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COURSE CONTENTS
Statistical data: types, sources and methods of collection. Presentation of data: tables,
charts and graphs. Error and Approximations. Frequency and cumulative distributions.
Measures of location, partition, dispersion, Skewness and kurtosis. Rates, Proportion
and index numbers.
READING LISTS
1. Adamu S.O and Johnson Tinuke L (1998): Statistics for Beginners; Book 1.
SAAL Publication. Ibadan. ISBN: 978-34411-3-2
2. Clark G.M and Cooke D (1993): A Basic course in statistics. Third edition.
London: Published by Arnold and Stoughton.
3. Olubosoye O.E, Olaomi J.O and Shittu O.I (2002): Statistics for
Engineering, Physical and Biological sciences. Ibadan: A Divine Touch
Publications.
4. Tmt. V. Varalakshmi et al (2005): Statistics Higher Secondary - First year.
Tamilnadu Textbook Corporation, College Road, Chennai- 600 006
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INTRODUCTION
In the modern world of information and communication technology, the
importance of statistics is very well recognised by all the disciplines. Statistics has
originated as a science of statehood and found applications slowly and steadily in
Agriculture, Economics, Commerce, Biology, Medicine, Industry, planning, education
and so on. As of today, there is no other human walk of life, where statistics cannot
be applied.
Statistics is concerned with the scientific method of collecting, organizing,
summarizing, presenting and analyzing statistical information (data) as well as
drawing valid conclusion on the basis of such analysis. It could be simply defined as
the “science of data”. Thus, statistics uses facts or numerical data, assembled,
classified and tabulated so as to present significant information about a given
subject. Statistic is a science of understanding data and making decisions in the face
of randomness.
The study of statistics is therefore essential for sound reasoning, precise
judgment and objective decision in the face of up- to- date accurate and reliable
data. Thus many researchers, educationalists, business men and government
agencies at the national, state or local levels rely on data to answer operations and
programs. Statistics is usually divided into two categories, which is not mutually
elution namely: Descriptive statistics and inferential statistics.
DESCRIPTIVE STATISTICS
This is the act of summarizing and given a descriptive account of numerical
information in form of reports, charts and diagrams. The goal of descriptive
statistics is to gain information from collected data. It begins with collection of data
by either counting or measurement in an inquiry. It involves the summary of specific
aspect of the data, such as averages values and measure of dispersion (spread).
Suitable graphs, diagrams and charts are then used to gain understanding and clear
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interpretation of the phenomenon under investigation keeping firmly in mind
where the data comes from. Normally, a descriptive statistics should:
i.
be single – valued
ii.
be algebraically tractable
iii.
consider every observed value.
INFERENTIAL STATISTICS
This is the act of making deductive statement about a population from the
quantities computed from its representative sample. It is a process of making
inference or generalizing about the population under certain conditions and
assumptions. Statistical inference involves the processes of estimation of
parameters and hypothesis testing. However, this concept is not in the context of
this course.
USES OF STATISTICS
Statistics can be used among others for:
1) Planning and decision making by individuals, states, business organizations,
research institution etc.
2) Forecasting and prediction for the future based on a good model provided
that its basic assumptions are not violated.
3) Project implementation and control. This is especially useful in on-going
projects such as network analysis, construction of roads and bridges and
implementation of government programs and policies.
4) The assessment of the reliability and validity of measurements and general
points significance tests including power and sample size determination.
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STATISTICAL DATA
Data can be described as a mass of unprocessed information obtained from
measurement of counting of a characteristics or phenomenon. They are raw facts
that have to be processed in numerical form they are called quantitative data. For
instance the collection of ages of students offering STS 102 in a particular session is
an example of this data. But when data are not presented in numerical form, they
are called qualitative data. E.g.: status, sex, religion, etc.
Definition: Statistical data are data obtained through objective measurement or
enumeration of characteristics using the state of the art equipment that is precise
and unbiased. Such data when subjected to statistical analysis produce results with
high precision.
SOURCES OF STATISTICAL DATA
1. Primary data: These are data generated by first hand or data obtained
directly
from
respondents
by
personal
interview,
questionnaire,
measurements or observation. Statistical data can be obtained from:
(i)
Census – complete enumeration of all the unit of the population
(ii)
Surveys – the study of representative part of a population
(iii)
Experimentation – observation from experiment carried out in
laboratories and research center.
(iv)
Administrative process e.g. Record of births and deaths.
ADVANTAGES
Comprises of actual data needed
It is more reliable with clarity
Comprises a more detail information
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DISADVANTAGES
Cost of data collection is high
Time consuming
There may larger range of non response
2. Secondary data: These are data obtained from publication, newspapers, and
annual reports. They are usually summarized data used for purpose other
than the intended one. These could be obtain from the following:
(i)
Publication e.g. extract from publications
(ii)
Research/Media organization
(iii)
Educational institutions
ADVANTAGES
The outcome is timely
The information gathered more quickly
It is less expensive to gather.
DISADVANTAGES
Most time information are suppressed when working with
secondary data
The information may not be reliable
METHODS OF COLLECTION OF DATA
There are various methods we can use to collect data. The method used depends
on the problem and type of data to be collected. Some of these methods include:
1. Direct observation
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2. Interviewing
3. Questionnaire
4. Abstraction from published statistics.
DIRECT OBSERVATION
Observational methods are used mostly in scientific enquiry where data are
observed directly from controlled experiment. It is used more in the natural
sciences through laboratory works than in social sciences. But this is very useful
studying small communities and institutions.
INTERVIEWING
In this method, the person collecting the data is called the interviewer goes to ask
the person (interviewee) direct questions. The interviewer has to go to the
interviewees personally to collect the information required verbally. This makes it
different from the next method called questionnaire method.
