UNIT I.

UNIT I

Variable: a variable is a value that may change within the scope of a given problem or set of operations. Data: The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data Collection: Data collection is a term used to describe a process of preparing and collecting data. Data are generally classified into following two groups: Internal data External data

Internal data: It comes from internal sources related with the functioning of an organization or firm where records regarding purchase, production, sales, profits etc. are kept on regular basis. External data: The external data are collected and published by external agencies. The external data can further classified as: Primary data Secondary data Primary Data: These are original and first hand information. Secondary data: these are one which are already been collected by a source other than the present investigator.

Sources of data External Primary Secondary Internal

The Collected data or raw data or ungrouped data are always in an unorganized form and need to be organized and presented in meaningful and readily comprehensible form in order to facilitate further statistical analysis. Classification: It is the process of arranging things in the groups according to their resemblances and affinities and gives expression to the unity of attributes that may subsist amongst a diversity of individuals. Or in simple words it is grouping of data according to their identity, similarity or resemblances. For eg. Letters in the post office are classified according their destinations viz., Delhi, Raipur, Agra, Kanpur etc.

Types of Classification Chronological or Temporal Classification
Geographical or Spatial Classification Qualitative classification Quantitative Classification

Example: The estimates of birth rates in India during (1970-79) are:
Chronological or Temporal classification: In Chronological classification, the collected data are arranged according to the order of time expressed in years, months, weeks etc. The data are generally classified in ascending order of time. Example: The estimates of birth rates in India during ( ) are: Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 Birth rate 36.8 36.9 36.6 34.6 34.5 35.2 34.2 33.0 33.3

When the names of countries are in alphabetical order:
Geographical or Spatial classification: In this type of classification the data are classified according to geographical region or place. The observations are either classified in the alphabetical order of the reference places or in the order of size of the observation. Example: When the names of countries are in alphabetical order: When observations are in descending order: Country America China Denmark France India Yield of wheat 1925 893 225 439 862 Country America China India France Denmark Yield of wheat 1925 893 862 439 225

Qualitative classification: In this type of classification data are classified on the basis of some attributes or quality like literacy, religion, employment etc. Such attributes cannot be measured along with a scale. When the classification is done w.r.t on attribute, which is dichotomous in nature, two classes were formed, one possessing the attribute and the other not possessing the attribute. This type of classification is called Simple or dichotomous classification.

The classification where two or more attributes are considered and several classes are formed, is called a manifold classification.

Population Urban Male Female Rural

Quantitative classification: The collected data are grouped with reference to the characterstics which can be measured and numerically described such as height, weight, sales, imports, age, income etc.

Example: Consider the marks of 50 students Ungrouped data
If data is arranged in ascending or descending order of magnitude then it is said to be an array. Example: Consider the marks of 50 students Ungrouped data Arranged in array 21 50 42 75 55 67 74 47 64 71 61 40 25 54 37 88 44 31 70 81 51 45 63 49 43 35 68 38 59 57 29 66 56 84 32 79 78 21 31 40 45 51 56 61 66 71 79 25 32 42 47 54 63 67 74 81 35 43 49 55 57 75 84 29 37 44 50 58 64 68 38 59 70 78 88

Diagrammatical representation: In this presentation we make use of geometric figures like bars, squares, rectangles, circles etc.

One Dimensional diagrams
Types of Diagrams One Dimensional diagrams Two dimensional diagrams Pictograms Cartograms

1. One Dimensional diagrams: One Dimensional diagrams are also called Bar diagrams, widely used diagrams for the visual presentation of data.

One Dimensional diagram Simple Bar Diagram
Multiple bar diagram Subdivided bar diagram Percentage bar diagram Deviation Bar Diagram Broken bars

i. Simple Bar-diagrams: It consists of number of rectangles and is used only for one-dimensional comparisons. It is generally used to show changes in the magnitudes of a phenomenon over time or space.

