CPT Section D Quantitative Aptitude Chapter 10 Dr. R. B. Tiwari

In this topic we shall study about the types of data and its various ways of representation

To have a broad overview of the subject and its application

Learning Objectives To have an idea about the primary and secondary data

Learning Objectives To have an idea about the textual, tabular format including the techniques of creating frequency distribution and determination of cumulative frequencies.

Learning Objectives To have an idea about the diagrammatic representation of the data, which includes Histogram, Frequency Polygone and Pie-Chart etc.

Statistics may be defined as a scientific method for collecting, analyzing and presenting data, leading finally to draw statistical inferences about some important characteristics.

In the present scenario Statistics has a wide application in the fields of Economics, Business Management, Commerce etc.

Statistical data can be defined as quantitative information about some particular characteristic under study. It can be classified as Primary Data Secondary Data

The data which are collected for the first time by an investigator or agency, are known as primary data

Methods for collection of primary data Interview method Mailed Questionnaire method Observation Method Questionnaires filled and sent by enumerators.

The data which are already collected by some individual or some agency, and used by some other individual or agency, is known as secondary data. The sources of secondary data may be various national and international agencies like WHO, ILO, World Bank, NSSO etc.

The data can be represented in textual, tabular and diagrammatic methods.

This is the classical method of representation of data, in which the data is written in the text form

Here the data is written in different columns with the corresponding values of the variables involved

This is the best possible method of data representation which includes various ways of graphs like Line Diagram, Bar Diagram, Histogram, Frequency Polygons and Ogives etc.

(c) 4 A line diagram is a one dimensional diagram in which the height of the line represents the frequency corresponding to the value of the item or factor.

When the data vary over time, we plot each pair of values (t, y t ), y t representing the time series at the time point t in the y t plain. The plotted points are then joined successively by line segments and the resulting chart is known as line-diagram.

If the profits in lacs of rupees of an industrial house for 2002, 2003, 2004. 2005. 2006. 2007 and 2008 are 5, 8, 9, 6, 12, 15 and 24 respectively. A Line Diagram may be give n as on the next slide.

20 18 Profit in Lacs 16 14 12 10 8 6 4 Series1 2 0 2004.5 2005 2005.5 2006 2006.5 2007 2007.5 2008 2008.5 Time in Years

Horizontal Bar Diagram Vertical Bar Diagram

Horizontal Bar Diagram is used qualitative data whereas the Vertical Bar Diagram is associated with quantitative data or time series data. Bars i.e. rectangles of equal width and usually of varying lengths are drawn either horizontally or vertically.

20 P r o f i t i n L a c s 18 16 14 12 10 8 6 4 2 Series1 0 2005 2006 2007 2008 Time in Years

Production in Metric Tones Year Wheat Rice 2005 12 25 2006 15 30 2007 18 32 2008 19 36

W h e a t a n d R i c e P r o d u c t i o n 60 50 40 30 20 10 0 2005 2006 2007 2008 Time in Years Wheat Rice

Production of Wheat and Rice 40 35 30 25 20 15 10 5 0 2005 2006 2007 2008 Time in Years Wheat Rice

A Pie-Chart is a circular diagram which is usually used for depicting the components of a single factor. The circle is divided into segments which are in proportion to the size of the components. They are shown by different patterns or colours to make them attractive.

Source Revenue in millions of rupees Customs 80 Excise 190 Income Tax 160 Corporate Tax 75 Miscellaneous 35

Miscellaneous 6% Customs 15% Corporate Tax 14% Income Tax 30% Excise 35%

Grouped Data Ungrouped Data

If the frequencies are assigned against the class intervals then we say that the data is grouped, whereas if frequencies are assigned to discrete values of the variable, then such a data is known as ungrouped data.

For a grouped data, corresponding to a class interval, the class limit is defined as the minimum value and the maximum value that the class interval may contain. The Minimum value is known as the lower class limit (LCL) whereas the maximum value is known as the upper class limit (UCL).