QUESTIONNAIRE
A set of questions or statement is assembled to get information on a variable (or a
set of variable). The entire package of questions or statement is called a
questionnaire. Human beings usually are required to respond to the questions or
statements on the questionnaire. Copies of the questionnaire can be administered
personally by its user or sent to people by post. Both interviewing and
questionnaire methods are used in the social sciences where human population is
mostly involved.
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ABSTRACTIONS FROM THE PUPLISHED STATISTICS
These are pieces of data (information) found in published materials such as figures
related to population or accident figures. This method of collecting data could be
useful as preliminary to other methods.
Other methods includes: Telephone method, Document/Report method, Mail
or Postal questionnaire, On-line interview method, etc.
PRESENTATION OF DATA
When raw data are collected, they are organized numerically by distributing
them into classes or categories in order to determine the number of individuals
belonging to each class. Most cases, it is necessary to present data in tables, charts
and diagrams in order to have a clear understanding of the data, and to illustrate the
relationship existing between the variables being examined.
FREQUENCY TABLE
This is a tabular arrangement of data into various classes together with their
corresponding frequencies.
Procedure for forming frequency distribution
Given a set of observation
, for a single variable.
1. Determine the range (R) = L – S where L = largest observation in the raw
data; and S = smallest observation in the raw data.
2. Determine the appropriate number of classes or groups (K). The choice of K is
arbitrary but as a general rule, it should be a number (integer) between 5 and
20 depending on the size of the data given. There are several suggested guide
lines aimed at helping one decided on how many class intervals to employ.
Two of such methods are:
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(a) K = 1 +3.322
(b) K =
3. Determine the width
where
= number of observations.
of the class interval. It is determined as
4. Determine the numbers of observations falling into each class interval i.e. find
the class frequencies.
NOTE: With advent of computers, all these steps can be accomplishes easily.
SOME BASIC DEFINITIONS
Variable: This is a characteristic of a population which can take different values.
Basically, we have two types, namely: continuous variable and discrete variable.
A continuous variable is a variable which may take all values within a given range.
Its values are obtained by measurements e.g. height, volume, time, exam score etc.
A discrete variable is one whose value change by steps. Its value may be obtained
by counting. It normally takes integer values e.g. number of cars, number of chairs.
Class interval: This is a sub-division of the total range of values which a
(continuous) variable may take. It is a symbol defining a class E.g. 0-9, 10-19 etc.
there are three types of class interval, namely: Exclusive, inclusive and open-end
classes method.
Exclusive method:
When the class intervals are so fixed that the upper limit of one class is the lower
limit of the next class; it is known as the exclusive method of classification. E.g. Let
some expenditures of some families be as follows:
0 – 1000, 1000 – 2000, etc. It is clear that the exclusive method ensures continuity
of data as much as the upper limit of one class is the lower limit of the next class. In
the above example, there are so families whose expenditure is between 0 and
999.99. A family whose expenditure is 1000 would be included in the class interval
1000-2000.
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Inclusive method:
In this method, the overlapping of the class intervals is avoided. Both the lower and
upper limits are included in the class interval. This type of classification may be used
for a grouped frequency distribution for discrete variable like members in a family,
number of workers in a factory etc., where the variable may take only integral
values. It cannot be used with fractional values like age, height, weight etc. In case of
continuous variables, the exclusive method should be used. The inclusive method
should be used in case of discrete variable.
Open end classes:
A class limit is missing either at the lower end of the first class interval or at the
upper end of the last class interval or both are not specified. The necessity of open
end classes arises in a number of practical situations, particularly relating to
economic and medical data when there are few very high values or few very low
values which are far apart from the majority of observations.
Class limit: it represents the end points of a class interval. {Lower class limit &
Upper class limit}. A class interval which has neither upper class limit nor lower
class limit indicated is called an open class interval e.g. “less than 25”, ’25 and
above”
Class boundaries: The point of demarcation between a class interval and the next
class interval is called boundary. For example, the class boundary of 10-19 is 9.5 –
19.5
Cumulative frequency: This is the sum of a frequency of the particular class to the
frequencies of the class before it.
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Example 1: The following are the marks of 50 students in STS 102:
48 70 60 47 51 55 59 63 68 63 47 53 72 53 67 62 64 70 57
56
48 51 58 63 65 62 49 64 53 59 63 50 61 67 72 56 64 66 49 52
62
71 58 53 63 69 59 64 73 56.
(a) Construct a frequency table for the above data.
(b) Answer the following questions using the table obtained:
(i) how many students scored between 51 and 62?
(ii) how many students scored above 50?
(iii) what is the probability that a student selected at random from the class will
score less than 63?
Solution:
(a) Range (R) =
No of classes
Class size
Frequency Table
Mark
Tally
frequency
7
51- 54
7
55 - 58
7
59 – 62
8
63 – 66
11
67 – 70
6
71 – 74
4
(b) (i) 22 (ii) 43 (iii) 0.58
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Example 2: The following data represent the ages (in years) of people living in a
housing estate in Abeokuta.
18 31 30 6 16 17 18 43 2 8 32 33 9 18 33 19 21 13 13 14
14
6 52 45 61 23 26 15 14 15 14 27 36 19 37 11 12 11
20 12 39 20 40 69 63 29 64 27 15 28.
Present the above data in a frequency table showing the following columns; class
interval, class boundary, class mark (mid-point), tally, frequency and cumulative
frequency in that order.
