NIOS Economics (318) Notes/Answer| Chapter-7|Statistical Methods

NIOS Economics (318) Notes/Answer| Chapter-7|Statistical Methods. Important questions for NIOS Economics (318) Questions Answers brings you latest queries and solutions with accordance to the most recent pointers SOS . Students will clear all their doubts with regard to every chapter by active these necessary chapter queries and elaborate explanations that area unit provided by our specialists so as to assist you higher. These queries can facilitate students prepare well for the exams thanks to time constraint . NIOS Economics (318) Notes/Answer| Chapter-7|Statistical Methods

HS 2nd years Solutions (English Medium)

NIOS Economics (318) Notes/Answer| Chapter-7|Statistical Methods

Intext Question

1. Fill in the blanks with appropriate word from the brackets: 

(a) Ratio is calculated by ________ (dividing, multiplying) the _______ (first, second) term by the (first, second) term. 

Ans. dividing

(b) In a ratio the term what is compared is called ________ (first, second) term. 

Ans. first

(c) In a ratio the term with which it is to be compared is called _______ (first, second) term.

Ans. second

(d) When we say ‘ratio of the first term to the second term’ we describe the ratio in _________ (words, symbol, fraction).

Ans. words

(e) When we say ‘first term: second term’ we describe the ratio in __________ (words, symbol, fraction). 

Ans. symbol

(f) Monthly income of A and B are respectively Rs.2000 and Rs. 1000. Answer the following:

(i) Income of A is _______ (2, 1/2) times that of B.

Ans. 2

(ii) Income of B is _______ (2, 1/2) times that of A.

Ans. ½

(g) Ratio 70:80 is _________ (the same, not the same) as ratio of 80: 70.

Ans. not the same

2. State whether the following statement are true or false:

(a) Rate is calculated in the similar way as a ratio.

Ans. True

(b) Rate is expressed per unit only.

Ans. False

(c) Percentage is quite different from rate or ratio

Ans. False

(d)Percentage is calculated with base ‘per’ capital. 

Ans. False 

3. Answer the following:

  1. Per capita income of the year 1995 and 1996 is respectively Rs. 1000 and Rs. 1200 (i) Find out how much percent is per capita income of the year 1996 as compared to 1995. 
  2. Find the percentage increase in the year 1996.

Ans.(i) Per capita income of the year 1996× 100 / Per capita income of the year  

1200 × 100 / 1000=120

(ii) Percentage increase in the year 1996 = 120-100=20

(b) In a city with a population of 10 lakh, the number of births and deaths during the year were 11,500 and 10,500 respectively.

Find out the following:

  1. Birth rate per unit 
  2. Birth rate per 100
  3. Birth rate per 1000
  4. Death rate per unit
  5. Death rate per hundred
  6. Death rate per thousand

Ans. 

  1. Birth rate per unit 11500 / 1000000 = 0.0115
  2. Birth rate per 100 = 11500 x 100 / 1000000 
  3. Birth rate per 1000 = x 1000 = 11.5
  4. Death rate per unit == 0.0105
  5. Death rate per hundred = x 100 = 1.05
  6. Death rate per thousand = x 1000 = 10.5

4. State whether the following statements are true or false:

(a) An average is representative of a set of data.

Ans. True

(b) The average is used only for calculating average marks of a student.

Ans. False

(c) Per capita income is an average.

Ans. True

(d) National income of India is an average

Ans. False

5. Fill in the blanks:

(a) Arithmetic mean or = or 

Ans. No. of observations

(b) The sign” is read as ___ In statistics and it means… 

Ans. Sigma, addition or ‘sum of’ similar term

(c) The marks of a student in 5 subjects were 10, 12, 12, 14, 11. The arithmetic mean of these marks is______

Ans. Arithmetic mean or =

= 11.5

(d) The monthly salaries of 4 men were 800, 500, 1000, 1200. The arithmetic means of their salaries is _______

Ans. 875

6. Fill in the blanks:

(a) Taking ’10’ as the assumed mean in a set of numbers 12, 13, 10, 15 and 17 the arithmetic mean is ______.

