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MADE BY
HAMZA SHAUKAT
I.A.S DEPARTMENT PUNJAB UNIVERSITY LAHORE PAKISTAN
Classifications of
DATA
BYSHAHZAD
Definition:
“The process of arranging data into classes or categories according to some common characteristics present in the data is called classification.”
Four important bases for classification : Qualitative Quantitative Geographical Chronological
Qualitative : “When data are classified by attributes it is said to be
qualitative.”
Example: religion , sex and marital status.
Quantitative :“When data are classified by quantitative characteristics
it is called quantitative classification.”
Examples : Height , weight , income etc .
Geographical :“When data are classified by geographical
regions or locations it is called geographical classification .”
Example : The population of a country ma be classified by provinces , districts , divisions and
towns .
Chronological or temporal :“When data are arranged by their time of
occurrence it is called chronological or temporal.”
(An arrangement of data by their time of occurrence is called a time series.)
Types of classification
Data may be classified by one , two ,three or more characteristics at a time .
One way Two way Three way Many way
One way classification :
“When data are classified by one characteristics , classification is said to be one way .”
Example :The population of a country may be classified as Muslims , Christians , Hindus etc .
Two way classification :
“When data are classified by two characteristics at a time, classification is said to be two way .”
Examples:
Three way classification :
“When data are classified by three characteristics at a time, classification is said to be three way .”
Example:
Many way classification :
“When data are classified by many characteristics at a time, classification is said to be many way .”
Example:
Classification of Qualitative Data
“In classifying qualitative data , we may divide a characteristic into two , three or many subclasses.”
Two fold division (Dichotomy):If the characteristic is divided into two subclasses, one
possessing the characteristic and the other not possessing it. This is called two fold division or
Dichotomy.Example:
If we are studying the literacy of a population, we may divide the population into two categories that is literate and illiterate.
Three fold division(trichotomy)
When we divide a characteristic into three subclasses it is called a three fold division or
trichotomy.example :
Manifold division :When we divide a characteristic into many sub classes
it is called a manifold division .Example : division on basis of religion like Muslims , Christians, Hindus and others.
Table : Def : “A table is a systematic arrangement of data
into vertical columns and horizontal rows”
Tabulation :The process of arranging data into rows and columns
is called tabulation .Types of tabulation : Simple Double Treble complex
Simple tabulation :“When tabulation corresponds to one way classification , it is called simple tabulation.”
example : Tabulation of data on population of a country classified by one characteristics (religion or marital status )
Double tabulation:When tabulation corresponds to two way classification
it is called double tabulation. example : tabulation of data classified by religion and sex or religion or marital status is an
example of double tabulation .
Complex tabulation :When tabulation corresponds to many way classification it is called complex tabulation.
example : tabulation of data on the population of a country classified by age , sex , religion , marital status etc.
statistical table
Statistical table has at least four parts The title The stub The box head The body in addition some tables have1. Prefatory note2. Foot note3. Source note
POPULATION OF PUNJAB AND BALOCHISTAN PROVINCES BY SEX FOR 1961
AND 1972 CENSUS
Prefatory note ( figures in thousands)
All areas including Gawadar foot note Source: population senses report, 1961&1972
Punjab Balochistan
Census Male Female Total Male Female Total
1961 13,643 11,938 25,581 640 521 1,161
1972 19,942 17,566 37,508 1,272 1,133 2,405
Title : A title is a heading at the top of the table describing
its contents. title is usually in capital through out. If the title requires two or more lines, it is arranged to
form an interval pyramid.Boxhead : The headings for various columns are called column
captions. The portion of the table containing column captions is called boxhead.
Stub:The headings for various rows are called row captions.
The portion of the table containing row captions is called stub.
Prefatory note :It is used to explain certain characteristics of the data .
the prefatory note appears between the title and the body of the table . it throws light on a table as a
whole.
Foot note:A foot note appears immediately below the body of table
it is used to explain a single fact or a part of the table.