Solution:
Range (R)
No of classes
Class width
Class interval Class boundary Class mark
Tally
Frequency
Cum.freq
2 – 11
1.5 – 11.5
6.5
7
7
12 – 21
11.5 – 21.5
16.5
21
28
22 – 31
21.5 – 31.5
26.5
8
36
32 – 41
31.5 – 41.5
36.6
7
43
42 – 51
41.5 – 51.5
46.5
2
45
52 – 61
51.5 – 61.5
56.5
2
47
62 – 71
61.5 – 71.5
66.5
3
50
Observation from the Table
The data have been summarized and we now have a clearer picture of the
distribution of the ages of inhabitants of the Estate.
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Exercise 1
Below are the data of weights of 40students women randomly selected in Ogun
state. Prepare a table showing the following columns; class interval, frequency, class
boundary, class mark, and cumulative frequency.
96
84
75
80
64
105
87
62
105
103 76 93
75 110 88
97
69
96
73
91
101
106
110 64 105 117
84
96
91
82
81
94
108
117 99 114 88 60 98
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Use your table to answer the following question
i.
How many women weight between 71 and 90?
ii.
How many women weight more than 80?
iii.
What is the probability that a woman selected at random from Ogun state
would weight more than 90?
GRAPHICAL PRESENTATION OF DATA
It is not enough to represent data in a tabular form. The most attractive way
of representing data is through charts or graphs.
PICTOGRAM
Pictograms or pictographs are representations in form of pictures. They convey
broad meanings and relationships among data. Also they are the simplest way of
presenting information. Pictograms are popularly used in newspapers by journalists
and advertisers.
EXAMPLE 1: In a certain secondary school, there are 4 Eng teachers, 3 math
teachers, 2 Biology teachers, 2 government teachers and 1 Physics teacher. Draw a
pictogram to represent this information.
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SOLUTION: Let us use
to represent a teacher
The different subject Teachers in a certain school
English Teachers
Maths teachers
Biology teachers
Government teachers
Physics teachers
PIE CHART
A pie chart is a circular graph in which numerical data are represented by sectors of
a circle. The angles of the sectors are proportional to the frequencies of the items
they represent
EXAMPLE 2: In ADAS international school, the lesson periods for each week are
given below.
English 7, Maths 10, Biology 3, Physics 4, Chemistry 3, others 9. Draw a pie chart to
illustrate this information.
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Solution: Total no of periods in a week
Subject
No of period
English
7
Maths
10
Biology
3
Physics
4
Chemistry
3
Others
9
Total
Angle of sector
36
The pie chart showing the lesson period in ADAS international school
English
Mathematics
Biology
Physics
Chemistry
Other
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BAR CHART
A bar chart is a statistical graph in which bars (rectangular bars) are drawn such
that their lengths or heights are proportional to the quantities or item they
represent. Each bar is separated by equal gaps.
Example 3: The allotment of time in minutes per week for some of the university
courses in second semester is;
Courses
Minutes
GNS 104
60
MTS 102
120
STS 102
180
ECO 102
120
BFN 108
120
PHS 192
140
Construct a bar chart to represent the above data.
Solution:
Bar chart showing the allotment of time in minutes
200
180
160
140
120
100
Series1
80
60
40
20
0
GNS 104 MTS 102
STS 102
ECO 102
BFN 108
16
PHS 192
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HISTOGRAM
A histogram is a graphical representation of a frequency distribution. It consists of a
number of rectangles. The area of each rectangular bar of a histogram is
proportional to the corresponding frequency. Unlike bar charts, the rectangles are
joined together for histogram.
Example 4: Draw a histogram to represent the data in the table below.
Height (cm)
Frequency
6
9
15
5
2
Solution:
174.5
164.5
154.5
144.5
134.5
124.5
Frequency
Histogram showing the heights
Height
CUMMULATIVE FREQUENCY CURVE (Ogive)
This curve is the plotting of cumulative freq. against upper class bounding of
the class intervals. The points, when joined, result usually in a smooth curve which
is the form of an elongated S.
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Example 5: Using the data below:
Weight (kg)
Frequency
Cumulative frequency
20 – 24
5
5
25 – 29
4
9
30 – 34
3
12
35 – 39
5
17
40 – 44
5
22
45 – 49
5
27
50 – 54
18
45
55 – 59
12
57
60 – 64
3
60
The cumulative frequency or Ogive will appears as follows:
Stem plots (Stem – and – leaf plots)
A stem-and-leaf plot is used to visualize data. To set up a stem-and-leaf plot,
we follow some simple steps. First we have a set of data. Then we identify the lowest
and greatest number in the data set and then we draw a vertical line. On the left
hand side of the line we write the numbers that correspond to the tens and on the
right hand side of the line, we will write the numbers that corresponds with the
units. The digits should be arranged in order.
Example : Draw the stem plot for the following data
(a) 13, 24, 22, 15, 33, 32, 12, 31, 26, 28, 14, 19, 20, 22, 31, 15.
(b) 46 59 35 41 46 21 24 33 40 45 49 53 48 54 61 36 70 58 47 12
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Solution: (a)
Stem
Leaf
1
2
0 2 2 4 6 8
3
(b)
2 3 4 5 5 9
1 1 2 3
Stem
Leaf
1
2
2
1 4
3
3 5 6
4
0 1 5 6 6 7 8 9
5
3 4 8 9
6
1
7
1
Exercise 2:
1. The population by classes of Adas international school is as shown below
Class
I
II
III
IV
V
VI
No of pupils
30
25
27
26
22
26
Draw a pictogram, pie chart and a bar chart to represent the information.
2. The age distribution (in years) of a group of 100 individual is given below.
Ages(x)
2–4
5–9
10
– 15 – 50 – 25
– 30
14
No.
of 8
11
19
24
29
34
21
20
17
10
10
– 35 – 39
individuals(f)
Draw a histogram and a cumulative frequency curve for the data.