Present the data with a suitable diagram
Example: Draw a bar diagram to represent the following data related to a school Present the data with a suitable diagram Year 1990 1991 1992 1993 1994 1995 No. of students 210 242 290 315 340 355

Represent the data by suitable diagram.
ii. Multiple Bar-diagrams: It is used when a comparison is to be made between two or more variables. These are also used for comparing magnitudes of one variable in two or three aspects. Example: Following data relate to the faculty-wise enrolment of students in a college: Represent the data by suitable diagram. Years 1993 1994 1995 No. of arts students 95 110 120 No. of science students 160 170 165 No. of commerce students 75

Represent the data by suitable diagram
iii. Subdivided Bar-diagrams: also known as components bar diagram, is useful in a situation when it is necessary to show and compare the breakup of one variable into several components. Example: Following data relate to year wise enrolment in a college, classified according to sex: Represent the data by suitable diagram Year No. of girls 810 825 844 780 820 No. of Boys 1215 1160 1325 1410 1480 Total 2025 1985 2169 2190 2300

iv. Percentage Bar-diagrams: The construction of percentage bar diagram is similar to the subdivided bar chart. The difference between the two is that, in subdivided bar diagram , the component parts are shown in absolute quantities, while in the percentage bar diagram, the component parts are transformed into percentages of the total. In this diagram all the bars are of equal heights. These bars are then divided in terms of percentages of the components.

Represent the data by percentage bar diagram.
Example: Following data relate to the faculty-wise enrolment of students in a college: Represent the data by percentage bar diagram. Years 1993 1994 1995 No. of arts students 95 110 120 No. of science students 160 170 165 No. of commerce students 75

Data can be represented as
Year % of students Total % Arts Science Commerce 1993 28.79 48.48 22.73 100 1994 29.33 45.34 25.33 1995 30.38 41.77 27.85

v. Deviation Bar-diagrams: These are used to show the magnitudes of a phenomenon, i.e. net profit, net loss, net exports or imports etc. Bars in these diagrams can assume both negative and positive values. Example: Depict the following data by a suitable diagram (Balance of trade=Export-Import) Year Export Import Balance of trade (Millions Rs.) 1993 98 115 -17 1994 110 140 -30 1995 96 +19 1996 120 100 +20

Example: The following data relate to sales in five firms A,B,C,D,E.
vi. Broken bars: It is used to represent series having wide variations in values. Example: The following data relate to sales in five firms A,B,C,D,E. Use a suitable bar diagram to represent the data. Firms A B C D E Sales (in Lakh Rs.) 25 38 300 200 56

2. Two dimensional diagrams: Such diagrams are useful in situations when the proportion between the magnitudes of the given values of the variable is quite large.

Two Dimensional diagram Rectangle diagram
Square and circle diagram Pie diagram Multiple pie diagram

i. Rectangle Diagrams: These diagrams are used for two dimensional comparisons. These rectangles vary in height as well as in the width, so that the areas of rectangles represent the magnitude of the variable over time or space or over some other characteristic of variation.

Example: The following data represent the expenditure of the two families on various items. Represent the data by a rectangle diagram. S. No. Items Expenditure (Rs.) Family A Family B 1 Food 1200 1700 2 Clothing 500 800 3 House Rent 600 900 4 Fuel and electricity 250 300 5 Miscellaneous 450 Total 3000 4500

Represent the data by a square and circle diagram.
ii. Squares and circle Diagrams: It is useful when the proportion between the magnitudes of the given value is quite large. For drawing squares, sides of squares are kept proportional to the magnitudes of the values and for circle diagrams, radii of the circles should be proportional. Example: The following data relate to the plan outlay of a country for three plans. Represent the data by a square and circle diagram. Five year plan I IV VII Outlay ( Rs. ‘000 crores) 196 2060 8820

Plan Outlay Side of square Or radius of circle Ratio I 196 14 0.7 IV 2060 45.39 2.26 V 8820 93.91 4.7

Square Diagram Plan I a=0.7” Plan IV a=2.26” Plan V a=4.7”

Circle Diagram Plan I r=0.7” Plan IV r=2.26” Plan V r=4.7”

Represent the data by a pie diagram.
iii. Pie- Diagrams: This diagram is generally used to compare the relations between various subdivisions of the value. Pie diagram is circle divided into sectors with areas equal to the corresponding components. A pie diagram shows the components or subdivisions in terms of percentages only and not in absolute terms. Example: The following data relate to faculty wise enrolment in a college Represent the data by a pie diagram. Faculty Science Arts Commerce Total No. of students 2010 1100 2390 5500

Faculty No. of students Angle in degree Science 2010 Arts 1100 Commerce 2390 Total 5500

iv. Multiple Pie Diagrams: A multiple pie diagram is used for two dimensional comparisons, where a variable value is shown over time, space or in terms of some other characteristic and the variable values are also broken into components.

