Bhubaneswar. STUDY MATERIAL 2012- 13. CLASS XI. ECONOMICS ...
Developing projects in Economics. 5. 50 ... Current Challenges Facing Indian
Economy.
KENDRIYA VIDYALAYA SANGATHAN
ZONAL INSTITUTE OF EDUCATION AND TRAINING Bhubaneswar STUDY MATERIAL 2012- 13 CLASS XI
ECONOMICS PATRON : Ms Usha Aswath Iyer MATERIAL PRODUCTION: Mr. Parsuram Shukla
[1]
STUDY MATERIAL ECONOMICS
CLASS. XI
For : 2012 - 2013 Syllabus: Units
Marks
Part. A. Statistics for Economics 1.
Introduction
03
2.
Collection, organization & presentation of Data
12
3.
Statistical Tools and interpretation
30
4.
Developing projects in Economics.
5 50
Part. B
Indian Economic Development
5.
Development Policies & Experiences (1947-90)
10
6.
Economic Reforms since 1991
08
7.
Current Challenges Facing Indian Economy
25
8.
Development Experience of India-A Comparison with neighbours
07 50
Total Marks : 50 + 50 = 100
[2]
UNIT – 3
STATISTICAL TOOLS AND INTERPRETATION :-
1. Measures of central Tendency. 2. Measures of Dispersion 3. Co-relation and Measures of correlation 4. Introduction to Index Numbers.
Measures of Central Tendency (Important Terms & Concepts) :1. Average or measures of central Tendency :- It is a value which is a typical or representative of a set of data. Averages are also called measures of central tendency, since they tend to lie centrally, with in a set of data arranged according to magnitude. 2. Functions of Average :
Average helps to get a representative value to the entire set of data.
It Facilitates Comparison.
It is a useful tool in decision- making.
3. Essentials of a Good Average / good measures of Central Tendency :
Simplicity in calculation
Easy to understand
Rigidly Defined
Precise Value
Based upon all observations
Unaffected by extreme values.
Capable for further statistical calculations.
4. Types of Averages / measures of central Tendency :- There are three averages, which are in common use- Arithmetic Mean, Median and Mode. [3]
5. Arithmetic Means ( X ):It is the most common type of measures of central tendency. It is obtained by dividing the sum of all observations in a series by the total numbers of observations. 6. Calculation of Arithmetic Mean :For Individual series / ungrouped Data :(i)
X
X N
Direct Method
(ii) X A
d
(iii) X A
d' i
N N
Assumed Mean Method / Short Cut Method. (step Deviation method.)
For Discrete and Continuous Series / Grouped Data: (i)
X
fx or fm f f
(ii) X A
fd
(iii) X A
fd ' i
N
N
( Direct method ) (Assumed Mean Method / Short-cut method.) (step deviation method)
7. Mathematical Properties Of Arithmetic Mean :The algebraic sum of deviations of items from arithmetic mean X is alwayss Zero, i.e.
X – X 0 .
The Sum of the squared deviations of the item from A.M. is minimum i.e. 2 X – X X – A 2
.
8. Merits of Arithmetic Mean:(i) (ii) (iii) (iv)
It is easy to calculate and simple to understand. It is rigidly defined. It is a calculated value not a positional value. It is based on all observations. [4]
9. Demerits of Arithmetic Mean: It is affected by presence of extreme values It cannot be calculated in open-ended series It cannot be ascertained graphically. It sometimes gives misleading and surprising results. 10. MEDIAN:It is defined as the middle value of the series when the data is arranged in ascending or descending order. In other words, median is that value of the distribution which divides the group into two equal parts, one part comprising all greater values and the other comprising values less than the median. 11. Calculation of median:For Individual and Discrete series : N 1 M Size of 2
th
item.
For Continuous series:N Median item Size of 2
th
item
N – c. f M l1 2 i F
12. Merits of median : It is easy to understand and easy to compute. It is not unduly affected by extreme observations Median can be located graphically with the help of ogives. It is the most appropriate average in case of open-ended classes. It is the most suitable average for qualitative measurement such as intelligence, beauty etc. It is a positional value and not a calculated value. [5]
13. Demerits of Median (I) It is not based on all observations of the series since it is a positional average. (II) It requires arrangement of data, but other averages do not need it. (III) It can not be computed exactly where the number of items in a series is even. 14. Related Measures Of Median- Quartiles & Percentiles:Quartiles are the measures which divide the data into four equal parts, each part contains equal number of observations. There are three quartiles - Q 1, Q2, and Q3. The 1st quartile denoted by Q 1 is called lower quartile and 25% of the item of the distribution are below it and 75% of the items are greater than it. The second quartile is known as median and is denoted by Q 2. It has 50% of the item above it. The third Quartile is known as Q 3 and it is also called upper Quartile and 75% of the items are below it and 25% of the items are above it. Thus , Q 1 and Q3 denote the two items with which central 50% of item lie. Percentiles divide the series into hundred equal parts. For any series, there are 99 percentiles denoted by P 1, P2, P3,……, P99. P50 is the median value. 15. Calculation of Quartiles and percentiles For individual and discrete series: N 1 Q1=Size of 4
th
item.
N 1 Q3= Size of 3 4
th
item.