Ans.

(b) A+, where

  1. stands for ________

Ans. Arithmetic mean

  1. A stands for ________

Ans. Assumed or guessed mean

  1. stands for ______

Ans. Sum of deviations taken from arithmetic mean

(iv) N stands for _______

Ans. Total number of items in the series

7. Fill in the blanks

Ans 6.6

(d) Out of 100 number 20 were 5’s; 30 were 6’s; 40 were 7’s and 10 were 8’s. The arithmetic mean of numbers.

Ans. 6.4

7. Fill in the blanks:

(a) The mid-value of limits_______

(i) 10-20 is _______

Ans. 15

(ii) 15-20 is _______

Ans. 17.5

(iii) 26-30 is _______

Ans.28

(b) Complete the following table

MarksNo.of solution (f) ( Mid point) Fx
5-102______________
10-153______________
15-205______________
Ef=Efx= 
MarksNo. Of students (f) (Mid-point) Fx
5-1027.515
10-15312.537.5
15-20517.587.5
Ef= 10Efx= 140

Note E = summation

(c) Give the formula for calculating arithmetic mean and insert the values from table of question No. 2 above.

8. A student’s final marks in Mathematics, English, Home Science and Economies are respectively 82, 86, 90, and 70. If respective credits received for these courses are 3, 5, and 1, complete the following:

Ans. = (3) (82)+(86)+(3)(90)+(1)(70) / 3+(5)+1

(b) 246+430+270+70 / _____

Ans 246+430+270+70 / 12

Ans.12

(c)____ / ____ = 84.66

Ans. 1016 / 12 = 84.66

Terminal Exercise

1.Explain the meaning of ‘Ratio’ with the help of an example.  

Ans. Ratio is the relation between two quantities. The two quantities are called terms. A ratio is found by dividing the first term by the second term.

In our example of two brands of pens we had two term, namely pen A priced at Rs. 6 and pen B priced at Rs.2. brand A priced at Rs. 6 is the first term because its price is intended to be compared with pen B. Brand B priced at Rs. 2 is the second term because it is in relation to the price of this pen that the brand A pen is intended to be compared. So ‘what is compared’ is the first term and ‘with which is to be compared’ is the second term.

2. Explain the significance of the first and second terms in calculation of ratio.

Ans. In our examples of pens the price of pen A is compared with the price of pen B. The point of interest was that how many times is the price of pen A higher than the price of pen B. Here the price of pen A was the first term, .e. one with which it is compared.

Suppose our point of interest shifts to pen B. We now want to know how many times the price of pen B is lower than the price of pen A. The Price of pen B is now the first term and the price of pen A is the second term. The value of ratio now is:

Rs.2: Rs. 6 = rs.2 / rs 6 = 1/3

The value of the ratio is now 1/3.It shows that pen B is priced 1/3(one third) of the price of pen A. Thus before attempting to calculate a ratio it is important to determine two things

  1. What is compared? (first term) and 
  2. with which it is compared (second term).If we interchange the two terms the value of ratio is reserved. Ratio of price of pen A to the price of pen B is 3 while the ratio of price of pen B to the price of pen A is 1/ 3 (the reserve of 3)

Sometimes it is said that the ratio between two quantities should be calculated only when both are expressed in the same units. It implies that both the first term and the second term must be expressed in the same unit of measurement like Rs.: R, s.. kilograms, metres: metres etc. For all practical purposes this rule is not observed in statistical calculations.

Whether both the terms are expressed in the same units or not, for a ratio to be meaningful it is necessary that both the terms must be related to each other. For example, take the value of the ratio of national income to population in a particular year. Here income is measured in rupees and population in numbers. Although the two terms are measured in different units, the value of the ratio gives us per capita income which is meaningful relation.