Example: The following data represent the expenditure of the two families on various items. Represent the data by a multiple pie diagram. S. No. Items Expenditure (Rs.) Family A Family b 1 Food 1200 1700 2 Clothing 500 800 3 House Rent 600 900 4 Fuel and electricity 250 300 5 Miscellaneous 450 Total 3000 4500

S. No. Items Expenditure (Rs.) Angles in degrees Family A Family B 1 Food 1200 1700 2 Clothing 500 800 3 House Rent 600 900 4 Fuel and electricity 250 300 5 Miscellaneous 450 Total 3000 4500

Production of bulbs (In millions)
3. Pictorial diagrams or Pictogram: Statistical data may be represented with the help of pictures also. Such a presentation is called pictorial diagram or pictogram. In pictograms, the magnitude of the values are explained with the help of pictures. In a pictogram, a symbolic picture represents the total magnitude of the values. Example: The following data relate to the production of electric bulbs in a factory. Represent the data by pictogram. Year 1992 1993 1994 1995 Production of bulbs (In millions) 32 57 79 89

90 80 70 60 50 40 30 20 10 1992 1993 1994 1995

4. Cartograms or Maps: Statistical data classified according to geographical regions are also representable with the help of suitable maps. The representation of statistical data by maps is called cartogram.

Graphical representation: It is used in the situations when we observe some functional relationship between the values of the variables. It provides us an accurate conception of the shape of a frequency distribution. There are many forms of graphs which can be broadly classified as: Graphs of frequency distribution Graphs of time series or line graphs

Graphs of frequency distributions Histogram
Frequency Polygon Frequency Curve Cumulative frequency curve or Ogives

Graphs of frequency distribution: The graphs representing a frequency distribution are:
Histogram Frequency Polygon Frequency curve Cumulative frequency curve or ‘Ogive’

Example: The table below given the distribution of the age of members in a sports club
Age Group (years) No. of members 15-19 11 20-24 36 25-29 28 30-34 13 35-39 7 40-44 3 44-49 2

The smoothened frequency distribution will be
Age groups (years) No. of members 11 36 28 13 7 3 2

Histogram for above data is represented as

The following chart shows the frequency polygon

The following chart shows the frequency curve

Measures of Dispersion

The extent or degree to which data tend to spread around an average is called the dispersion or variation. Measures of dispersion help us in studying the extent to which observations are scattered around the average or central value.

Types of Dispersion: There are two types of measures of Dispersion
Absolute measure of Dispersion: These are expressed in the same unit in which the observations are given. Thus, absolute measures of dispersion are useful for comparing variation in two or more distributions where units of measurement is the same. Such measures are not suitable for comparing the variability of the distributions expressed in different units measurement.

b) Relative measure of dispersion: These are expressed as ratio or percentage or the coefficient of the absolute measure of dispersion. Relative measures are useful for comparing variability in two or more distributions where units of measurement may be different.

Various measures of Dispersion
The following are some important measures of dispersion: Range Interquartile Range and Quartile Deviation Mean Deviation or average deviation Standard Deviation

Range: Range is the simplest measure of Dispersion
Range: Range is the simplest measure of Dispersion. For a given set of observations, the range is the difference between the largest and the smallest observation. Thus Range=R=L-S Where L=the largest observation S= the smallest observation R= the Range In case of grouped data, the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Coefficient of Range: Range is an absolute measure of dispersion which is unsuitable for comparing variation in two or more distributions expressed in different units. So a relative measure of dispersion called the coefficient of range is defined as: Coefficient of range=