For continuous series :N Q1= Size of 4
th
item
N – c. f Q1 = L1 4 i f
N Q3=Size of 3 4
Q3 = L1 +
th
item
N 3 – c. f 4 i f [6]
16 MODE (Z) :It is defined as the value which occurred most frequently in a series. In other words, it is the value which has highest Frequency in a distribution. For example : Mode in the series: 20,21,23,23,23,23,25,26,26 would be 23 as this value occurs most Frequently than any other value. There is greatest concentration of items around this value. 17. CALCULATION OF MODE :For Individual Series :(i) To identify the value that occurs most frequently in a series. (ii) By conversion into discrete series and then identify the value corresponding to which there is highest frequency. For Discrete Series:(i) By Inspection method. (ii) Grouping method.; By preparing grouping table and then preparing analysis table. For Continuous Series :(I) Determination of modal class interval by inspection method or grouping table and analysis table. (II) Applying the Formula: Z L1
f1 – f 0 i 2 f1 – f 0 – f 2
Where, l1= Lower limit of modal class. F1= frequency of the modal class. F2= frequency of the class succeeding modal class. F0= frequency of the class preceding modal class. i-= size of the class interval.
[7]
18. Merits of mode :(i) It is easy to understand and simple to calculate (ii) It is not affected by the presence of extreme values . (iii) It can be located graphically with the help of histogram. (iv) It can be easily calculated in case of open-ended classes. 19. Demerits of mode:(i) It is not rigidly defined. (ii) When frequencies of all items are identical, It is difficult to identify the Modal Value. (iii) It is not based on all observations. (iv) Mode is not capable of further algebraic treatment. 20. Relative position of mean, median and mode:The relative position of X , M and Z depends upon the shape of the frequency distribution which is discussed below. (i)
In case of symmetrical distribution, mean median and mode are identical i.e. XMZ.
(ii)
In a moderately asymmetrical (skewed) distribution, mean median and mode are not equal, i.e. X M Z .
(a)
When the distribution is positively skewed, i.e. skewed to the right, then XMZ.
(b)
When the distribution is negatively skewed, i.e. skewed to the left, then XMZ.
Note :- The median (m) always lie between arithmetic mean X and mode (Z). 21. Empirical Relationship between x,m and z: In a moderately asymmetrical distribution, the values of mean, median and mode are observed to have the following relationship : Mode = 3 median – 2 mean. [8]
22. Questions with Answer 1
Mark Questions :-
1. Define an average? Ans.
An average is a single value that represents the whole group.
2. Name the measures of central tendency? Ans.
Three important types of statistical averages are :Arithmetic Mean, median and mode.
3. What is median? Ans.
It is defined as the middle value of the series when arranged either in ascending order or in descending order.
4. What is mode ? Ans.
It is defined as the value which occurs most frequently in a series.
5. Can mode be graphically located ? Ans :
Mode can be located graphically with the help of histogram
6. Average daily wage of 50 workers of a factory was Rs. 200. Each worker is given a raise of Rs 20. What is the new average daily wage ? Ans.
Increase is wages of each worker=Rs20. Total increase in wages = 50x20= Rs 1000. Total wages before increase in wages = 50x200=Rs 10,000. Total wages after increase in wages= 10,000+1000= Rs 11000. New average wages
X 11,000 Rs. 220 N
50
Thus mean wage will increase by Rs 20. 7.
What relationship exists between mean, median and mode in case of a symmetrical distribution ?
Ans.
In a symmetrical distribution, X M Z [9]
8.
What relationship exists between X M and Z in moderately negativee swewed distribution ?
Ans.
In a moderately negative skewed distribution. X M Z . 3 / 4 marks Questions : -
9.
Explain the characteristics of a good average?
Ans.
Characteristics of a good average/measures of central tendency
(i) It should be easy to understand. (ii) It should be simple to compute. (iii) It should be rigidly (well) defined. (iv) It should be based on all the observations. (v) It should not be unduly affected by the extreme values. (vi) It should be capable of further algebraic treatment. 10.
“Arithmetic mean is affected by very large and very small values but median and mode are not affected by them.” Explain.
Ans.
Median is the value of the middle item of a series arranged in ascending or in descending order of magnitude. Mode only takes values at the points around which the items tend to be most heavily concentrated. Arithmetic mean takes into account the value of all items (i.e. very large and very small) in a series. Thus it is only the arithmetic mean which is affected by extreme values in the series.
11.