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ERRORS AND APPROXIMATION
Rounding of statistical data is the method of approximating a number such
that the last digits affected are changed to zero and the number become clearer.
Since in reality most measurement is not exact, hence offence is not much
committed when data are rounded up.
Methods of rounding are rounding to specific units. E.g. 4,234,120 to the
nearest million is 4,000,000. Also rounding to specific significant figures e.g. to 5
significant figures will give 4,235,100. Again when an amount, say N750.3 is used to
denote the average salary per person in a month, the figure in this wise is rounded
to one place of decimal and the actual figure is between the ranges N750.25 to less
than N 750.35. This implies rounded figures have errors.
Error: For any given figure the correct figure lies in a certain range. Half of this
range is the Error.
Suppose
correct figure;
Absolute error:
Percentage Error
rounded figure e = error;
. Relative error
. i.e. percentage Error
Example: The length of a pole is measured as 10meters to the nearest meter. What
is the range of its actual length? Calculate the percentage error.
Solution The actual length of the pole will be between 9.5 and 10.5
Error
Percentage Error
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MEASURES OF LOCATION
These are measures of the centre of a distribution. They are single values that
give a description of the data. They are also referred to as measure of central
tendency. Some of them are arithmetic mean, geometric mean, harmonic mean,
mode, and median.
THE ARITHMETIC MEAN (A.M)
The arithmetic mean (average) of set of observation is the sum of the
observation divided by the number of observation. Given a set of a numbers
, the arithmetic mean denoted by
is defined by
Example 1: The ages of ten students in STS 102 are
determine the mean age.
Solution:
.
If the numbers
times respectively, the
(
for short.)
Example 2: Find the mean for the table below
Scores (x)
2
5
6
8
Frequency (f)
1
3
4
2
Solution
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.
Calculation of mean from grouped data
If the items of a frequency distribution are classified in intervals, we make the
assumption that every item in an interval has the mid-values of the interval and we
use this midpoint for .
Example 3: The table below shows the distribution of the waiting items for some
customers in a certain petrol station in Abeokuta.
Waiting
time(in 1.5 – 1.9
2.0 – 2.4
2.5 – 2.9
3.0 – 3.4
3.5 – 3.9
4.0 – 4.4
10
18
10
7
2
mins)
No. of customers
3
d the average waiting time of the customers.
Solution:
Waiting (in min) No of customers Class mark mid-value(X)
1.5 – 1.9
3
1.7
5.1
2.0 – 2.4
10
2.2
22
2.5 – 2.9
18
2.7
48.6
3.0 – 3.4
10
3.2
32
3.5 – 3.9
7
3.7
25.9
4.0 – 4.4
2
4.2
8.4
= 2.84
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Use of Assume mean
Sometimes, large values of the variable are involve in calculation of mean, in
order to make our computation easier, we may assume one of the values as the
mean. This if A= assumed mean, and d= deviation of
from A, i.e.
Therefore,
since
.
If a constant factor C is used then
.
Example 4: The exact pension allowance paid (in Nigeria) to 25 workers of a
company is given in the table below.
Pension in N
25
30
35
40
45
No of person
7
5
6
4
3
Calculate the mean using an assumed mean 35.
Solution
Pension in N frequency
25
7
25 – 35 = -10 - 70
30
5
30 – 35 = -5
- 25
35 A
6
35 – 35 = 0
0
40
4
40 – 35 = 5
20
45
3
45 – 35 = 10 30
25
- 45
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Example 5: Consider the data in example 3, using a suitable assume mean, compute
the mean.
Solution:
Waiting time
1.5 – 1.9
3
1.7
-1
-3
2.0 – 2.4
10
2.2
-0.5
-5
2.5 – 2.9
18
2.7 A
0
0
3.0 – 3.4
10
3.2
0.5
5
3.5 – 3.9
7
3.7
1
7
4.0 – 4.4
2
4.2
1.5
3
50
7
NOTE: It is always easier to select the class mark with the longest frequency as the
assumed mean.
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ADVANTAGE OF MEAN
The mean is an average that considers all the observations in the data set. It is single
and easy to compute and it is the most widely used average.
DISAVANTAGE OF MEAN
Its value is greatly affected by the extremely too large or too small observation.
THE HARMONIC MEAN (H.M)
The H.M of a set of numbers
is the reciprocal of the arithmetic
mean of the reciprocals of the numbers. It is used when dealing with the rates of the
type
per
(such as kilometers per hour, Naira per liter). The formula is expressed
thus:
H.M
If
has frequency , then
H.M
Example: Find the harmonic mean of
.
Solution:
H.M
.
Note:
(i) Calculation takes into account every value
(ii) Extreme values have least effect
(iii) The formula breaks down when “o” is one of the observations.
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THE GEOMETRIC MEAN(G.M)
The G.M is an analytical method of finding the average rate of growth or
decline in the values of an item over a particular period of time. The geometric mean
of a set of number
is the
root of the product of the number. Thus
G.M
If
is the frequency of
, then
G.M
Example: The rate of inflation in fire successive year in a country was
. What was the average rate of inflation per year?
Solution:
G.M
Average rate of inflation is
Note: (1) Calculate takes into account every value.
(2) It cannot be computed when “0” is on of the observation.
Relation between Arithmetic mean, Geometric and Harmonic
In general, the geometric mean for a set of data is always less than or equal to
the corresponding arithmetic mean but greater than or equal to the harmonic mean.
That is,
H.M
G.M
A.M
The equality signs hold only if all the observations are identical.