3. Explain the meaning of ‘rate’ by giving an example. 

Ans. In economics we often talk in terms of rates like rate of economic growth, rate of growth of population, birth rate, death rate, agriculture yield rates etc. When we see how these rates are calculated, we will find that the process of calculation is either the same or nearly the same. Let us take some examples to clarify what we have said. For example, take the rate of yield per hectare of a crop.

Rate of yield (in kg.) per hectare of a crop

 = Total production of crop (kgs) / Total area(hectares) undercrop

In the above example we find that the process of calculation is the same as that in ratio. Thus, rate and ratio are the same in this example. Here yields rate is nothing but the ratio of production to area during a particular year. Rate is thus a ratio between two magnitudes shown over a period of time.

4. In what respects rate’ is sometimes distinguished from ‘ratio’?

Ans. Although the method of calculation of rate and ratio is the same, the meaning conveyed by rate is somewhat different. Rate is a ratio between two magnitudes shown over a period of time. In our example above, the yield rate per hectare is the ratio of production of crop’ to ‘total area under the crop’ during the year.

Rate is different from ratio in another respect. Ratios are generally expressed per unit, while rate can be expressed per unit, per 100 units, even higher. In calculating rates any arbitrary figure can be taken as ‘base’ depending upon which base is more suitable for comparison. In economic studies we find that 100 is the most common base adopted.

5. Show with the help of an example the need to adopt a higher base rather than ‘per unit’ base in calculation of a ‘rate’.

Ans. Let us now explain that why is there a need to adopt an arbitrary base different from per unit base. The need for a base like 100 or 1000 or 10,000 arises because ‘the value of ratio per unit’ sometime is so small that it fails to convey the significance of the rate or ratio. We take an example to explain this point.

Suppose there is a town with a population of 1000,000. Further suppose that total number of births during the year 1995 in the town is 2340. Let us calculate the birth rate per unit of population.

Birth rate per unit of  population

= Number of births during the year 1995 / population

2340 / 100000 = 0.0234

The above calculation reveals that there is.0234 birth per unit of population. Here the value of ratio is so small that it may make the comparison difficult. Suppose during the next year, i.e. 1996 total number of births are 2520.

Birth rate per unit of population is = 2520/ 10000

= .0252

To say that birth rate per unit of population in the year 1996 is.0252 as compared to .0234 in the year 1995. It is very difficult to realise from these figures how higher is the birth rate is in 1996 as compared to 1995. There is, therefore, a need to raise the base.

Suppose we raise the base to 100. The two birth rates in the year 1995 and 1996 would be:

Birth rate per 100 in 1995

= Births during 1995  x100 / population

=x100=2.34

Birth rate per 100 in 1996=100=2.52 Comparison between the two birth rates is now more convenient. Still there are some difficulties in comparisons. First the two rates are so small that one may fail to realise the difference and may treat the differences as insignificant. Second, it is better to avoid population figures in fractions we have to round them. Rounding a fraction when it is small may significantly affect the rate. So to avoid fractions also it is necessary to raise the base. Let us raise the base to 1000. The two birth rates are now as follows:

Birth rate per 1000 in 1995=x1000=23.4 

Birth rate per 1000 in 1996-x1000=25.2

The difference between the two birth rates is now more clearly visible. Secondly the fraction can be avoided by rounding 23.4 to 23 and 25.2 to 25 without significantly affecting the absolute difference between the two rates. This is why in practical statistics birth rates are calculated per 1000 of population.

6. Show with the help of an example how is percentage calculated? How it is distinguished from rate?

Ans. By now it must be clear to you that percentage is a type of rate or ratio with base 100. Every ratio per unit when multiplied by 100 is converted into percentage. It is calculated as follows:

Percentage=x100

Let us come back to the example of pen A and B pen respectively priced Rs. 6 and Rs.2. If the period of interest is that how much percent is the price of A in relation to the price of pen B, we calculate the percentage x100-100-300%

According to the calculation, the price of pen is 300 percent of the price of pen B. The main step in the calculation of percentage is the same as in ratio plus an additional step. The additional step is to multiply the ratio by 100. The reason behind taking the additional step of raising the base from ‘per unit’ to ‘per 100’ is already discussed in the paragraphs on rates.