Which average would be suitable in the following cases? (i) Average size of readymade garments. (ii) Average intelligence of students in a class. (iii) Average Production in a factory per shift. (iv) Average wages in an industrial concern. (v) When the sum of absolute deviations from average is least. (vi) In case of open-ended frequency distribution
Ans. (i) Mode, (ii) Median, (iii) Mean(iv) Mean, (V) Median, (vi) Median or mode. [ 10 ]
23. Some Numerical Questions : 1. Calculate Arithmetic mean from the following data using direct and short cut method / Assumed mean method: Size:
10
20
30
40
50
60
7
8
12
15
5
3
Frequency
2. Calculate Arithmetic mean from the following data using step- deviation method:Size
20-29 30-39
Frequency
10
40-49
8
50-59
6
60-69
4
2
3. Find median of the following observations: 20,15,25,28,18, 16,30 4. Calculate median of the following data. Marks : No of Students:
11-15 16-20 21-25 26-30 31-35 7
10
13
26
36-40 41-45 46-50
35
22
11
5
5. Calculate Q1 and Q3 From the following data Marks:
10
20
30
40
50
60
No of students:
4
10
20
8
6
3
6. Calculate the value of median and Q 1 from the following data : Marks:
0-10
No of Students:
10-20
5
20-30
8
10
30-40
40-50
4
3
7. Calculate the mode of the following data: 4,6,5,7,9,8,10,4,7,6,5,8,7,7,9 8. Calculate mode from the following data Marks: No of Students :
0-10 2
10-20
20-30 30-40
5
8 [ 11 ]
10
40-50 8
50-60 5
60-70 2
MEASURES OF DISPERSION :1. Dispersion refers to the variation of the items around an average. According to Dr Bowley : “Dispersion is the measure of variations of items.” To quote CONNOR : - “Dispersion is a measure of the extent to which the individual items vary.” 2. Objectives of Dispersion : (i) To determine the reliability of an average (ii) To compare the variability of two or more series (iii) It serves the basis of other statistical measures such as correlation etc. (iv) It serves the basis of statistical quality control. 3. Properties of a good measure of dispersion: (i) It should be easy to understand. (ii) It should be simple to calculate (iii) It should be uniquely defined. (iv) It should be based on all observations. (v) It should not be unduly affected by extreme items. 4. Measures of dispersion may be either absolute or relative. Absolute measures of dispersion are expressed in the some units in which data of the series are expressed i.e., rupees kgs, tons etc. where as relative measures of dispersion are independent of the units of measurement. They are expressed in percentage these are used to compare two or more series which are expressed in different units. 5. Absolute measures of dispersion are:(i) Range (ii) Quartile Deviation (iii) Mean Deviation (iv) Standard deviation and variance. [ 12 ]
6. Relative measures of dispersion are:(i) Coefficient of Range. (ii) Coefficient of Quartile Deviation (iii) Coefficient of mean Deviation (iv) Coefficient of standard Deviation (v) Coefficient of variation 7. Besides the above measures of dispersion there is a graphic method of studying dispersion, known as Lorenz curve. 8. Range is the simplest measure of dispersion : It is the difference between the largest and smallest value of the distribution. Computation of range:- It is calculated as Range = L - S Coefficient of Range =
L–S . LS
9. Merits of Range :(i) It is simple to understand and easy to calculate (ii) It is widely used in statistical quality control. 10. Demerits of Range= (i) It is affected by extreme values in the series. (ii) It can not be calculated in case of open-ended series. (iii) It is not based on all the items of the series. 11.Inter quartile range and quartile deviation are another measures of dispersion. Inter-quartile range is the difference between the upper quartile Q 3 and lower quartile Q1. Quartile deviation is half of the difference between the upper quartile and lower quartile i.e. half of the inter-quartile range. Computation of Inter-quartile Range and Quartile Deviation:Inter- quartile Range:- Q 3 – Q1 [ 13 ]
Quartile Deviation (Q.D) : =
Q3 – Q1 2
Q3 – Q1
Co. efficient of Q.D : = Q Q 3 1 12. Merits of Quartile Deviation:(i) It is easy to compute. (ii) It is less affected by extreme items (iii) It can be computed in open-ended series. 13. Demerits of Quartile Deviation :(i) It ignores half i.e. 50% of the items. (ii) It is useful only for rough study. (iii) It is not based on all observations. 14. Mean deviation :- It is defined as the arithmetic average of the absolute deviations (ignoring signs) of the various items from a measure of central tendency ; i.e. mean or median. Generally, mean deviation is calculated from median because the sum of the absolute deviations taken from median is minimum or least. 15. Computation of mean Deviation :- It is computed as:Individual series /ungrouped data:MD
D N
Discrete/Continuous series :- MD
f D N
MD
Coefficient of M.D. = MD X or M 16. Merits of mean Deviation:(i) It is based on all observations. (ii) It is least affected by extreme items. (iii) It is simple to understand and easy to calculate. [ 14 ]
17. Demerits of mean Deviation:(i) It ignores ± signs in deviations (ii) It can not be computed with open - ended series. (iii) It is not well defined measure because it is calculated from different averages (Mean, Median & Mode) (iv) It is difficult to compute when X or M comes in fractions. 18. Standard deviation =- It is the most widely used measure of dispersion. It is defined as the positive square root of the arithmetic average of the squares of deviations taken from the mean. Variance is another measure of dispersion. It 2 = 2 i
s
t
h
e
s
q
u
a
r
e
o
f
t
h
e
s
t
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
i
.
e
,
V
a
r
i
a
n
c
e
=
(
S
Computation of standard Deviation: It is computed as:For Individual series/un-grouped Data:(i)
X – X
(ii)
d 2 – d
(iii)
d '2 – d '
(iv)
X 2 – X
2
Actual mean method
N 2
Assumed mean method
n
n
2
i
n
n
2
n Or
n
Step deviation method
X 2 – X
2
n
Direct Method/Actual Mean method For Discrete continuous series:-
F X – X
2
(i)
n
Actual mean method
[ 15 ]
.
D
.