THE MEDIAN
This is the value of the variable that divides a distribution into two equal
parts when the values are arranged in order of magnitude. If there are
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observation, the median
location of the median is
But if
is even, the median
is the center of observation in the ordered list. The
th item.
is the average of the two middle observations in the
ordered list.
i.e.
Example 1: The values of a random variable
are given as
Find the median.
Solution: In an array:
.
is odd, therefore
The median,
Example 2: The value 0f a random variable
and
are given as
. Find the median.
Solution:
is odd.
Median,
Calculation of Median from a grouped data
The formula for calculating the median from grouped data is defined as
Where:
Lower class boundary of the median class
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Total frequency
Cumulative frequency before the median class
Frequency of the median class.
Class size or width.
Example3: The table below shows the height of 70 men randomly selected at Sango
Ota.
Height
118-126
127-135 136-144 145-153
154-162
163-171
172-180
No of rods
8
10
9
7
4
14
18
Compute the median.
Solution
Height
Frequency
Cumulative frequency
118 – 126
8
8
127 – 135
10
18
136 – 144
14
32
145 – 153
18
50
154 – 162
9
59
163 – 171
7
66
172 – 180
4
70
70
. The sum of first three classes frequency is 32 which therefore means
that the median lies in the fourth class and this is the median class. Then
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.
ADVANTAGE OF THE MEDIAN
(i) Its value is not affected by extreme values; thus it is a resistant measure of
central tendency.
(ii) It is a good measure of location in a skewed distribution
DISAVANTAGE OF THE MEDIAN
1) It does not take into consideration all the value of the variable.
THE MODE
The mode is the value of the data which occurs most frequently. A set of data
may have no, one, two or more modes. A distribution is said to be uni-model,
bimodal and multimodal if it has one, two and more than two modes respectively.
E .g: The mode of scores 2, 5, 2, 6, 7 is 2.
Calculation of mode from grouped data
From a grouped frequency distribution, the mode can be obtained from the
formula.
Mode,
Where:
lower class boundary of the modal class
Difference between the frequency of the modal class and the class
before it.
Difference between the frequency of the modal class and the class
after it.
Class size.
Example: For the table below, find the mode.
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Class
11 – 20
frequency 6
21 – 30
31 – 40
41 - 50
51 – 60
61 – 70
20
12
10
9
9
Calculate the modal age.
Solution:
Mode,
ADVANTAGE OF THE MODE
1) It is easy to calculate.
DISADVANTAGE OF THE MODE
(i) It is not a unique measure of location.
(ii) It presents a misleading picture of the distribution.
(iii) It does not take into account all the available data.
Exercise 3
1. Find the mean, median and mode of the following observations: 5,
6,10,15,22,16,6,10,6.
2. The six numbers 4, 9,8,7,4 and Y, have mean of 7. Find the value of Y.
3. From the data below
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Class
21 – 23
Frequency 2
24 – 26
27 – 29
30 – 32
33 – 35
36 – 38
37 – 41
5
8
9
7
3
1
Calculate the (i)Mean (ii)Mode (iii) Median
MEASURES OF PARTITION
From the previous section, we’ve seen that the median is an average that
divides a distribution into two equal parts. So also these are other quantity that
divides a set of data (in an array) into different equal parts. Such data must have
been arranged in order of magnitude. Some of the partition values are: the quartile,
deciles and percentiles.
THE QUARTILES
Quartiles divide a set of data in an array into four equal parts.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude.
Then
First Quartiles:
Second Quartile:
Third Quartile:
For a grouped data
Where
The quality in reference
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Lower class boundary of the class counting the quartile
Total frequency
Cumulative frequency before the
The frequency of the
Class size of the
class
class
class.
DECILES
The values of the variable that divide the frequency of the distribution into
ten equal parts are known as deciles and are denoted by
the fifth
deciles is the median.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude. Then
For a grouped data
Where
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PERCENTILE
The values of the variable that divide the frequency of the distribution into
hundred equal parts are known as percentiles and are generally denoted by
The fiftieth percentile is the median.
For ungrouped data, the distribution is first arranged in ascending order of
magnitude. Then
For a grouped data,
Where
Example: For the table below, find by calculation (using appropriate expression)
(i) Lower quartile,
(ii) Upper Quartile,
(iii) 6th Deciles,
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(iv) 45th percentile of the following distribution
Mark
20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99
Frequency 8
10
14
26
Solution
Marks
frequency cumulative frequency
20 – 29
8
8
30 – 39
10
18
40 – 49
14
32
50 – 59
26
32
60 – 69
20
58
70 – 79
16
78
80 – 89
4
98
90 – 99
2
100
100
(i) Lower quartile,
w
(ii) Upper Quartile,
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(iii)
(iv)
MEASURES OF DISPERSION
Dispersion or variation is degree of scatter or variation of individual value of a
variable about the central value such as the median or the mean. These include
range, mean deviation, semi-interquartile range, variance, standard deviation and
coefficient of variation.
THE RANGE
This is the simplest method of measuring dispersions. It is the difference
between the largest and the smallest value in a set of data. It is commonly used in
statistical quality control. However, the range may fail to discriminate if the
distributions are of different types.
Coefficient of Range
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SEMI – INTERQUARTILE RANGE
This is the half of the difference between the first (lower) and third quartiles
(upper). It is good measure of spread for midrange and the quartiles.
THE MEAN/ABSOLUTE DEVIATION
Mean deviation is the mean absolute deviation from the centre. A measure of
the center could be the arithmetic mean or median.
Given a set of
the mean deviation from the arithmetic mean is defined
by:
In a grouped data
Example1: Below is the average of 6 heads of household randomly selected from a
country. 47, 45, 56, 60, 41, 54 .Find the
(i) Range
(ii) Mean
(iii) Mean deviation from the mean
(iv) Mean deviation from the median.