Source note:Source note is placed immediately below the table but
after the foot note, if any. Every table must have the source note unless data are original.
Use of zeros:Zero should not be used in a table. When no case have been
found to exist or when the value of an item is zero, this is indicated by means of dots (…..) or short dashes (-----).
Raw data:Def. “collected data which have not been organized
numerically are called raw data”.67 63 57 85 67 60 75 55 67 68 51 54 45 57 64 68 67 86 63 60 98 83 76 70 56 50 74 74 67 77 61 85 66 66 60 61 58 56 56 57 60 60 63 64 85 80 75 75 57 58 59 58 58 61 62 91 74 72 57 73 61 86 64 91 64 64 61 62 69 5781 66 65 81 82 76 77 81 76 66 62 63 62 63 60 60 60 72 72 79 70 70 58 78 58 71 76 60 60 65 60 73 73 71 73 66 73 76 68 69 68 73
73 68 74 68 67 76 52 79
“An arrangement of raw numerical data in ascending or descending order of magnitude is called an Array”
45 50 51 52 54 55 56 56 56 57 57 57 57 57 57 58 58 58 58 58 58 59 60 60 60 60 60 60 60 60 60 61 61 61 61 61 62 62 62 62 63 6363 63 63 64 64 64 64 64 65 65 65 65 66 66 66 66 66 66 67 67 67 67 67 67 67 68 68 6868 68 68 69 69 70 70 70 71 71 72 72 72 73 73 73 73 73 73 74 74 74 74 75 75 75 76 76
76 76 76 77 77 78 79 79 80 81 81 81 82 83 85 85 85 86 86 91 91 98
Class limits and class boundaries:
Def: “each class is defined by two numbers, these numbers are called class limits. The smaller number is called lower class limit and larger number is called upper class limit”
Example: 45 and 49
Lower class limit Upper class limit As measurements are seldom are exact so, 45kg is interpreted as (weight lying between 44.5kg & 45.5kg)Similarly, 49kg is interpreted as (weight lying between 48.5kg & 49.5kg) The values 44.5 and 49.5 are called true class limits or class
boundaries.
Open end classes:
Some times frequency tables are formed in which a class has either no lower class limit or no upper class limit.
Example: In the class “below 5” there is no lower class limit and in the class “25 and above” there is no upper class limit. Such class is called an open end class.
Class mark or mid point: The class mark or mid point is that value which
divides a class into 2 equal parts.
Size of Class interval
BY RAFIA HAMEED
Size of Class interval
“The size of class interval is the difference between the upper class boundary and the lower class boundary.”
More about Class Interval Class intervals are generally equal in width and
are mutually exclusive. The ends of a class interval are called class
limits, and the middle of an interval is called a class mark.
Class interval is generally used to draw histogram.
In following table, the class interval for data 49.5_44.5=50-45=52-47= 5
Weight Class boundaries
Frequency
45_49 44.5_49.5 1
50_54 49.5_54.5 4
55_59 54.5_59.5 17
60_64 59.5_64.5 28
65_69 64.5_69.5 25
70_74 69.5_74.5 18
75_79 74.5_79.5 13
80_84 79.5_84.5 6
85_89 84.5_89.5 5
90-94 89.5-94.5 2
95-99 94.5-99.5 1
Formation of frequency distribution
“The organization take raw data in table form with classes and frequencies”.
Determine the greatest and smallest value and find range.
e.g in weight of 120 students Greatest no=98 Smallest no=45 Range = 98-45=53 Decide on the number of classes. e.g its alright to have 5 to 20 classes no hard and
fast rules.