Some of the important percentages used in economics are : (1) percentage rate of economic growth (2) percentage rate of interest (3) percentage rate of tax (4) percentage rate of capital formation etc.

7. What is arithmetic mean? Explain the method of calculating in case of ungrouped frequency data through both direct and indirect methods.

Ans. Arithmetic mean is the most commonly used average or measure of the central tendency applicable only in case of quantitative data; it is also simply called the “mean”. Arithmetic mean is defined as:

“Arithmetic mean is a quotient of sum of the given values and number of the given values”. 

1. Direct Method or Long Method (Ungrouped Data): In this method the mean is calculated directly from the given

series. In this method we can calculate mean from the ungrouped data and the formula for calculating mean from un grouped data.

The formula for calculate mean from ungrouped data is:

Or = = =

Where, = Mean

n=No of items

X = Score

“=Sum total of

Example:

A random sample of 10 boys had the following intelligence quotients (I.Q’s). 70, 120, 110, 101, 88, 83, 95, 98, 105, 100 Find the mean I.Q.

Solution:

In this example, the ungrouped data ranges from 70 to 120. Therefore, 95 a neat round value in the middle of 70 and 120, may be taken as assumed mean, i.e., A.M. = 95.

Deviations and sum of deviations needed in formula (03) may be calculated in a table given below:

Or = = =

Where, = Mean

n=No of items

X = Score

“=Sum total of

Example:

Calculate the mean of following data. Marks obtained by 6 students given: 20, 15, 23, 22, 25, 20.

Solution:

= or = 

= = = 20.83

NIOS Class 12th Economics (318) Notes/Question Answer

Chapter Chapters NameLink
Chapter 1Economy and Its ProcessClick Here
Chapter 2Basic Problems of an EconomyClick Here
Chapter 3Economic Development and Indian EconomyClick Here
Chapter 4Statistics: Meaning and ScopeClick Here
Chapter 5Making Statistical Data MeaningfulClick Here
Chapter 6Presentation of Statistical DataClick Here
Chapter 7Statistical MethodsClick Here
Chapter 8Index Numbers (Meanings and Its Construction)Click Here
Chapter 9Index Numbers (Problem and Uses)Click Here
Chapter 10Income FlowsClick Here
Chapter 11National Income: ConceptsClick Here
Chapter 12National Income: MeasurementClick Here
Chapter 13Uses of National Income EstimatesClick Here
Chapter 14What micro EconomicsClick Here
Chapter 15What affects demandClick Here
Chapter 16What affects supplyClick Here
Chapter 17Price determinationClick Here
Chapter 18CostClick Here
Chapter 19RevenueClick Here
Chapter 20Profit maximizationClick Here
Chapter 21Government budgetingClick Here
Chapter 22Money supply and its regulationClick Here
Chapter 23Need for planning in IndiaClick Here
Chapter 24Achievements of planning in IndiaClick Here
Chapter 25Recent economic reforms and the role of planningClick Here

Optical Module – I

Chapter 26AgricultureClick Here
Chapter 27IndustryClick Here
Chapter 28Independence of Agriculture and IndustryClick Here
Chapter 29Transport and CommunicationClick Here
Chapter 30EnergyClick Here
Chapter 31Financial InstitutionsClick Here
Chapter 32Social Infrastructure (Housing, Health and Education)Click Here

Optical Module – II

Chapter 33Direction and composition of India’s Foreign tradeClick Here
Chapter 34Foreign exchange rateClick Here
Chapter 35Balance of trade and balance of paymentsClick Here
Chapter 36Inflow of capital (Foreign Capital and Foreign Aid)Click Here
Chapter 37New trade policy and its implicationsClick Here
Chapter 38Population and economic developmentClick Here
Chapter 39Population of IndiaClick Here

2. Indirect method (Ungrouped Data):

By choosing an assumed mean and calculating deviations of the given varieties or observations from it makes the calculation of mean simpler. This average is usually chosen to be a neat round number in the middle of the range of the observations, so that deviations can be easily obtained by subtraction.