Solution:
(i) Range
(ii)
Mean ( )
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(iii)
Mean Deviation
(iv)
In array: 41, 45, 47, 54, 56, 60
Median
Example2: The table below shown the frequency distribution of the scores of 42
students in MTS 101
Scores
0–9
No of student 2
10 – 19
20 – 29
30 – 39
40 – 49
50 – 59
60 – 69
5
8
12
9
5
1
Find the mean deviation from the mean for the data.
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Solution:
Classes
midpoint
0–9
4.5
10 – 19
14.5
5
72.5
-19.52
19.52
97.60
20 – 29
24.5
8
196
-9.52
9.52
76.16
30 – 39
34.5
12
414
0.48
0.48
5.76
40 – 49
44.5
9
400.5
10.48
10.48
94.32
50 – 59
54.5
5
272.5
20.48
20.48
102.4
60 – 69
64.5
1
64.5
30.48
30.48
30.48
42
1429
2
9
-29.52
29.52
59.04
465.76
THE STANDARD DEVIATION
The standard deviation, usually denoted by the Greek alphabet
(small
signal) (for population) is defined as the “positive square root of the arithmetic
mean of the squares of the deviation of the given observation from their arithmetic
mean”. Thus, given
as a set of
observations, then the standard deviation
is given by:
(Alternatively,
. )
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For a grouped data
The standard deviation is computed using the formula
If A= assume mean and
Note: We use
is deviation from the assumed mean, then
when using sample instead of the population to obtain
the standard deviation.
MERIT
(i)
It is well defined and uses all observations in the distribution.
(ii)
It has wider application in other statistical technique like skewness,
correlation, and quality control e.t.c
DEMERIT
(i)
It cannot be used for computing the dispersion of two or more
distributions given in different unit.
THE VARIANCE
The variance of a set of observations is defined as the square of the standard
deviation and is thus given by
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COEFFICIENT OF VARIATION/DISPERSION
This is a dimension less quantity that measures the relative variation between two
servers observed in different units. The coefficients of variation are obtained by
dividing the standard deviation by the mean and multiply it by 100. Symbolically
The distribution with smaller C.V is said to be better.
EXAMPLE3: Given the data 5, 6, 9, 10, 12. Compute the variance, standard deviation
and coefficient of variation
SOLUTION
Hence C.V
EXAMPLE4: Given the following data. Compute the
(i) Mean
(ii) Standard deviation
(iii) Coefficient variation.
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Exercise 4
The data below represents the scores by 150 applicants in an achievement text for
the post of Botanist in a large company:
Scores
10 – 19
20 – 29
30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99
Frequency
1
6
9
31
42
32
17
10
Estimate
(i)
The mean score
(ii)
The median score
(iii)
The modal score
(iv)
Standard deviation
(v)
Semi – interquartile range
(vi)
D4
(vii) P26
(viii) coefficient of variation
SKEWNESS AND KURTOSIS
Skewness means “lack of symmetry”. We study skewness to have an idea
about the shape of the curve which we can draw with the help of the given data. If in
a distribution mean in median
mode, then that distribution is known as
symmetrical distribution. If in a distribution, mean
median
mode, then it is not
a symmetrical distribution and it is called a skewed distribution and such a
distribution could either be positively skewed or negative skewed.
(a) Symmetrical distribution:
mean
median
42
mode
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Mean, median and mode coincide and the spread of the frequencies is the
same on both sides of the centre point of the curve.
(b) Positively Skewed distribution:
mode
median mean
In a positive skewed distribution, the value of the mean is the maximum and
that of the mode is the least, and the median lies in between the two. The
frequencies are spread out over a grater range of values on the right hand side
than they are on the left hand side.
(c) Negatively Skewed distribution:
mean median mode
In a negatively skewed distribution, the value of the mode is the maximum
and that of the mean is the least. The median lies in between the two. The
frequencies are spread out over a greater range of values on the left hand side
than they are on the right hand side.
Measures of Skewness
The important measures of skewness are:
(i)
Karl-Pearson’s coefficient of skewness
(ii)
Bowley’s coefficient of skewness
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(iii)
Measures of skewness based on moments.
Karl-Pearson’s coefficient of skewness
This is given by:
Karl-Pearson’s Coefficient Skewness
.
In case of mode is ill-defined, the coefficient can be determined by the formula:
Coefficient of Skewness
.
Bowley’s coefficient of skewness
This is given by:
Bowley’s Coefficient of Skewness
.
Measure of skewness based on moment
First, we note the following moments:
1st moment about mean:
;
.
2nd moment about mean:
;
.
3rd moment about mean:
;
.
4th moment about mean:
;
.
Then the measure of skewness based on moments denoted by
is given by:
.
KURTOSIS
The expression ‘Kurtosis’ is used to describe the peakedness of a normal
curve. The three measures – central tendency, dispersion and skewness, describe
the characteristics of frequency distributions. But these studies will not give us a
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clear picture of the characteristics of a distribution. Measures of kurtosis tell us the
extent to which a distribution is more picked or more flat topped than the normal
curve, which is symmetrical end bell – shaped, is designated as Mesokurtic. If a
curve is relatively more narrow and peaked at the top, it is designated as
Leptokurtic. If the frequency curve is more flat than normal curve, it is designated
as Platykurtic.
L
M
P
Measures of Kurtosis
The measure of kurtosis of a frequency distribution based on moment is denoted by
and is given by:
.
If
, the distribution is said to be normal and the curve is Mesokurtic.
If
, the distribution is said to be more peaked and the curve is Leptokurtic.