Determine the approximate class interval size. i.e dividing range by desirable class no. e.g class interval size=53 now 53/11=4.8 or say 5
Decide what should be the lower class limit, it should cover smallest value in raw data.
e.g here it is 44.5
Find the upper class boundary by adding the class interval size in lower class boundary.
e.g lower class boundary=44.5 class interval size=5 Upper class boundary=44.5+5=49.5
Distribute the raw data into classes and determine the the cases falling in each class i.e the class frequencies. There are two methods:-
a) By listing the actual values. b) By using tally marks. Weight
(lb) Tally marks Frequenc
y
45_49 / 1
50_54 //// 4
55_59 //// //// //// // 17
60_64 //// //// //// //// //// /// 28
65_69 //// //// //// //// //// 25
70_74 //// //// //// /// 18
75_79 //// //// /// 13
80_84 //// / 6
85_89 //// 5
90-94 // 2
95-99 / 1
Cumulative frequency distribution
“The number of values less than the upper class boundary for the current class. This is a running total of the frequencies”.
e.g. cumulative frequency of class 50_54 is 1+4=5 cumulative frequency of class 50_59 is 1+4+17=22 It means 22 students have weight less then 59.5
and 5 students have weight less then 54.5.
Weight Class
boundariesFrequency Cumulative
frequency
45_49 44.5_49.5 1 1
50_54 49.5_54.5 4 1+4=5
55_59 54.5_59.5 17 5+17=22
60_64 59.5_64.5 28 22+28=50
65_69 64.5_69.5 25 50+25=75
70_74 69.5_74.5 18 75+18=93
75_79 74.5_79.5 13 93+13=106
80_84 79.5_84.5 6 106+6=112
85_89 84.5_89.5 5 112+5=117
90-94 89.5-94.5 2 117+2=119
95-99 94.5-99.5 1 119+1=120
Relative frequency distribution
“The frequency of a class divided by the total frequency is called relative frequency”. This gives the percent of values falling in that class.
e.g relative frequency of class 70_74 is (18/120)=15%
Weight Relative frequency
45-49 1/120=0.0083 or 83%
50-54 4/120=0.0333 or 3.33%
55-59 17/120=0.147 or 14.17%
60-64 28/120=1.2333 or 23.33%
Relative cumulative frequency
“The running total of the relative frequencies or the cumulative frequency divided by the total frequency is called relative frequency”.
Gives the percent of the values which are less than the upper class boundary.
e.g. relative cumulative frequency of weight less then 69.5 is (75/120)*1oo=62.5%
which means 62.5% of students have weight less then 69.5kg.
weight Relative cumulative frequency
Less than 44.5 0%
Less than 49.5 1/120=0.0083 or 83 %
Less than 54.5 5/120=0.0417 or 4.17%
Less than 59.5 22/120=0.1833 or 18.33%
Less than 64.5 50/120=0.4167 or 41.67%
Less than 69.5 75/120=0.6250 or 62.5%
Less than 74.5 93/120=0.7750 or 77.5%
Less than 79.5 106/120=0.8833 or 88.33%
Less than 84.5 112/120=0.9933 or 93.33%
Less than 89.5 117/120=0.9750 or 97.5%
Less than 94.5 119/120=0.9917 or 93.17%
Less then 99.5 120/120=1 or 100%
Bivariate frequency Distribution
“Constructing frequency distribution by taking two variables is called bivariate frequency”.
e.g we have height in inches and weights in pounds of 50 students at a certain college.
Weight Height
60-62 63-65
66_68 69-71 72-74
100_104 ////
105-109 //// //// ////
// /
110-114 /// //// / /
115_119 / //// // //// / //
WEIGHT
HEIGHT
60-62 63-65 66-68 69-71 72-74 TOTAL
100_104 5 - - - - 5
105-109 5 10 2 1 - 18
110-114 - 4 6 1 - 11
115_119 - 1 7 6 2 16
TOTAL 10 15 15 8 2 50
GRAPHS
BY UMMEFAQIHA
Graphs:
o A graph consists of curve or straight lines.o Data can be effectively presented by means of
graphso They provide a very good method of showing
fluctuations and trends in statistical datao Only disadvantage is that they don’t convey
accurate information
Important rules for drawing graphs:
o Every graph must have a clear and comprehensive title
o Source of the data must be giveno X-axis and Y-axis should be taken on horizontal
axis and vertical axis respectivelyo Axes should be labeled correctly and graph must
not be over crowded.