)
(ii) Assumed Mean Method
f d 2 – fd
n
n
2
(iii) Step - deviation Method
f d '2 – fd '
n
n
2
C
(iv) Direct Method.
f X 2 – fX
n
n
2
19. The important properties of standard deviation are:(i) The standard deviation of 1st n natural number is given by
1 n2– 1 12
(ii) The standard deviation is computed from A.M. because the sum of squares of the deviations taken from the A.M. is least. (iii) If a constant ‘a’ is added or subtracted from each item of a series then S.D. remains unaffected i.e. S.D. is independent of the change of origin. (iv) If each item of a series is multiplied or divided by a constant ‘a’ the S.D. is affected by the same constant i.e. S.D. is affected by change of scale. 20. Merits of standard Deviation :(i) It is rigidly defined. (ii) It is based on all observations where as range and quartile- deviations are not based on all items. (iii) It takes algebraic signs in consideration where as these are ignored in meanDeviation. (iv) It can be algebraically manipulated , i.e. we can find the combined S.D. of two or more series. (v) It serves the basis of other measures like correlation etc. [ 16 ]
21. Demerits of standard deviation:(i) As compared to range and quartile deviation, it is difficult to understand and compute. (ii) It gives more importance to extreme items. 22.
Coefficient of variation is a relative measure of dispersion. It is used in comparing the variability of two or more series. Computation of coefficient of variation:- It is computed as: Coefficient of variation (C.V.)=
23.
X
100
Lorenz curve:- It is a graphical method of measuring dispersion. It has great utility in the study of degree of inequality in the distribution of income and wealth between the countries. It is also useful for comparing the distribution of wages, profits etc over different business groups. It is a cumulative percentage curve in which the percentage of frequency (persons or workers) is combined with the percentage of other items such as income, profits, wages etc.
Selected Questions 1. What do you mean by dispersion ? 2. What is range? 3. What is meant by quartile deviation? 4. What do you mean by mean deviation? 5. What do you mean by standard deviation? 6. What is variance ? 7. What is relative measure of dispersion? 8. What is coefficient of variation?” 9. What is a Lorenz curve? 10. If Q1=41, Q3=49, find the value of coefficient of Quartile deviation. 11. Name the important absolute and relative measures of dispersion. 12. Why standard deviation is measured from the mean?” 13. Find out the standard deviation, if variance is, 1444 ? 14. Write the formula of calculating mean deviation from mean. [ 17 ]
15. Distinguish between absolute and relative measures of dispersion. 16. Name the various measures of dispersion. Explain the merits and demerits of any two . 17. From the following data, calculate range and coefficient of range. Marks:
10
20
30
40
50
60
70
No of students
8
12
7
30
10
5
2
18. Calculate quartile deviation and coefficient of quartile deviation from the data given below. 320, 400 450
530
550
580
600
610
700
780
800
19. Find out mean deviation of the following data (use median method) Item
12
18
25
35
47
55
62
75
Frequency
8
12
15
23
16
18
31
12
20. Calculate mean and standard deviation from the following data:Class interval:
0-10
Frequency:
10-20
8
20-30
13
16
30-40
40-50
8
5
21. Draw a Lorenz curve from the data given below. Income No. of persons:
100
200
400
500
800
80
75
50
30
20
22. Explain the characteristics of a good measure of dispersion. 23. Find the mean deviation from the median and its coefficient for the following data : Class interval: Frequency:
10-19 3
20-29 4
30-39
40-49
6
5
50-59 2
24. Calculate mean and standard deviation from the following data : C.I.
10-20
20-30
30-40
40-50
F:
13
16
8
5
[ 18 ]
Correlation Analysis: 1.
Meaning of correlation : It studies and measures the intensity of relationship between two or more variables. If the two variables, X and Y change (vary) in such a way that with a change in value of one variable the values of the other variable also change, then they are said to be correlated.
2.
Significance of correlation: correlation has immense utility in statistics. i.
It helps in determining the degree of relationship between variables.
ii. We can estimate the value of one variable on the basis of the value of another variable correlation serves the basis of regression. iii. Correlation is useful for economists. An economist specifies the relationship between different variables like demand and supply, money supply and price level by way of the correlation. 3.
Correlation and causation: It measures co-variation, not causation. It should never be interpreted as implying cause and effect relationship between two variables. The presence of correlation between two variables X and Y simply means that when one variable is found to change in one direction, the value of the other variable is found to change either in same direction or in the opposite direction.
4.
Positive and Negative Correlation :Correlation is classified into positive and negative correlation when two variables move in the same direction, i.e. if the value of Y increases ( or decreases) with an increase (or decrease) in the value of X, they are said to be positively related. On the other hand when two variables move in the opposite direction i.e. if the value of variable ‘X’ increase (or decrease) with the decrease or increase in the value of Y variable, they one said to be negatively correlated.
5.
Linear and Non- linear correlation:Correlation may be linear or non-linear . If the amount of change in one variable tends to have a constant relation with the amount of change in the other variable then the correlation is said to be liner. It is represented by a straight line. On the otherhand if the amount of change in one variable does not have constant proportional relationship to the amount of change in the other variable, then the correlation is said to be non-linear or curvi-linear. [ 19 ]
6. Simple , multiple and partial correlation :Correlation may also be simple, multiple and partial correlation. When two variables are studied to determine correlation, it is called simple correlation on the other hand when more than two variables are studied to determine the correlation it is called multiple correlation. When correlation of only two variables is studied keeping other variables constant, it is called partial correlation. 7. Methods of studying correlation :- The correlation between the two variables can be determined by the following three methods:(a) Scatter diagram (b) Karl Pearson’s method of correlation coefficient (c) Spearman’s method of Rank correlation. 8. Scatter Diagram: It is a graphic (or visual) method of studying correlation. To construct a scatter diagram, x. variable is taken on X axis and Y Variable is taken on Y-axis. The cluster of points so plotted is referred to as a scatter diagram. In a scatter diagram, the degree of closeness of scatter points and their overall direction gives us an idea of the nature of the relationship:(i)
If the dots move from left to the right upwards, correlation is said to be positive where as the movements of dots from left to right downward indicates negative correlation.