If
, the distribution is said to be flat peaked and the curve is Platykurtic.
Example 1: Calculate Karl – Pearson’s coefficient of skewness for the following data:
25, 15, 23, 40, 27, 25, 23, 25, 20.
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Solution: Computation of mean and standard deviation using assume mean method:
Size
Deviation from A
25
25
0
0
15
-10
100
23
-2
4
40
15
225
27
2
4
25
0
0
23
-2
4
25
0
0
20
-5
25
Mean
Mode
, as this size of item repeats three times.
Karl – Pearson’s coefficient of skewness
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Example 2: Find Karl – Pearson’s coefficient of skewness for the given distribution:
Class 0 – 5
6 – 11
12 – 17
18 – 23
24 – 29
30 – 35
36 – 41
42 – 47
F
5
7
13
21
16
8
3
2
Solution: Mode lies in 24 – 29 group which contains the maximum frequency.
Mode
Computation of mean and standard deviation:
Class
Mid Point ( )
0–5
2.5
2
5
-23.28
541.96
1083.92
6 – 11
8.5
5
42.5
-17.28
298.59
1492.95
12 – 17
14.5
7
101.5
-11.28
127.24
890.68
18 – 23
20.5
13
266.5
-5.28
27.88
362.44
24 – 29
26.5
21
556.5
0.72
0.52
10.92
30 – 35
32.5
16
520
6.72
45.16
722.56
36 – 41
38.5
8
308
12.72
161.79
1294.32
42 – 47
44.5
3
133.5
18.72
350.44
1051.32
Mean
Standard deviation
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.
Therefore,
Karl – Pearson’s coefficient of skewness
Example 3: Find the Bowley’s coefficient of skewness for the following series:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22.
Solution: The given data in order: 2, 4, 6, 10, 12, 14, 16, 18, 20, and 22.
size of
size of
th item
th item
size of 3rd item
size of
th item
size of
th item
size of 9th item
Median
size of
th item
size of
th item
size of 6th item
Bowley’ coefficient of skewness
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Since skewness
, the given series is a symmetrical data.
Example 4: Calculate
(measure of skewness based on moment) and
(measure
of kurtosis based on moment) for the following data.
X
0
1
2
3
4
5
6
7
8
F
5
10
15
20
25
20
15
10
5
Solution:
First Moment:
where
Second Moment:
Third Moment:
Fourth Moment:
. That is, symmetrical curve.
. The value of
, hence the curve is Platykurtic.
Example 5: For the data below, determine the
and
Mark
30 – 33
Frequency 2
.
34 – 37
38 – 41
42 – 45
46 – 49
50 – 53
4
26
47
15
6
Solution:
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Exercise 5
Calculate the
and
for the data below and interpret your result.
Class 10 – 14
15 – 19
20 – 24
25 – 29 30 – 34 35 – 39 40 – 44 45 – 49
f
4
8
19
1
35
20
7
5
Rates, Proportion and Index Numbers
Proportion
Proportion is the ratio of a number of items with certain characteristics (X)
and the total number of items exposed to such characteristics (N). It is defined as
.
The above expresses the chance of occurrences of such characteristics (i.e.
Probability of event X).
Example: If the voting age population (people 18 years and above) in Ifo ward I
consists of 550 males and 600 females. What is the proportion of males?
Solution:
, Total population
Proportion of males,
.
.
Rates
When proportion refers to the number of events or cases occurring during
certain period of time, it becomes a rate and is usually expressed as so many per
1000. Thus we refer to birth rate as the number of birth per 1000 population in a
year. So also we have death rate, migration rate, marriage rate etc.
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INDEX NUMBERS
There are various types of index numbers, but in brief, we shall discuss three
kinds, namely: (a) Price Index (b) Quality Index (c) Value Index.
(a)
Price Index
For measuring the value of money, in general, price index is used. It is an
index number which compares the prices for a group of commodities at a certain
time as at a place with prices of a base period. There are two price index numbers
such as whole sale price index numbers and retail price index numbers. The
wholesale price index reveals the changes into general price level of a country, but
the retail price reveals the changes in the retail price of commodities such as
consumption of goods, bank deposit etc.
(b)
Quantity Index
This is the changes in the volume of goods produced or consumed. They are
useful and helpful to study the output in an economy.
(c)
Value Index
Value index numbers compare the total value of a certain period with total
value in the base period. Here a total vale is equal to the price of commodity
multiplied by the quantity consumed.
NOTATION: For any index number, two time periods are needed for comparison.
These are called the Base period and the Current year. The period of
the year which is used as a basis year and the other is the current year.
The various notations used are as given below:
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Price of current year
Price of base year
Quantity of current year
Quantity of base year
Definition (The weight): Different weights are used in different part of the country
for a particular commodity. For instance, “congo” – western part of Nigeria, “mudu”
– northern part etc.
Problems in the construction of Index numbers
No index number is an all purpose index number. Hence, there are many
problems involved in the construction of index numbers, which are to be tackled by
an economist or statistician. They are:
1. Purpose of the index numbers
2. Selection of base period
3. Selection of items
4. Selection of source of data
5. Collection of data
6. Selection of average
7. System of weighting
METHOD OF CONSTRUCTION OF INDEX NUMBERS
Index numbers may be constructed by various methods as shown below:
INDEX NUMBERS
Un weighted
Simple
aggregate
index
numbers
Weighted
Simple
average of
price
relative
53
Weighted
aggregate
index
number
Weighted
average of
price
relative
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Simple Aggregate Index Number
This is the simplest method of construction of index numbers. The prices of
the different commodities of the current year are added and the sum is divided by
the sum of the prices of those commodities by 100. Symbolically,
S.A.P.I,
Where
total prices for the current year
total prices for the base year
And we noted that Price Relative Index
.