Line chart:
o On this chart the two coordinates of a point measured along two perpendicular axes X and Y from a fixed point called the origin , are taken to represent numerical values of two characteristics of an individual
o Usually x-axis represents qualitative characteristics. e.g. time, age etc
YEAR PRODUCTION
1959 928
1960 1088
1961 1326
1962 1456
1964 1767
year
Cigarettes(crore tons)
1959 1960 1961 1962 19640
200
400
600
800
1000
1200
1400
1600
1800
2000Line chart showing production of
cigarrete in pakistan during 1959 to 1964
PRODUCTION
Graphs of frequency distributions
The important graphs of frequency distributions are Histogram frequency polygon frequency curve cumulative frequency polygon or Ogive
Histograms
A histogram consists of a set of adjacent rectangles having bases along the x-axis with centers at the class marks (i.e. marked off by class boundaries) and areas proportional to the class frequencies
To draw a histogram (for equal class intervals) ,class boundaries are marked along the x-axis and frequencies are marked along on y-axis.
47 52 57 62 67 72 77 82 87 92 970
5
10
15
20
25
30
Histogram for the frequency dis-tribution of weight of 120 stu-
dents
weight(kg)
no o
f stu
den
ts (
fre
qu
en
cy)
Frequency polygon:
A frequency polygon is a many sided closed figure. it is constructed by plotting the class marks(mid-points) and then joining the resulting points by means of straight lines
A frequency polygon can also be obtained by joining the mid points of the tops of rectangles in the histogram
To construct it, we mark the class marks along the x-axis and class frequencies along y-axis
After plotting points, they are joined by straight lines. Extra classes are taken on both ends to make frequency polygon a closed figure
Frequency polygon for frequency distribution of weight of 120 students
Weight (kg)
Relative frequency histogram and relative frequency polygon
Graphic representation of relative frequency distribution can be obtained from the histogram or frequency polygon simply by changing the y-axis from frequency to relative frequency on a graph
The resulting graphs are called relative frequency polygon or percentage frequency polygon respectively
Cumulative frequency polygon or Ogive
A graph showing the cumulative frequencies plotted against the upper class boundaries is called a cumulative frequency polygon or an ogive
If we use relative cumulative frequencies in place of cumulative frequencies, the resulting graph is called a relative cumulative frequency polygon or percentage ogive
The graphs corresponding to a “less than” and an “or more” cumulative frequency distributions are called “less than "and “or more” ogives respectively
44.5
- 49
.5
54.5
- 59
.5
64.5
- 69
.5
74.5
-79.
5
84.5
- 89
.5
94.5
- 99
.50
20406080
100120140
Ogive for the less than commulative frequency distribution of weight of
120 students
COMMULATIVE FREQUENCY
Frequency curve
Smoothed curve by joining the lowest and highest points of frequency polygon is frequency curve
47 52 57 62 67 72 77 82 87 92 9705
1015202530
frequency curve for the frequency distribution of weight of 120 stu-
dents
no o
f stu
den
t (f
req
uen
cy)
Common shapes of frequency curve
Common shapes of frequency curve The frequencies curves arising in practice take
on certain characteristics shapes and are generally classified as
1) Symmetrical or bell shaped curve2) Moderately asymmetrical curve3) J shaped and reverse j shaped curves4) U shaped curves5) Bimodal & multimodal curve
Symmetrical or bell shaped curve (observations are equidistant from the central maximum)
Moderately asymmetrical curve (in these curves , the tail of the curve to one side of the central maximum is longer than that to the other)
J shaped and reverse j shaped curve (a j shaped curve starts at a low point on the left hand and goes higher and higher towards extreme right and reverse j shaped curve starts with a high point on the right and goes to the extreme left)
J-shaped curve
Reverse J-shaped curve
U shaped curve (a frequency curve with a low spot on middle and high spots at both curves)
Bimodel and multimodel frequency curve (a bimodel curves has 2 maximas while a multimodel frequency curve has more than 2 maximas)
BAR CHARTS
BY HAMZA
BAR CHART
A bar graph is a visual display used to compare the amounts of occurrence of different characteristics of data.