Then, a formula, based on deviations from assumed mean, for calculating arithmetic mean becomes:

=A+ 

Where, A. = Assumed Mean,

= Sum total of,

dx = The deviation of each value of the variable from the Assumed Mean,

N=Total Nos. Example:

A random sample of 10 boys had the following intelligence quotients (I.Q’s). 70, 120, 110, 101, 88, 83, 95, 98, 105, 100 Find the mean I.Q.

SL.no.I.Q (X) Deviation A=95 (X-A) =d
170-25
212025
311015
41016
588-7
683-12
7950
8983
910510
101005
Σdx = 20

Solution:

In this example, the ungrouped data ranges from 70 to 120. Therefore, 95 a neat round value in the middle of 70 and 120, may be taken as assumed mean, i.e., A.M. = 95. Deviations and sum of deviations needed in formula (03) may be calculated in a table given below:

XDeviation (X-A) =dDx
5050-55-5
5252-55-3
5555-550
6060-555
6565-5510
Σdx =7

8. Calculate the arithmetic mean marks of 5 students in Economics who scored 50, 52, 55, 60, and 65 mark by:

(a) direct method and (b) indirect method.

=050+52+55+60+65 / 5

= 56.4

(b)

9. The annual salaries of four men were Rs. 5000, Rs. 6000, Rs.6500 and Rs. 30000. 

  1. Find the arithmetic mean of their salary.
  2. Is this average typical of their salaries?
Marks xNo.of students FFxDx=X-A where A = 40fdx f x dx
208160-20-160
3012360-10-120
402080000
501050010100
60636020120
70428030120
Σf=60Σfx=2460Σfdx=60

Ans 

Rs(5000+6000+6500+30000) /4

= 47500/4

=Rs11,875

10. Following are the marks obtained by 60 students of a class. Calculated the arithmetic mean by (a) direct and (b) indirect methods.

Marks in Economic (out of 100) 20, 30, 40, 50, 60, 70 

No. of students 8, 12, 20, 10, 6, 4

Ans.

Same as above

= 41 marks

  1. = A+ =40+=41

11. How is arithmetic mean of a grouped frequency distribution calculated? Explain both direct and indirect methods.

Ans. A frequency distribution classifies the data into groups. In such case also there are two methods: (i)Direct method (ii) Indirect method.

(i) Direct Method:

There is an additional step in this method in comparison to frequency array case. To get the value of ‘fx’ we multiply x with f. X, in case of frequency distribution, is a group and not an individual item. It is not possible to multiply the group (say 1-3) with f. So we take first the mid value of the group and then multiply with ‘f’. The mid value is obtained by taking simple average of lower () and upper limits of the group. For example:

Mid value of group (1-3)==2 Now this mid value of group is taken as x. All other steps are the same as in case of frequency array. The main steps are:

  1. Take the mid value of each group as the value of x.
  2. Multiply x with to obtain fx. 
  3. Take t4he sum of fx to obtain “fx.
  4. Divide 4″fx by “f to get 

(ii) Indirect Method:

There are two versions of indirect method. In one version we take assumed mean as an additional step. In another version we take a further additional step in form of step deviation. The two versions are explained below.