Compare the range of marks in the two subjects.
Example: Marks of 10 students in Mathematics and Statistics are given below: Compare the range of marks in the two subjects. Compare the coefficients of range for both the subjects. Marks in Mathematics 25 40 30 35 21 45 23 33 10 29 Marks in Statistics 39 42 20 18 19

Solution: Highest Marks in Mathematics = 45
Lowest marks in Mathematics = 10 Range of marks in Mathematics Coefficient of Range= Highest Marks in Statistics = 42 Lowest marks in Statistics = 18 Range of marks in Statistics The range as well as the coefficient of range for marks in Mathematics are higher than that of marks in Statistics.

Example: Find the range and coefficient of range from the following
Solution: Class limits are ( ), ( ),…………., ( ) Range Coefficient of range Mid value 5 10 15 20 25 30 35 Frequency 7 8 12 9

Interquartile Range and Quartile Deviation: Interquartile range includes the middle fifty percent of the distribution or it is the difference between the third quartile (Q3) and the first quartile (Q1). Interquartile Range= Q3-Q1 Quartile Deviation or semi interquartile range is defined as the average amount by which the two quartiles differ from the median. Quartile deviation or semi interquartile range=(Q3-Q1)/2

Quartile deviation is an absolute measure of dispersion
Quartile deviation is an absolute measure of dispersion. For comparing two or more distributions in respect of variation, the coefficient of quartile deviation is defined as Coefficient of Q.D.= Example: From the following information of wages of 15 workers, find interquartile range, quartile deviation and coefficient of Q.D. S.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Wages(Rs.) 520 550 440 580 450 620 470 680 400 490 420 480 500

Solution: Arrange the wages in ascending order
Interquartile Range = Quartile deviation= Coefficient of Quartile Deviation= S.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Wages(Rs.) 400 420 440 450 470 480 490 500 520 550 580 620 680

Example: Calculate quartile deviation and its coefficient from the following distribution:
Weekly income (Rs.) 58 59 60 61 62 63 64 65 66 No. of workers 2 3 6 15 10 5 4 1

Solution: Quartile deviation= Coefficient of Quartile Deviation=
Weekly income (Rs.) 58 59 60 61 62 63 64 65 66 No. of workers 2 3 6 15 10 5 4 1 Cumulative Frequency 11 26 36 41 45 48 49

Example: The following is the age distribution of 799 workers.
Find Quartile deviation and its coefficient. Solution: Age Group 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 No. of workers 50 70 100 180 150 120 59 Age Group 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 No. of workers 50 70 100 180 150 120 59 Cumulative Frequency 220 400 550 670 740 799

Quartile Deviation= Coefficient of Quartile deviation=

Mean Deviation or Average deviation: Mean deviation of a series is the arithmetic mean of the absolute deviations of various items from some central value, such as mean, median, mode. 1. For ungrouped data a) Mean deviation from mean b) Mean deviation from median c) Mean deviation from mode

2. For grouped data a) Mean deviation from mean b) Mean deviation from median c) Mean deviation from mode

Coefficient of Mean deviation: Mean deviation is an absolute measure of dispersion. The corresponding relative measure called coefficient of mean deviation, is obtained by dividing mean deviation by the average or central value used for calculating it. Coefficient of M.D.=

Example: Compute mean deviation from mean and its coefficient from the following data relating to the marks obtained by a batch of 11 students in a class test: Marks 10 70 50 53 20 95 55 42 60 48 80

Solution: Mean deviation= Coefficient of mean deviation=
Marks(X) 10 43 70 17 50 3 53 20 33 95 42 55 2 11 60 7 48 5 80 27 583 190

Example: Calculate mean deviation from median from the following data
Example: Calculate mean deviation from median from the following data. Also compute the coefficient of M.D. Size 2 4 6 8 10 12 14 16 Frequency 5 3 1

Solution: Median(Md)= Mean deviation= Coefficient of mean deviation=
X F C.F. 2 6 12 4 8 5 13 10 3 16 18 14 1 19 20 32 56