(ii)
Dots in a straight line indicate perfect correlation.
(iii)
Scattered dots indicate no-correlation. The following diagrams illustrate the idea:-
(iv)
Dots falling close to each other in a straight line indicate high degree of correlation.
(i) (a) Y
O
(i) (b) Y
O
X
Positive correlation
X
Negative correlation
[ 20 ]
(ii) (a) Y
(ii) (b) Y
O
O
X Perfect Positive correlation
(iii)
X
Perfect Negative correlation
Y
O
X No correlation
9 Karl pearson’s coefficient of correlation:- Karl pearson’s coefficient of correlation is an important and widely used method of studying correlation. Karl pearson has measured the degree of relationship between the two variables with help of correlation coefficient. Coefficient of correlation measures the degree of relationship between the two variables. Computation of Karl pearsons coefficient of correlation :- The various formulae used to calculate coefficient of correlation (r) are : (i) When deviations are taken from mean r
XY X 2 . Y 2
The formula is also expressed as : r
X – X Y – Y 2 2 X – X Y – Y
[ 21 ]
(ii)
When deviations are taken from assumed mean r
N dxdy – dx . dy
N . dx 2 – dx
N . dy 2 – dy
2
2
where dx = X –Ax, dy = Y – Ay. (iii)
When step deviations are taken r
N dx' dy ' – dx'. dy ' N . dx'2 –
Where dx' (iv)
dx'
2
N . dy '2 –
dy'
2
dx dy , dy ' ix iy
When actual data is used (product moment correlation-Direct method) r
N xy – x . y
N . x 2 – x
2
N . y 2 – y
2
10. Properties of coefficient of correlation :- some of the important properties of karl- pearson’s coefficient of correlation are : (i) The correlation coefficient is independent of the units of measurement of the variables:(ii) The value of co-relation coefficient(r) lies between +1 and -1. (iii) The correlation coefficient is independent of the choice of both origin and scale of observations. (iv) The correlation coefficient of the variables x and y is symmetric, i.e; rxy ryx . 11. Advantages of karl Pearson’s method:Karl person’s method assumes a linear relationship between two variables x and y. If r = 0, it simply means there is no linear correlation between x and y. There may exist quadratic or cubic relationship between x and y. The most important advantage of this method is that it gives an idea about co-variation of the values of two variables and also indicates the direction of such relationships. [ 22 ]
12. Rank Correlation :- Charles Edward spearman evolved another method of finding out correlation between different qualitative attributes of a variable. This is known, as rank correlation coefficient. When a group of individuals are arranged according to their degree of possession of a character (say, beauty, intelligence etc), they are said to be ranked. Spearman’s formula for ranks correlation coefficient in as follows:(A) When Ranks are not repeated :R 1 –
6 D 2 N3 – N
(B When Ranks are repeated
1 6 D 2 M 3 – M ........ 12 R 1 – 3 N –N
Where D = Difference between the ranks of the two series and N = number of individuals in each series. 13. The most important advantage of rank correlation method is that it can be used when quantitative measurement is not possible. Important Questions :(1) What is correlation? (2) When are the two variables said to be in perfect correlation? (3) Define karl- Pearson’s coefficient of correlation (4) Mention any two properties of karl Pearson’s coefficient of correlation. (5) Define covariance ? (6) Can simple correlation coefficient measure any type of relationship? (7) What is the difference between liner and non-liner correlation? (8) What is scatter Diagram method and how is it useful in the study of correlation ? (9) State the merits of Spear Man’s Rank - Correlation ? [ 23 ]
(10) Name various methods of studying correlation. Describe any one. (11) Draw a scat ter Diagram and indicate the nature of correlation
(12)
X
10
20
30
40
50
60
70
80
Y
5
10
15
20
25
30
35
40
C
o
m
p
u
t
e
k
a
r
l
Person’s coefficient of correlation from the following data.
Marks in Economics
15
18
21
24
27
Marks in Business Studies
25
25
27
31
32
(13) Calculate karl- person’s coefficient of correlation between the two variable x and y and interpret the result. X
24
26
32
33
35
30
Y
15
20
22
24
27
24
(14) Calculate the Coefficient of Spear Man’s rank correlation from the given data : X:
20
10
70
60
45
29
50
Y:
60
63
26
35
43
59
37
(15) Calculate coefficient of rank correlation from the following data:X:
48
33
40
9
16
16
65
24
16 27
Y:
13
13
24
6
15
4
20
9
6
19
Index Numbers : MEANING OF INDEX NUMBERS : Index numbers are devices which measure the change in the level of a phenomenon with respect to time, geographical location or some other characteristic. An index number is a statistical device for measuring changes in the magnitude of a group of related variables. It is a measure of the average change in a group of related variables over two different situations. [ 24 ]
Definitions of Index Numbers : In the words of Edge worth, ‘Index number shows by its variation the changes in a magnitude which is not susceptible either of a accurate measurement in itself or of direct valuation in practice.’ In the words of Tuttle, ‘ An index number is a single ratio (usually in percentage)which measures the combined (i.e., averaged) change of several variables between two different times, places or situations.’ Features/Characteristics of Index Numbers:Index numbers are specialized averages. Index numbers are expressed in percentages. Index numbers measure the effect of changes in relation to time or place. PROBLEMS IN CONSTRUCTION OF INDEX NUMBERS:
Purpose of Index Numbers.
