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About Statistics:

Statistics is a subject that deals with

collection, tabulation, presentation,

analysis and interpretation of the

numerical data. Statistics assignment

are generally used in our day-to-day

life to make decisions that may be

related to our own business or job,

personal work, and/or for some other

reasons, where these decisions are mostly

taking under the conditions of

uncertainty. For example, a businessman,

who is having sufficient experience in his

own business, can estimate the sales of a

particular product on the forthcoming week based on the sales of the same product during

the previous months. In this example, the businessman using statistics for estimating the

sales of a product only based on his experience, but not based on any standard statistical

models. Similarly, the usage of statistics is widely increased in each and every field that

deals with qualitative or quantitative data or both. In particular, due to a phenomenal

growth in science and technology, the application of statistical theory and its methods are

highly increasing in various disciplines such as Actuarial Science, Biostatistics, Business

Analytics, Econometrics, Epidemiology, Reliability Engineering, and Statistical Finance.

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Statistics Homework Sample Questions & Answers

Illustration: 1

From the following data. Fit a trend line, and determine the trend values by the method of

semi average. Also, forecast the trend value for the year 2002.

Year

Output (in’

000 units)

1999

20

200

16

2001

24

2002

30

2003

28

2004

32

Solution (a) Computation of Semi Averages.

Year Outputs Semi totals Semi averages Semi average

points

1999

2000

2001

2002

2003

2004

20

16

24

30

28

32

60

90

20

30

𝑋 1

𝑋 2

With the above semi averages of 20 for 2000 and 30 for 2003 the trend line is fitted to the

given series as under:

Graphic Representation of the Trend Line by method of Semi averages.


(b) Computation of trend values by location on the trend line

Years :

Outputs (in

’000 units)

1999

15.67

2000

20

2001

23.33

2002

26.67

2003

30

2004

33.33

2005

36.67

Alternative

Computation of the trend values by the averages change

The average change is given by

𝑋 𝑐 = 𝑋 2− 𝑋 1

𝑁

= 30−20

3 = 3.33

Let the average of origin be 𝑋 1 i.e. 20.

Thus the trend values will be computed as under:


Year Time dvn.from the time of

origin

= 2000

Trend Values

T = 𝑿 𝟏 + 𝑿 𝟐− 𝑿 𝟏

𝑵 . x

(T)

1999

200

2001

2002

2003

2004

2005

-1

0

1

2

3

4

5

20 + 3.33(-1) – 16.67

20 + 3.33 (0) = 20.00

20 + 3.33 (1) = 23.33

20 + 3.33 (2) = 26.67

20 + 3.33 (3) = 30.00

20 + 3.33 (4) = 33.33

20 + 3.33 (5) = 36.67

From the above, it must be noticed that the required trend value for 2002 is 26.67.

Further, it must be seen that the trend values computed as above are the same as those

located from the trend line shown above.

Illustration 2.

(On odd number of data)

Fit a trend line to the following series by the method of semi averages. Also, determine the

trends values both by location and by computation of the average change.

Day :

Sales in

$:

Sun

125

Mon

130

Tue

135

Wed

110

Thu

105

Fri

110

Sat

115


Solution: Computation of the Semi Averages

Days Sales in $. Semi totals Semi averages Semi averages

points

Sun

Mon

Tue

Wed

Thu

Fri

Sat

125

130

135

110

105

110

115

390

330

130

110

𝑋 1

𝑋 2

With the above semi averages of 130 for Monday, and 110 for Friday, the trend line is fitted

to the given date as under:

Graphic Representation of the Trend Line by the method of Semi Average


(b) Computation of the Trend values

(i) By location on the trend line

Days :

Trend

value

S

135

M

130

T

125

W

120

TH

115

F

110

SA

105

(ii) By the average change

The average change 𝑋 𝑐 = 𝑋 2−𝑋 1

𝑁 =

110−130

𝑁 = -5

Here, N = 4 days, because there is a gap of 4 days between Friday and Monday

representing 𝑋 2 and 𝑋 1 respectively. Taking the average of origin to be 𝑋 2 i.e., 110. The

trend values are computed as under:


Days Time dvn. From the

average of origin i.e.

Friday

Trend values

Y = 𝑿 𝟐 + 𝑿 𝟐− 𝑿 𝟏

𝑵 .x

Sun

Mon

Tue

Wed

Thu

Fri

Sat

-5

-4

-3

-2

-1

0

1

110 + ( -5 × -5 ) = 135

110 + ( -5 × -4) = 130

110 + ( -5 × -3) =125

110 + (-5 × -2) =120

110 + (-5 × -1 ) = 115

110 + (-5 × -10 ) = 110

110 + (- 5 × 1) = 105

(iv) Moving Average Method

Under this method, the trend line is fitted to a series on the basis of its moving averages

which represent the trend value of the series. The technique of calculating the moving

averages have been discussed at length in the chapter entitled “Measures of different

groups of a set of figures are computed in a moving manner. Each group ends with the last

item, and at each advancing step the group is included in the next group is left aside, and

the first group begins with the first item, and the last group ends with the last item, and at

each advancing step the first item of the preceding group ends with the last item, and at

each advancing step the first item of the preceding group is left aside, and one more item

that succeeds the group is included in the next group to get the moving averages thereof.

Each group may consist of 2,3,4,5,6,7,9 or 12 numbers of items depending on the size of

the series and purpose of the study. Symbolically these may be computed as follows :

In case of 3 tons:

MA1 = 𝑋1+ 𝑋2+ 𝑋3

3 , 𝑀𝐴2 =

𝑋2+ 𝑋3+ 𝑋4

3

MA3 = 𝑋3+ 𝑋4+ 𝑋5

3 and so on

In case of 5 tons: M.A1 = (𝑋1+ 𝑋2𝑋3+ 𝑋4+ 𝑋5)

5 ,

MA2 = (𝑋2+ 𝑋3+𝑋4+ 𝑋5+ 𝑋6)

5, and son on

The total, and the average figure of each group are placed against the middle item of the

group in the adjacent column respectively. If the number of items in the groups is even,

then the total and average of groups are placed, in the first instance, just against the

middle points of the respective groups, and then the averages of the adjacent averages are

placed against the mid item of the respective groups. This process of averaging the


averages is called centering, which is very much done in case of even numberly moving

averages.

It may be noted that the greater the number of items taken in the groups for averaging, the

greater will be the blank of moving averages for the extreme items of a series. For example,

in case of 9 yearly moving averages, we will not have any average figures for the first 4,

and last 4 items of the series. This is because, in such cases, the averages figures are

placed only against the middle item i.e. the 5th item leaving aside the 4 items before, and 4

items after in all the groups. Hence, unless the series is abnormally a lengthy one, it is

advisable to find the moving averages of smaller groups consisting of 3, or 5 items only so

that we may cover the entire series by the moving averages leaving aside only one, or two

items at each of the extremes. Further, for these few blank spaces we can estimate the

average values with greater degree of accuracy with reference to the nature of increase, or

decrease in the adjacent average figures.


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