Also find the coefficient of M.D.
Example: Compute the mean deviation (M.D.) from mean from the following data. Also find the coefficient of M.D. Classes 0-20 20-40 40-60 60-80 80-100 Frequency 5 50 84 32 10 6

Solution: Mean= Mean deviation= Coefficient of mean deviation=
Classes Frequency (f) Mid point(x) fd 0-20 5 10 -3 -15 41 205 20-40 50 30 -2 -100 21 1050 40-60 84 -1 -84 1 60-80 32 70 19 608 80-100 90 39 390 6 110 2 12 59 354 187 -177 2691

Standard Deviation: The standard deviation is defined as the positive square root of the arithmetic mean of the squares of deviations of the observations from the arithmetic mean. For ungrouped data For grouped data or frequency distribution

Variance: The square of standard deviation is known as variance.
For ungrouped data For grouped data or frequency distribution

Example: Calculate Standard deviation from the following set of observations:
10 11 17 25 7 13 21 12 14

Solution: Mean = Standard deviation=
X X-14 (X-14)2 10 -4 16 11 -3 9 17 3 25 121 7 -7 49 13 -1 1 21 12 -2 4 14 140 274

Example: Calculate standard deviation of the following discrete frequency distribution
Size(X) 4 5 6 7 8 9 10 Frequency 12 15 28 20 14

Solution: Mean = Standard deviation=
Size(X) Frequency (f) d=X-7 fd fd2 4 6 -3 -18 54 5 12 -2 -24 48 15 -1 -15 7 28 8 20 1 9 14 2 56 10 3 45 100 238

Moments: Moments are used to describe the characteristics of a distribution. The moments of a distribution are the arithmetic mean of the various powers of the deviations of items from some given numbers.

Moments about mean (Central moment):
For an individual series: If be the n observations in a data set with mean then rth moment about the mean of a variable is defined as

b) For grouped data or frequency Distribution:
Let be the n observations in a data set with corresponding frequencies respectively. Then rth moment about the mean of a variable is defined as where

In particular

Moments about an arbitrary point (Raw moment):
For an individual series: If be the n observations in a data set then rth moment about arbitrary point A is defined as

b) For grouped data or frequency Distribution:
Let be the n observations in a data set with corresponding frequencies respectively. Then rth moment about arbitrary point A is defined as where

In particular

Moment about zero or origin:
Let be the n observations in a data set with corresponding frequencies respectively. Then rth moment about origin is defined as where

In particular

Relation between and In particular

Example: Calculate first four moments about mean from the following distribution:
1 2 3 4 5 6 7 8 F 28 56 70

Solution: Mean = X Frequency(f) fx X-4 f(X-4) f(X-4)2 f(X-4)3 f(X-4)4
1 -4 16 -64 256 8 -3 -24 72 -216 648 2 28 56 -2 -56 112 -224 448 3 168 -1 4 70 280 5 6 224 7 24 216 64 1024 512 2816

Example: The first three moments of a distribution about the value 2 of the variable are 1,16 and -40. Show that the mean is 3, the variance is 15, the third moment about mean is -86. Also show that the first three moments about the origin are 3,24 and 76.

Solution: Given that

Skewness: Skewness means lack of symmetry
Skewness: Skewness means lack of symmetry. A frequency distribution of the set of values that is not symmetrical is called asymmetrical or skewed. In a skewed distribution, extreme values in a data set move towards one side of a distribution. When extreme values moves towards the upper or right tail, the distribution is positively skewed. When extreme values moves towards the lower or left tail, the distribution is negatively skewed. The basic purpose of measuring skewness is to estimate the extent to which an distribution is distorted from perfectly symmetrical distributions.

Symmetrical distribution
Positively skewed distribution Negatively skewed distribution

Mean=Median=Mode Mean<Median<Mode Mean>Median>Mode

Measure of Skewness: The degree of skewness in a distribution can be classified as follows:
Absolute measure of skewness Relative measure of skewness

Absolute measure of skewness: Skewness can be measured in absolute terms by finding the difference between the mean and the mode or mean and median. Skewness = Mean-Mode Skewness = Mean-Median Skewness = Q3+Q1-2Median

Relative measure of skewness: The Relative measure of skewness is known as coefficient of skewness is obtained by dividing the absolute measure of skewness by any of the measure of dispersion.