Selection of Base Year. Base year is the reference year. It is the year with which price of the Current year are compared. (i) The base period should be a normal year. (ii) The difference between base year and current year should not be too long. (iii) Fixed Base or Chain Base.
Selection of number of goods and services. (i) The commodities selected should be representative of the tastes, habits and customs of the people for whom the Index is meant. (ii) The total number of items should be neither too small nor too large, because if it is too small, then the index number will not be representative and if it is too large, the more representative shall be the index but at the same time the greater shall be the cost and the time taken. (iii) The standardized or graded commodities should be selected to arrive at meaningful and valid comparisons. (iv) Selection of Prices of the goods and services (Price Quotations) V) There are two methods, in which price can be quoted:- (Price Quotations) (i) Money prices (ii) Quantity prices Selection of the Average [ 25 ]
V) Selection of appropriate weights : - There are several Methods of according weight i.e. (i) Fisher’s Method
(ii) Paasche’s Method (iii) Laspeyre’s Method.
Vi) Selection of an appropriate Formula Index numbers can be constructed with the help of the many Formulae. Such as : (1) (3)
Laspeyres Method. (2)
Paasche’s Method
Fisher’s ideal Method.
TYPES OF INDEX NUMBERS:i)
Consumer Price Index (CPI)
(iv) Index of Agricultural Production
ii) Wholesale Price Index (WPI)
v) Sensex (Stock Market)
iii) Index of Industrial Prodution (IIP) Price Index Numbers:i.
Wholesale Price Index Numbers
The wholesale price index number reflects the general price level for a group of items taken as a whole. In India, it is the most popular price index used in the business industry and policy market. It acts as an indicator of the rate of inflation. ii.
Retail Price Index Numbers
The retail price index number reflects the general changes in the retail prices of various items including food, housing, clothing, and so on. The Consumer Price Index, a special type of retail price index, is the primary measure of the cost of living in a country. METHODS OF CONSTRUCTING PRICE INDEX NUMBERS i.
Unweighted Index Numbers
ii.
Weighted Index Numbers.
iii.
Both of these methods of constructing index numbers are further classified as: (i) Aggregative Method; (ii) Average of Relatives Method. [ 26 ]
Method of Constructing Index Numbers 1. Simple (Unweighted) Index Numbers (a) Simple Aggregative Method
2. weighted Index Numbers (a) Weighted Aggregative Method
(b) Simple Average of Price Relatives Method
(b) Wighted Average of Price Relatives Method
Simple Aggregative Of Actual Price Method This is the simplest method of constructing index numbers. In this method, aggregate prices of all the selected commodities in the current year are divided by the aggregate prices in the base year and Multiplied by 100 to get Index. The steps in the constructions of such an index are: (i)
Add up the current year prices of various commodities and denote by
(ii)
Add up the base year prices of various commodities
(iii)
Use the following formula: P01
P
O
.
PI 100 P0
Where, P01= Index number of the current year.
P1 =Total of the current year’s price of all commodities. P0 =Total of the base year’s price of all commodities. Limitations of Simple Aggregative of Actual Price Method:i.
It is influenced by the magnitude of the prices
ii.
Equal weights are assigned to every item
iii.
Prices of various commodities may be quoted in different units
[ 27 ]
P1 .
Simple Average of Price Relatives Method : This index is an improvement over the simple aggregative price index because it is not affected by the unit in which prices are quoted. Price relative : A price relative is percentage ratio between price of commodity in the current year and that in the base year Pr ice – relative ( P01 )
Current Year Pr ice P1 100 BaseYear Pr ice P0
Price Index number of the Current year find out by using the following formula : P
P01
P1 100
0
N P1
Here P 100 Pr ice relatives. 0 N = Number of goods P = Current Year Value 1
P = Base Years Value 0
Merits and Demerits of Average Price Relative Index Merits This index has the following advantages over the simple aggregate price index: (I)
The value of this index is not affected by the units in which prices of commodities are quoted. The Price relatives are pure numbers and, therefore, are independent of the original units in which they are quoted.
(II) Equal importance is given to each commodity and extreme commodities do not influence the index number.
[ 28 ]
Demerits (i) As it is an unweighted index, each price relative is given equal importance. However, in actual practice, a few price relatives are more important than others. (ii) Difficulty is faced with regard to the selection of an appropriate average. WEIGHTED INDEX NUMBERS Weighted index numbers can be constructed by two methods: (i) Weighted Aggregative Method; and (ii) Weighted Average of Price Relatives Method. Weighted Aggregative Method 1. Laspeyre’s method 2. Paasche’s Method 3. Fisher’s Ideal Method 1.
Laspeyre’s Method
Mr. Laspeyres in 1871 gave an weighted aggregated index, in which weights are represented by the quantities of the commodities in the base year. It helps in answering the question that, if the expenditure in the base year on a basket of commodities was Rs.100, then, how much should be that expenditure in the current period on the same basket of commodities. Formula: P01
P1q0 100 P0 q0
Steps: 1. Multiply the current year prices (p 1) by base year quantity weights (q 0) and total all such products to get
p1q0 .