This type of display allows us to:• compare groups of data, and• to make generalizations about the data quickly.
TYPES OF BAR CHART
BAR CHART
SIMPLE BAR CHARTCOMPONENT BAR
CHART
SIMPLE BAR CHART Simple bar chart includes
Single bar
chart
Multiple bar chart
Rules of making simple bar chats.
1 •Vertical bars are used to represent data classified on quantitative or chronological basis .
2 •Horizontal bars are used to represent data classified on qualitative or geographical basis.
3 •The bar should neither be short and wide nor very long and narrow.
4 •Bars should be separated by spaces which are not less than half the width of a bar and greater than the width of a bar.
SINGLE BAR CHART
Single bar charts includes Horizontal and vertical bars.
Example Horizontal bar chartYear 1981 198
21983 1984 1985
Production of wheat(Lac tons)
115 113 124 109 117
1981 1982 1983 1984 1985
100
110
120
130
Horizontal bar chart showing production of wheat in Pakistan for the year 1981 to
1985Production (Lac tons)
produc-tion(Lac tons)
years
EXAMPLE
CHINA
INDIA
INDONESIA
JAPAN
PAKISTAN
100 300 500 700 900 1100
CHINA INDIA IN-DONE-
SIA
JAPAN PAK-ISTAN
POPU-LA-TION(MIL-LION)
1088 816 175 123 106
Vertical bar chart showing population of different countries
country
population (million)
MULTIPLE BAR CHART
It is the extension of simple bar chart. It consists of multiple bars used to
represent data.
EXAMPLEYEAR IMPORTS(IN
CRORES OF RUPEES)
EXPORTS (IN CRORES OF RUPEES)
1982 – 83 6815 3444
1983 – 84 7671 3733
1984 - 85 8978 3798
1982 – 83 1983 – 84 1984 - 85 0
2000
4000
6000
8000
10000
Multiple bar chart showing imports and exports of Pakistan for year 1982 – 83
to 1984 - 85
IMPORTS(IN CRORES OF RU-PEES)
Cro
re r
u-
pee
s
year
COMPONENT BAR CHART
This bar chart consists of horizontal or vertical bars which are sub-divided into two or more parts.
EXAMPLEYEAR MAJOR
CROPSMINOR CROPS
OTHERS
TOTAL
1972 – 73
1235 283 672 2190
1973 – 74
1533 378 897 2808
1974 – 75
1827 490 1047 3364
1972 – 73 1973 – 74 1974 – 75 0
500
1000
1500
2000
2500
3000
3500
4000
Component bar chart showing ma-jor, minor and other crops from
1972 - 73 to 1974 - 75
OTHERS
MINOR CROPS
MAJOR CROPS
Crops in(crore rupees)
year
EXAMPLEITEMS FAMILY A FAMILY B
FOOD 480 1200
CLOTHING 240 850
SERVICES 100 400
MISCELLANEOUS
120 300
TOTAL 940 2750
To find the % of each component we use the following formula
% of food = 480940
100×
ITEMS FAMILY A FAMILY B
FOOD 51 % 44 %
CLOTHING 25 % 31 %
SERVICES 11 % 14 %
MISCELLANEOUS
13 % 11 %
TOTAL 100 % 100 %
FAMILY A FAMILY B 0%
10%20%30%40%50%60%70%80%90%
100%
% Component bar chart showing expenditure of family A and family
B
MISCELLANEOUS SERVICES CLOTHING FOOD
PIE CHARTS
IMRAN NADEEM
Pie chart Used to compare the relation between
the whole and its components. Difference between bar chart and pie
chart.. Circles are drawn proportional to the
square root of the quantities to be represented.