Mutually Inclusive type class intervals Mutually Exclusive type class intervals

If the upper class limit of any class does not coincide with the lower limit of the next class, then such type of class intervals are known as mutually inclusive type of class intervals, whereas if the upper limit of a class is same as the lower limit of the next class then such type of class intervals are said to be exclusive type class intervals.

Note that for an exclusive type grouped data the UCB and UCL are same and the LCB and LCL are equal, whereas for Inclusive type data the values of UCB UCL are different and similarly in this case the values of LCB and LCL are also different.

In case of inclusive type class intervals the upper class boundaries are obtained by adding half of the difference of the lower class limit of any class and the upper class limit of the previous class (i.e. D/2), and similarly the Lower class boundaries are obtained by subtracting the same from the corresponding lower class limits.

The width of a class interval may be defined as the difference between the UCB and the LCB of that class interval. i.e. Class Width = UCB - LCB

The cumulative frequency corresponding to a value for a discrete variable and corresponding to a class boundary for a continuous variable may be defined as the number of observations less than or equal to the class boundary. This is called the less than type cumulative frequency. Similarly upper than type cumulative frequency may be defined.

Class Intervals Frequency Less than type cumulative frequency greater than type cumulative frequency 0-10 7 7 64 10-20 12 19 57 20-30 25 44 45 30-40 14 58 20 40-50 6 64 6

The frequency density of a class interval may be defined as the ratio of the frequency of that class interval to the corresponding class length.

Relative frequency of a class interval may defined as the ratio of the class frequency to the total frequency, whereas the percentage frequency of a class may be defined as the ratio of the class frequency to the total frequency, expressed in percentage.

Histogram or Area Diagram Frequency Polygone Ogives or cumulative frequency graphs

A comparison among the frequencies for different class intervals is possible in this diagrammatic representation. To draw a histogram the class limits are first converted to the corresponding class boundaries and a series of adjacent rectangles, one against each class interval, with the class interval as breadth and the frequency as length or altitude, is erected.

A frequency polygon is plotted between the mid values of the class interval (i.e. X i ) against the values of the corresponding frequencies (f i ) for i=1,2,,n

Class Intervals Frequencies 0-20 0 20-40 170 40-60 400 60-80 200 80-100 180 100-120 190 120-140 260 140-160 100 160-180 50 180-200 30

Mid Points No. of Students 46 3 51 4 56 5 61 7 66 9 71 8

frequency Polygon 10 f r e q u e n c y 9 8 7 6 5 4 3 2 1 y 0 46 51 56 61 66 71 Weight (mid-values)

Ogives are plotted between the cumulative frequencies against the class boundaries. These are of two types:

Less than type Ogive: Between less than type cumulative frequencies and upper class boundaries.

Lower Boundary Upper Boundary Frequency Less Than Type C.F. Greater Than Type C.F. 0 10 6 6 100 10 20 10 16 94 20 30 15 31 84 30 40 35 66 69 40 50 27 93 34 50 60 7 100 7

less than type graph less than type c.f. 120 100 80 60 40 20 0 10 20 30 40 50 60 upper class boundaries less than

Between upper than type cumulative frequencies and lower class boundaries.

upper than type Ogive upper than type c.f. 120 100 80 60 40 20 0 0 10 20 30 40 50 lower class boundaries greater than type c.f.

Note that the intersection point of two types of ogives gives us median.

A frequency curve is a smooth curve for which the total area is taken to be unity. It is obtained by drawing a smooth and free hand curve through the mid-points of the upper sides of the rectangles forming the histogram. There are four types of frequency curves.