2. Similarly , multiply the base year prices p0 by base year quantity weightss
q0 and obtain the total to get p0q0 .
3. Divided p1q0 by p0q0 and multiply the quotient by 100. This will be the index number of the current year. [ 29 ]
2.
Paasche’s Method : Paasche uses current year’s quantities (q ) as weights. 1
The German statistician Paasche in 1874 constructed an index number, in which weights are determined by quantities in the given year. It helps in answering the question that, if the current period basket of commodities was consumed in the base period and if we were spending Rs 100 on it, how should be the expenditure in current period on the same basket of commodities. Formula: P01
P1q1 100 P0q1
Steps : 1. Multiply the current year prices (p 1) by current year quantities (q 1) and total all such products to get
p1q1 .
2. Similarly , multiply the base year prices (p 0) by current year quantities (q 1) and obtain the total to get
p0q1 .
3. Divide p1q1 by p0 q1 and multiply the quotient by 100. This will be the index number of the current year. 3.
Fisher’s Method
Fisher has combined the techniques of Laspeyres and Paasches Method. He used both base year as well as Current Year quantities (q , q ) as weight. 0
1
Prof. Irving fisher has given a number of formulae for constructing index numbers and of these, he calls one as the ‘ideal’ index. The Fisher’s Ideal Index is given by the following formula: P01
P1q0 P1q1 P0q0 P0q1
100
From the above formula, it is clear that Fisher’s Ideal Index is the geometric mean of the Laspeyre and Paasche indices.
[ 30 ]
Fisheris Method is considered as ideal Method because 1. It is based on variable weights. II. It takes into consideration the price and quantities of both the base year and current year. III. It is based on geometric mean which is regarded as best mean for calculating Index number. IV. It satisfies both the time reversal test and Factor reversal test. CONSUMER PRICE INDEX (CPI) Meaning : The consumer price index numbers are also called (i) Cost of Living Index Numbers, (ii) Retail Price Index Numbers, or (iii) Price of Living Index Numbers. They are designed to measure effects of change in prices of a basket of goods and services on purchasing power of a particular section of the society during any given (current) period with respect to some fixed (base) period. Consumer Price Index reflects the average increase in the cost of the commodities consumed by a class of people so that they can maintain the same standard of living in the current year as in the base year. In India, the consumer price Indices are constructed for the following consumer groups. I.
Industrial Workers (IW)
II. Urban- Non Manual Employees (UNME) III. Agricultural Labourers (AL) Methods of Constructing CPI:i.
Aggregate Expenditure Method or Weighted Aggregate Method;
ii. Family Budget Method or Method of Weighted Average of Price Relatives. Aggregate Expenditure Method This method is similar to the Laspeyre’s method of constructing weighted index. To apply this method, the quantities of commodities by the particular group in the base year are estimated and these figures are used as weights. Then, the total expenditure on each commodity for each year (base and current) are calculated. [ 31 ]
Consumer Price Index (CPI)
p1q0 100 . p0q0
Where P = Price of the Commodities in the Current Year 1
Where P = Price of the Commodities in the base Year O
q = Quantity consumed in base year 0
The steps involved in this method are: 1.
Multiply prices of the base year (p 0) with quantities of the base year (q 0) and add it to get aggregate expenditure for the base year
p0q0 ;
2.
Multiple prices of the current year(p 1) with quantities of the base year (q 0)
3.
Divide aggregate current year’s expenditure
and add it to obtain aggregate expenditure of the current year p1q0 ;
of base year
p1q0 by aggregate expendituree
p0q0 and multiply it by 100 to get consumer price index number..
Family Budget Method In this method, the family budgets of a large number of people , for whome the index is meant, are carefully studied. Then, the aggregate expenditure of an average family on various commodities is estimated. These values constitute the weights. Consumer Price Index
RW W
.