Draw a circle with some suitable radius(square root of the total)
To show the components by sectors we calculate angles for each sector by the
formula… component part × 360
total
Construction of a pie chart
Example: Draw a pie chart to show the distribution of Punjab Govt. employees by their academic qualifications
ACADEMIC QUALIFICATIONS
NO. OF EMPLOYEES
ANGLE OF SECTOR(DEGREES)
PERCENTAGE
NO EDUCATION 47 47/296*360=57 15.9%
PRIMARY 25 25/296*360=2010
8.4%
MIDDLE 63 63/296*360=12628
21.3%
METRIC 97 97/296*360=33276
32.8%
INTERMEDIATE 26 26/296*360=10048
8.8%
BACHELOR 23 23/296*360=9576
7.8%
MASTER 15 15/296*360=6480
5%
TOTAL 296 360 100
ComparisonWith the help of pie chart, we can find
comparison between any two things.Example: Construct pie chart to compare
the budgets of two families A & B
Items of Expenditure
Family A Family B
Food 600 1050
Clothing 150 750
House rent 180 600
Education 150 450
Miscellaneous 120 150
Total 1200 3000
Item of expenditur
e
Family A Family B
Angle of sector
Percentage Angle of sector
Percentage
Food 600/1200*360=180
50% 1050/3000*360=126
35%
Clothing 150/1200*360=45
12.5% 750/3000*360=90
25%
House rent 180/1200*360=54
15% 600/3000*360=72
20%
Education 150/1200*360=45
12.5% 450/3000*360=54
15%
Miscellaneous
120/1200*360=36
10% 150/3000*360=18
5%
Family A Family B
The Stem- and -Leaf Plot
Suppose the data are represented by x1,x2,…xn and that each number xi consist of at least two digits. To construct a stem-and-leaf plot, we divide each number xi into two parts: a stem consisting of one or more of the leading digits; and a leaf
consisting of the remaining digits. The following example will illustrate the
construction of a stem-and-leaf plot.
Example
The following table represent weight measurements in kilogram of forty individual in a locality. Construct a
stem-and-leaf display of these measurements.
48 53 49 52 51 52 63 60 53 64 59 58 47 49 45
64 79 65 62 60 68 65 73 88 69 83 78 81 86 92
75 85 81 77 82 76 75 91 73 92.
First select the values 4, 5, 6, 7, 8 and 9 as stems.
The resulting table is given..Stems Leaf Frequen
cy
4 8 9 7 9 5 5
5 3 2 1 2 3 9 8 7
6 3 0 4 4 5 2 0 8 5 9 10
7 9 3 8 5 7 6 5 3 8
8 8 3 1 6 5 1 2 7
9 2 1 2 3
In this display we use stems twice in two lines.. We use one line for leaves o, 1, 2, 3, 4 and the other for 5, 6, 7, 8, 9. the
stems are repeated with the * shown for the leaves 0, 1, 2, 3, 4 and • for the
leaves 5, 6, 7, 8, 9.
Example: Construct a stretched stem-and-leaf plot ..
Stretched Stem-and-Leaf Plot
48 53 49 52 51 52 63 60 53 64 59 58 47 49 45
64 79 65 62 60 68 65 73 88 69 83 78 81 86 92
75 85 81 77 82 76 75 91 73 92.
Stretched stem-and-leaf plot for 40 measurements:
Stem Leaf frequency
4* 0
4• 8 9 7 9 5 5
5* 3 2 1 2 3 5
5• 9 8 2
6* 3 0 4 4 2 0 6
6• 5 8 5 9 4
7* 3 3 2
7• 9 8 5 7 6 5 6
8* 3 1 1 2 4
8• 8 6 5 3
9* 2 1 2 3
9• 0
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