Example 1: Calculate index numbers from the following data by simple aggregate
method taking prices of 2007 as base.
Commodity
Price per unit (in N)
2007
2013
A
80
95
B
50
60
C
90
100
D
65
97
Solution:
Commodity
Price per unit (in N)
2007
2013
A
80
95
B
50
60
C
90
100
D
65
97
285
352
Total
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S.A.P.I,
Simple Average Price Relative Index
In this method, first calculate the price relative for the various commodities
and then average of these relative is obtained by using arithmetic mean and
geometric mean. When arithmetic mean is used for average of price relative, the
formula for computing index is
S.A. of P. R. by arithmetic mean
where n is the number of items.
When geometric mean is used, the following formula is adopted:
Example 2: From the following data, construct an index for 2012 taking 2011 as
base by the average of price relative using (a) arithmetic mean (b) geometric mean.
Commodity
Price in 2011
Price in 2012
Beans
50
70
Elubo
40
60
Rice
80
100
Garri
20
30
Solution:
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Commodity Price in 2011 ( ) Price in 2012( )
Beans
50
70
140
2.1461
Elubo
40
60
150
2.1761
Rice
80
100
125
2.0969
Garri
20
30
150
2.1761
Total 565
8.5952
(a) Simple average of price relative index by arithmetic mean
(b) Price relative index number using geometric mean
.
Weighted Aggregate Index Number
In order to attribute appropriate importance to each of the items used in an
aggregate index number, some reasonable weights must be used. There are various
methods of assigning weights and consequently index numbers have been devised
of which some of the most important ones are:
1. Laspeyre’s method
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2. Paasche’s method
3. Fisher’s Ideal method
4. Bowley’s method
5. Marshall – Edgeworth method
6. Kelly’s method
1. Laspeyre’s method
This is a weighted aggregate price index, where the weights are determined
by quantity in the based period and is given by:
Laspeyre’s price index
.
2. Paasche’s method
It is a weighted aggregate price index in which the weight are determined by
the quantities in the current year. The formula for constructing the index is given
by:
Paasche’s price index
.
3. Fisher’s Ideal method
This is the geometric mean of the Laspeyre and Paasche indices. Symbolically,
Fisher’s Ideal index method
.
4. Bowley’s method
This is the arithmetic mean of Laspeyre’s and Paasche’s method. Symbolically,
Bowley’s price index number
.
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5. Marshall – Edgeworth method
In this method, the current year as well as base year prices and quantities are
considered. The formula in using to get this is given as follows:
Marshall – Edgeworth price index
.
6. Kelly’s method
Kelly has suggested the following formula for constructing the index
number:
Kelly’s price index number
where
. Here the average of the quantities of two years is used as
weights.
Example 3: Construct price index number from the following data by applying (a)
Laspeyre’s method (b) Paasche’s method and (c) Fisher’s ideal method.
Commodity
2000
Price Quantity
2012
Price Quantity
A
2
8
4
5
B
5
12
6
10
C
4
15
5
12
D
2
18
4
20
Solution:
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Commodity
Base year
Current year
Quantity Price
Quantity
Price
A
10
3
8
4
B
20
15
15
20
C
2
25
3
30
Solution:
Commodity
A
10
3
8
4
30
24
40
32
B
20
15
15
20
300
225
400
300
C
2
25
3
30
50
75
60
90
500
422
380
324
(a) B.P.I.N
.
(b) M.E.P.I.N
.
Example 5: Calculate a suitable price index from the following data.
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Commodity
Quantity
Price
1999
2000
X
20
2
4
Y
15
5
6
Z
8
3
2
Solution: Here the quantities are given in common and so we can use Kelly’s index
price number.
K.P.I.N
.
Remark:
1. For Quantity or Volume index number, interchange
and
in the formula for
Laspeyre, Paasche and Fisher’s ideal.
2. For Consumer Price index number:
(a) Aggregate Expenditure method:
Consumer Price Index Number
(based on Laspeyre)
(b) Family Budget method or Method of Weighted Relative
Consumer Price Index Number
weight i.e.
where
and
value
.
Exercise 6
1. Five feed components are to be used in the construction of an animal
feedstuff index number. From the figures given in the following table,
calculate
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(i)
simple average of price relative index by arithmetic mean and by
geometric mean.
(ii)
Laspeyre’s price index number
(iii)
Paasche’ price index number
(iv)
Fisher’s ideal index number
(v)
Bowley’s price index number
(vi)
Marshall – Edgeworth price index number
(vii) Kelly’s price index number
Component
1914
Price per ton
2014
Consumption(tons)
Price per ton Consumption
A
40
3600
41
2750
B
39
2750
53
1500
C
38
2050
35
2350
D
37
500
30
750
E
36
1475
24
2850
2. From the following data, compute quantity indices by
(i) Laspeyre’s method (ii) Paasche’s method (iii) Fisher’s method.
2008
Commodity
Price
2010
Total value
Price
Total value
X
10
100
12
180
Y
12
240
15
450
Z
15
225
17
340
Hint: Quantity
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3. Construct the consumer price index number for 2012 on the basis of 2002
from the following data using Aggregate expenditure method.
Price in
Commodity
Quantity consumed
2002
2012
A
100
8
12
B
25
6
7
C
10
5
8
D
20
15
18
4. Calculate consumer price index by using Family Budget method for year 2013
with 2012 as base year from the following data.
Price in
Items
Weights
2012 (in N) 2013 (in N)
Food
35
150
140
Rent
20
75
90
Clothing
10
25
30
Fuel
15
50
60
Miscellaneous
20
60
80
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