(I) Based on assumed mean

The main steps are:

  1. Take mid value of (x) of each group. 
  2. Take some value of x as assumed mean (A).
  3. Deduct A from x to get dx.
  4. Multiply dx by f to get fdx.
  5. Take sum of all value of fdx to obtain “fdx.
  6. Divide “fdx by “f to get .s A+
(ii) Step Deviation Method

This is another version of indirect method. In this we take an additional step to make the calculation easy. The step is called step deviation. If the value of x is high,value of dx=(X A) is also likely to be high. To make the calculation simple we first find a common figure by which all the values of dx can be divided. It will reduce the value of dx and make further calculation easy. This common factor by which the values of dx are divided is termed as ‘i’. At a latter stage value of dx are again multiplied by this common factor so that final result of arithmetic mean is not affected. The main steps are :

  1.  Take mid value of group (x).
  2. Take some assumed mean (A).
  3. subtract A from X to get dx.
  4. Take a common factor (i) from amongst the values of dx and divide each value of dx.
  5. Multiply dxi by fto get fdxi.
  6. Take the sum of fdxi to get “f dxi.
  7. Apply the following formula to obtain arithmetic mean =A+x i

12. Following are the marks of student in History. Calculate the arithmetic mean marks by: (a) Direct method and (b) Indirect methods. (take 35 as assumed mean)

=33 marks

  1. =A+=35+ = 33 marks
Marks in historyNo.of students
0-105
10-2010
20-3025
30-4030
40-5020
50-6010

Ans. 

MarksMid value xNo.of students FFxDx=X-A where A=35Fdx (f x dx)
0-105525-30-150
10-201510150-20-200
20-302525625-10-250
30-403530105000
40-50452090010200
50-60551055020200
Σf=100Σfx=3300Σfdx=-200

13. Outline the need for a weighted arithmetic mean. How is it calculated?

Ans. While calculating the arithmetic mean we have given equal emphasis to each item in the series. This equal emphasis may be quite misleading if individual items have different importance, as in the following example:

Example: supposing a shopkeepers sells, say, brand A pens for Rs.5, brand B pens for Rs. 15 and brand C pens for Rs. 25 each.

Here, N = 3 and =Rs. 15.00

In this example we find that some pens are very cheap and others costly. In addition, the shopkeeper may sell different quantities of different brands. For example, he may sell 100 pens of brand A, 40 pens of brand B and 20 pens of brand C only. As such different brands have different relative importance. In statistics this importance is known as weights (w).

It may be noted that quantities in numbers or kilograms or any other unit, is not the only basis of assigning weights. We can use other methods also namely, value weight. It combines both quantity as well as price. That is price (p) multiplied by quantity (q) gives us value (pq).

In order to find out weighted arithmetic mean the following steps should be taken.

First, multiply each quantity (x) by its weight (w) to obtain

different products (wx),i.e. Second, all these products are added to get Twx, i.e.

TMWX =

Third, this sum of products (TMwx) is then divided by the sum of the weights (TMw) to obtain the required weighted arithmetic mean. Thus,

Weighted arithmetic mean w

Let us now apply this result to our example of three brands of pens. 

Using the formula for weighted arithmetic mean we get,

14. Calculate the weighted mean of Number 12, 29, 14, 41

Brand of pensPrice (x)Quantity (q)Wx
A5100500
B1540600
C2520
Σw=17
500
ΣWx=1600

Weight 6,4,5,2

Ans

XWWx
12672
294116
14570
41282
Σw=17ΣWx=340

15. State the advantages and disadvantages of arithmetic mean as a measure of central tendency.

Ans. It has the following advantages: 

  1.  It is easily understandable.
  2. It is easily and exactly computable.
  3. It is suitable for algebraic treatment. 
  4. It remains free from sampling fluctuations.
  5. Its calculation involves all the values of the given variable
  6. It can rigidly be defined.
  7. It is safely usable and expressible in all situations. However, arithmetic mean is not free from any

disadvantages. Some of them are mentioned below: 

  1. The arithmetic mean from a set of given observations is not identifiable only through observations.
  2. It is not correctly computable when a single observation is missing or rejected from the given data set. 
  3. It can never be calculated correctly unless the entire observations on the given variable are supplied at a time.
  4. The arithmetic mean attaches greater importance to bigger items while smaller items receive lesser attention.
  5. It is to be kept in mind that it is not a locational average as such. It needs only mathe-matical calculations.
  6. Finally, the simple arithmetic mean suggests only a numerical figure and nothing else.

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