Relative measure of skewness Karl Pearson coefficient of skewness
Bowley coefficient of skewness Kelly’s coefficient of skewness Method of moments

Karl Pearson’s coefficient of skewness: Karl Pearson’s coefficient of skewness is based on the difference between mean and mode and is given by

2. Bowley coefficient of skewness: This method is based on the fact that in a symmetrical distribution, the quartiles are equidistant from the median.

3. Kelly’s coefficient of skewness: Kelly’s coefficient of skewness is based on percentile and deciles.

4. Method of moments: It is denoted by Skm

Example: Calculate Karl Pearson’s coefficient of skewness from the following:
Marks above 10 20 30 40 50 60 70 80 No. of students 150 140 100 14

Solution: Median= Class Frequency(f) Cumulative frequency
Mid point (x) d=(x-45)/10 fd fd2 0-10 10 5 -4 -40 160 10-20 40 50 15 -3 -120 360 20-30 20 70 25 -2 80 30-40 35 -1 40-50 45 50-60 120 55 1 60-70 16 136 65 2 32 64 70-80 14 150 75 3 42 126 -86 830

Mean= Standard deviation= Coefficient of skewness(Skp)=

Example: From the following distribution, calculate the first four moments about mean, and coefficient of skewness based on moments: Income(Rs) 0-10 10-20 20-30 30-40 Frequency 1 3 4 2

Solution: Mean = X Frequency(f) Mid point(x) fx X-22 f(X-22) f(X-22)2
0-10 1 5 -17 289 -4913 83521 10-20 3 15 45 -7 -21 147 -1029 7203 20-30 4 25 100 12 36 108 324 30-40 2 35 70 13 26 338 4394 57122 10 220 810 -1440 148170

Coefficient of skewness(Skm)=

Example: Calculate Bowley’s coefficient of skewness from the following:
Solution: Wages (Rs.) 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of persons 1 3 11 21 43 32 9 Wages (Rs.) 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. of persons 1 3 11 21 43 32 9 Cumulative frequency 4 15 36 79 111 120

Median= Coefficient of skewness (SkB)=

Kurtosis: The measure of kurtosis describes the degree of concentration of observed frequencies in a given data. Kurtosis is used to test how near a frequency distribution conforms to normal curve or it is the degree of peakedness of a distribution, usually taken in relative to a normal distribution.

Measures of Kurtosis: Karl Pearson’s coefficient of kurtosis is defined as
The kurtosis of a distribution is also defined as If , the distribution is leptokurtic If , the distribution is platykurtic If , the distribution is mesokurtic

Example: Calculate the coefficient of skewness and kurtosis from the following data:
Profit(Rs. In lakh) 10-20 20-30 30-40 40-50 50-60 No. of companies 18 20 30 22 10

Solution: Class interval Frequency(f) Mid point(x) d=(x-35)/10 fd fd2
10-20 18 15 -2 -36 72 -144 288 20-30 20 25 -1 -20 30-40 30 35 40-50 22 45 1 50-60 10 55 2 40 80 160 100 -14 154 -62 490

Example: Prove that the frequency distribution curve of the following frequency distribution is leptokurtic: Class 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 Frequency 1 4 8 19 35 20 7 5

Solution: Class interval Frequency(f) Mid point(x) d=(x-32.5)/5 fd fd2
10-15 1 12.5 -4 16 -64 256 15-20 4 17.5 -3 -12 36 -108 324 20-25 8 22.5 -2 -16 32 128 25-30 19 27.5 -1 -19 30-35 35 32.5 35-40 20 37.5 40-45 7 42.5 2 14 28 56 112 45-50 5 47.5 3 15 45 135 405 50-55 52.5 64 120 212 1520

Example: The first four moments about the working mean 28
Example: The first four moments about the working mean 28.5 of a distribution are 0.294, 7.144, and Calculate the moments about mean. Also evaluate β1 β2 and comment upon skewness & Kurtosis of the distribution.

Solution: Given that

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