Where R = Current years Price relatives of various items W = weights of various items The steps involved in this method are: 1. Calculate price relatives for the current year (p 1/p0x100) and denote it by R; 2. Multiplying the price in the base year (p 0) with quantity in the base year (q 0) to calculate the weight of a commodity, i.e. to get W; [ 32 ]
3. Multiply the price relatives (R) with weight (W) of each commodity and obtain its total to get
RW ;
4. Obtain the sum total of weights to get
W ;
5. Apply the formula: Consumer Price Index =
RW W
Uses of Consumer Price Index (CPI) Number 1. Consumer price index numbers helps in wage negotiations, formulation of wage policy, price policy, rent control, taxation and general economic policy formulation. 2. The government and business units use the consumer price index numbers to regulate the Dearness allowance (D.A.) or grant of bonus to the employees in order to compensate them for increased cost of living due to price rise. 3. The CPI are used to measure purchasing power of the consumer in rupees. The purchasing power of the rupee is the value of a given year as compared to a base year. The formula for calculating the purchasing power of the rupee is: Purchasing Power=1/ Consumer Price Index X 100 It indicates that money purchasing power is the reciprocal of the price index. Accordingly, if the consumer price index for a given year is 140, then purchasing power of a rupee is 1/ 140 x 100 = 0.71. That is, the purchasing power of a rupee in given year is 71 paisa as compared to the base year. 4. With the increase in prices, the amount of goods and services which money wages can buy (or the real wages) goes on decreasing. Index numbers tell us the change in real wages. Real wages can also be determined, in the following manner: Real Wages = Money Wages/Consumer Price Index X 100 5. Consumer Price index numbers are also used for analyzing markets for particular kinds of goods and services. INDEX OF INDUSTRIAL PRODUCTION Index numbers of industrial production have become fairly common these days. The index number of industrial production measure change in the level of industrial production comprising many industries. [ 33 ]
q
Index Number of Industrial Production =
q1 W
0
W
100
Where, q1 = Level of production in the current year q0 = Level of production in the base year w = Weight or relative importance of industrial output USES OF INDEX NUMBERS:1. Helps in Policy Formulation 2. Index numbers act as Economic Barometers 3. Help in studying trends and forecasting demand and supply 4. To measure and compare changes 5. Index numbers help to measure purchasing power 6. Index numbers help in deflating various values 7. Indicator of rate of Inflation. LIMITATIONS OF INDEX NUMBERS : 1. Provides relative changes only 2. Lack of Perfect Accuracy 3. Difference between purpose and method of construction 4. Ignores qualitative changes 5. Manipulations are possible WHOLESALE PRICE INDEX NUMBERS: (WPI) Wholesale price index numbers are those price index numbers which measure the general changes in the wholesale prices of goods in a country. Groups of Commodities for Wholesale Price Index (WPI):1. Primary Articles : Ex : Rice, Fruits, Pukes, vegetables and non - food articles like cotton, Jute. 2. Energy Articles : Ex. : LPG, Electricity, Petroleum of Coal. 3. Manufactured Articles : Ex. : Textiles, Sugar, Paper Machinery & Chemicals. Utility of Wholesale Price Index Number:1. Indicator of Inflaction 2. Forecasting Demand and Supply 3. Helps in determining real changes in aggregates 4. Useful in Cost Evaluation of various projects [ 34 ]
INFLATION AND INDEX NUMBERS:Inflation is described as a situation characterized by a sustained increase in the general price level over a period of time. Index Number of Agricultural Production:It is a weighted average of quantity relatives. It provides a ready reckoner of the performance of agricultural sector. Its base period is the triennium ending 198182. In 2003-02, the index number of agricultural production was 179.5. It means that agricultural production has increased by 79.5 percent over the average of the three years 1979-80, 1980-81 and 1981-82. Sensex:The sensex, short from of the Bombay stock Exchange (BSE)- Sensitive Index, is a market capitalization weighted index of 30 stocks, representing a sample of large, well established and financially sound companies. It is the oldest index in India and has acquired a unique place in the collective consciousness of investors. The index is widely used to measure the performance of the Indian stock markets. Sensex is considered to be the pulse of the Indian stock markets. The base value of the Sensex is 100 on April 1, 1979, and the base year of BSE-Sensex is 1978-79. Sensex is a useful guide for investors in the stock market. If the sensex rises, it indicates that the market is doing well and investors are optimistic of the future performance of the economy. Human Development Index:The Human Development Index (HD) is an combining normalized measures of life expectancy, literacy, educational attainment, and GDP per capita for countries worldwide. It is claimed as a standard means of measuring human development. The basic use of HDI is to rank countries by level of ‘human development’, which aims to determine whether a country is a developed, developing, or underdeveloped country. Producer Price Index:Producer Price Index measures the average change of the selling prices of producers who sell goods. To compute this index, the mean of all changes over a year is usually taken. It measures the price changes according to the producer’s perspective. It concentrates on the area of industry based production and stage of processing based companies. [ 35 ]
Some selected questions:Short Answer type Questions (3-4 Marks each): 1. What are the desirable properties of the base period? 2. Why is it essential to have different CPI for different categories of consumers? 3. Discuss the limitations of simple aggregative of actual price method. 4. Discuss the merits and demerits of average price relative index. 5. Mention the steps involved for calculating index number by Laspeyre’s Method. 6. Why Fisher’s method is considered to be an ideal method? 7. Mention the difficulties in construction of consumer price index. 8. Write a short note on index of industrial production. 9. Explain clearly the classification of commodities in the formation of wholesale price index. 10.Write a short note on inflation and index Numbers. Very Short Answer type Questions (1 Mark each):1. Define index number. 2. State any one feature of index numbers. 3. Define base period. 4. What are three types of index numbers? 5. What is the difference between a price index and quantity index? 6. State any one limitation of index number? 7. Mention the types of price index numbers? 8. What is the difference between unweighted and weighted index numbers? 9. What is meant by price relative? 10. What does a Consumer price index for industrial workers measure? 11. Whether change in price is reflected by price index number? 12. State any one use of index number. 13. Mention one problem in constructing index numbers. 14. What does consumer price index number reflect ? [ 36 ]
Long Answer Questions : 1. Construct cost of living Index on the basis of the following data Items
Price
Weights
Wheat
241
10
Rice
150
4
Pulses
170
2
Oil
125
2
Milk
40
2
2. Construct Price Index Number of the following data by using. (i)
Laspeyre’s Method
(ii)
Poasche’s Method
(iii)
Fisher’s Method
Items
Base Year
Current Year
Quantity
Price
Quantity
Price
A
3
5
2
8
B
7
4
5
6
C
4
7
3
10
D
6
6
5
7
3. Construct cost of living Index for 2006 based on 2011 from the following data. Group
Food Housing Clothing
Group Index No. of 2011 (based on 2006)
122
140
112
116
106
Weights
32
10
10
6
42
[ 37 ]
Food & Light Miscellaneous
