sample the entire population is divided in relatively homogeneous group. For example all the students of a school may be divided in groups of boy and girls. Once this is done random sample from each of such groups is drawn independently. This approach is suitable when there ate identifiable subgroups exist within the population that differ significantly in respect of characteristic under study.
In cluster sampling the population is divided into groups or clusters, a sample of these clusters may be drawn. For example, a city may be divided in a cluster of small localities, and a sample of these localities may be drawn using random sampling methods. The all the households within each of the locality may be studied for the research. A research based on a well designed cluster sampling can often give better result than a research based on simple random sample with same time and cost of research.
SECTION  C
7. Distinguish between any two of the following: 5+5
(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.
(b) Primary and secondary data
Ans: Difference between Primary Data and Secondary Data:
(a) Primary data are those which are collected for the first time and thus original in character While Secondary data are those which are already collected by someone else.
(b) Primary data are in the form of rawmaterial, whereas Secondary data are in the form of finished products.
(c) Primary data are collected directly from the people related to enquiry while Secondary data are collected from published materials.
(d) Data are primary in the hands of institutions collecting it while they are secondary for all others.
(c) Discrete variable and continuous variable
Ans: Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable. Some examples will clarify the difference between discrete and continuous variables.
Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds.
Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.3 heads. Therefore, the number of heads must be a discrete variable.
(d) Dispersion and skewness
Ans: Dispersion: In statistics, the dispersion is the variation of a random variable or its probability distribution. It is a measure of how far the data points lie from the central value. To express this quantitatively, measures of dispersion are used in descriptive statistic. Variance, Standard Deviation, and Interquartile range are the most commonly used measures of dispersion.
Skewness: In statistics, skewness is a measure of asymmetry of the probability distributions. Skewness can be positive or negative, or in some cases nonexistent. It can also be considered as a measure of offset from the normal distribution. If the skewness is positive, then the bulk of the data points is centred to the left of the curve and the right tail is longer. If the skewness is negative, the bulk of the data points is centred towards the right of the curve and the left tail is rather long. If the skewness is zero, then the population is normally distributed.