(a) Bell Shaped Curve (b) U-Shaped Curve

Question Time

a) Economics b) Business Management c) Commerce and Industry d) Always Answer: (d) always

(a) A qualitative characteristics (b) A Quantitative Characteristics (c) A Measurable Characteristics (d) All These Answer: (a) A qualitative characteristics

(a) An attribute (b) A discrete variable (c) A continuous variable (d) (b) or (c) Answer: (b) A discrete variable

(a) An attribute (b) A discrete variable (c) A continuous variable (d) None of these Answer: (b) A discrete variable

(a) An attribute (b) A continuous variable (c) A discrete variable (d) (a) or (c) Answer: (a) An attribute

(a) Primary data (b) Secondary data (c) Sample data (d) (a) or (b) Answer: (b) Secondary data

(a) Interview method (b) Observation method (c) Questionnaire Method (d) All these Answer: (d) All these

(a) A telephonic method (b) Mailed Questionnaire method (c) Direct Interview method (d) All these Answer: (b) Mailed Questionnaire method

(a) Textual, Tabular and diagrammatic (b) Tabular, Internal and external (c) Textual, Tabular and internal (d) Tabular, Textual and Internal Answer: (a) Textual, Tabular and diagrammatic

X: 0-10 10-20 20-30 30-40 40-50 f: 3 5 7 4 1 (a) 20.5 (b) 21.5 (c) 22.5 (d) 23.5 Answer: (c) 22.5

11. The Mode can be graphically calculated with the help of (a) Less Than Type Ogive (b) Greater Than Type Ogive (c) Histogram (d) Frequency Polygon Answer: (c) Histogram

(a) Diagrammatic method (b) Tabulation (c) Textual presentation (d) None of these Answer: (b) Tabulation

(a) Qualitative data (b) Data varying over time (c) Data varying over space (d) (a) or (c) Answer: (d) (a) or (c)

(a) The data are qualitative (b) The data are quantitative (c) When the data vary over time (d) (a) or (c) Answer: (b) The data are quantitative

(a) Comparing different components and their relation to the total (b) Representing qualitative data in a circle (c) Representing quantitative data in a circle (d) (b) or (c) Answer: (a) Comparing different components and their the total relation to

(a) Grouped frequency distribution (b) Simple frequency distribution (c) (a) or (b) (d) (a) and (b) Answer: (a) Grouped frequency distribution

(a) Discrete variable (b) Continuous variable (c) An attribute (d) All these Answer: (a) Discrete variable

(a) Discrete variable (b) Continuous variable (c) An attribute (d) Any of these Answer: (b) Continuous variable

(a) An upper limit to LCL (b) A lower limit to LCL (c) (a) or (b) (d) (a) and (b) Answer: (b) A lower limit to LCL

(a) The difference between the UCB and LCB of that class (b) The difference between the UCL and LCL of that class (c) (a) or (b) (d) Both (a) and (b) Answer: (a) The difference between the UCB and LCB of that class

(a) Lies between 0 and 1 (b) Lies between 0 and 1, both inclusive (c) Lies between -1 and 0 (d) Lies between -1 to 1 Answer: (a) Lies between 0 and 1

(a) Histogram (b) Less than type ogive (c) More than type ogive (d) Frequency polygone Answer: (a) Histogram

The number of accidents for seven days in a locality are given below: No. of accidents: 0 1 2 3 4 5 6 Frequency: 15 19 22 31 9 3 2 What is the number of cases when 3 or less accidents accurred? (a) 56 (b) 6 (c) 68 (d) 87 Answer: (d) 87

(a) Pictogram (b) Histogram (c) Bar Diagram (d) Line diagram Answer: (d) Line diagram

(a) Graph (b) Frequency (c) Statistical Table (d) Distribution Answer: (c) Statistical Table

(a) Histogram (b) Frequency Polygone (c) Ogive (d) None Answer: (a) Histogram

(a) Histogram (b) Ogive (c) Both (d) None Answer: (b) Ogive

(a) Density (b) frequency (c) Both (d) None Answer: (b) frequency

(a) Two Types of Ogive (b) Frequency Polygone (c) Histogram (d) None Answer: (a) Two Types of Ogive

(a) Left part of the table describing the columns (b) Right part of the table describing the colums (c) Right part of the table describing the rows (d) Left part of the table describing the rows Answer: (d) Left part of the table describing the rows

(a) 5 (b) 0 (c) 10 (d) none Answer: (b) 0

(a) 0 (b) 19 (c) 9.5 (d) none Answer: (c) 9.5