class 11 Statistics all Pakistani boards, KPK textbook boards, Punjab textbook boards, Sindh textbook boards, and Baluchistan textbook boards Statistics Chapter 2 (Collection and presentation of Data).
(Collection and presentation of Data)
Table of Contents
Q.1) What do you understand by the term classification? What are different types of classification?
Answer:
Classification is the process of arranging data into sequences and groups according to their common characteristics, or separating them into different but related parts. For example, we may arrange the marks into groups of 60 marks each e.g. 300 to 359, 360 to 419, 420 to 479 and so on. The people may be divided into different age groups like 10-20, 20-30, 30-40 etc.
Types of classification:
Classification is determined by characteristics possessed by the individual units of a group. They are of two types.
a) Descriptive b) Numerical
a) Descriptive classification:
When the data are classified based on qualities or attributes, which are incapable of quantitative measurement, the classification is said to be descriptive or according to attributes. For example, sex, marital status, educational standard, etc. If more than one attribute is used as a basis of classification, it is called a manifold classification. For example, classification according to the sex and marital status divide the population into six classes. These classes are:
I) Male married ii) Male unmarried iii) Male widowed,
iv) Female married v) Female unmarried VI) Female widowed
b) Numerical classification:
This type of classification is applicable to quantitative data only. Data relating to height, weight, income, production, etc. are quantitative characteristics of a group of objects. Their values are expressed in terms of numbers with appropriate units. Hence, they are also known as numerical characteristics.
Q.2) What is meant by tabulation of statistical data? Describe the main parts of the table and the general consideration to be kept in mind in order to prepare a good table.
Answer:
Tabulation simply means presenting of data through tables. It is the next of classification in the process of statistical investigation. To be more precise ’Tabulation is an orderly arrangement of data in columns and rows’.
In very general terms, tabulation may be distinguished as simple and complex. A simple table contains data regarding one characteristic only, information relating to other characteristics being ignored. Complex tabulation shows the division of the data into two or more categories. Tabulation is also classified as simple, double, or manifold. For example, marks and number of students provides of a simple tabulation. Tables with two criteria of classification, e.g. sex and marital status or height and weight, etc. are examples of double tabulation. An example of manifold tabulation is the presentation of a population of a country by age, by sex, by literacy, by marital status, etc.|
The table is a frequency distribution of weights (recorded to the nearest Kg.) of 120 students at the Government College Lahore.
| Weights (Kg) | No. of students |
| 45-49 | 1 |
| 50-54 | 4 |
| 55-59 | 17 |
| 60-64 | 28 |
| 65-69 | 25 |
| 70-74 | 18 |
| 75-79 | 13 |
Q.3) What is a frequency distribution? What points should be noted in preparing a frequency distribution?
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Answer:
Frequency distribution:
A large mass of data possessing different characteristics is grouped into different classes. Then observations are determined in each class. The arrangement of these classes into tabular form makes frequency distribution. The number of observation falling in a class makes a class frequency. Data organized and summarized in the form of a frequency distribution are often called grouped data.
Main points of preparing a frequency distribution:
In the preparing of frequency distribution, the following points should be taken into account.
Number of classes and their lengths:
A frequency distribution should not have too few or too many large. Depending upon a particular data, the number of classes should not exceed 25 and should not be less than 6. The ideal number of classes is that which gives maximum information about the data. A general rule for obtaining the proper number of classes is to divide the range i.e. the difference between the highest and the lowest values of the distribution by the class- interval. As far as possible, the class-intervals should be of equal lengths.
Class limit:
The limit of the classes should be so fixed that the midpoint of each class-interval fall on an integer and not a fraction. Thus if the value of varieties start from 14 and magnitude of the class-interval is taken as 5, a convenient class-interval can be taken as 12.5—17.5 with 12.5 and 17.5 as class limits giving 15 as the mid points. The class limits may be written in any one of the ways explained below:
i) 0-10 10-20 20-30 30-40 etc.
ii) 0-9.9 10-19.9 20-29.9 30-39.9 etc.
iii) 0 and under 10, 10 and under 20, 20 and under 30etc
The best method of classification amongst the three is the first one. In this upper class-limit, a given class falls in the immediately next class.
For example, 10 falls in class 10 — 20 and not in 0 —10. This brings in continuity of the class-intervals.
Class boundaries:
If one have grouped frequency distribution with class-limits having a gap between the upper class-limit of one class and the lower class-limit of the next class as in (ii) above. We can make it continuous by subtracting half the length of the gap from the lower class-limit and adding the half the length of the gap to the upper class-limit. Thus in case (ii) there is a gap of 10—9.9 is 0.1 between two successive class-intervals. The class can be made continuous by subtracting and adding 0.1/2 = 0.05 from the lower class-limit and the upper class-limit respectively of each class. We thus get a corresponding continuous grouping in the form.
0-9.95, 9.95-19.95, 19.95-29.95
Class marks:
It will be seen later on that for further statistical analysis, each class will have to be represented by one number only. Taking the mid-value of that class does this. These mid-values of the classes are called the class- marks.
Thus in the above illustration, the class marks are 4.95, 14.95, 24.95, 34.95, 44.95 and 54.95 respectively.
Class-interval in terms of class width/length:
The class-interval in terms of class width or length is the difference between the upper class boundary and the lower class boundary. If all the class-interval of a frequency distribution is of equal size, the class width is denoted by h. The class-interval of a frequency distribution may be fractional but it is advisable to prefer a class-interval, which is an integer or a whole number.
Class frequency:
The frequency of a class-interval is the total number of items falling in that class-interval. The class frequencies are counted by writing the class- intervals, on a sheet of paper (called tally sheet) and for each items falling in a class-interval, a stroke is marked against it. Usually, after every four lines in a class, the fifth item is marked by a horizontal or slanted line across the strokes.
Q.4) Calculate class-intervals, class-marks and class-boundaries from (a) 2-4 (b) 2.1-2.4 (c) 2.10-2.22
Answer:
Part A:
Class interval=3
Class marks=3(add both the class limits and divide it by 2)
Class boundaries=1.5-4.5
Part B:
Class interval=0.4
Class marks=2.25(add both the class limits and divide it by 2)
Class boundaries=2.05-2.45
Part C:
Class interval=0.3
Class marks=2.16(add both the class limits and divide it by 2)
Class boundaries=2.095-2.225
Q.5) Arrange the data given below in an array and construct a frequency distribution using a class interval of 5.00. Indicate the class-boundaries and class-limit clearly.
79.4, 71.6, 95.5, 73.0, 74.2, 81.8, 90.6, 55.9, 75.2, 81.9, 68.9, 74.2, 80.7, 65.7, 67.6, 82.9, 88.1, 77.8, 69.4, 83.2, 82.7, 73.8, 64.2, 63.9, 58.3, 48.6, 83.5, 70.8, 72.1, 71.6, 59.4, 77.6
Answer:
The value of the variety range from 48.6 to 95.5.We take class interval of length is 5.
No of classes=range/class interval=95.5-48.6/2=9.38+1=10
| Class limits | Frequency | Class boundaries |
| 48.6-53.5 | 1 | 48.55-53.55 |
| 53.6-58.5 | 2 | 53.55-58.55 |
| 58.6-63.5 | 1 | 58.55-63.55 |
| 63.6-68.5 | 4 | 63.55-68.55 |
| 68.6-73.5 | 8 | 68.55-73.55 |
| 73.6-78.5 | 5 | 73.55-78.55 |
| 78.6-83.5 | 8 | 78.55-83.55 |
| 83.6-88.5 | 1 | 83.55-88.55 |
| 88.6-93.5 | 1 | 88.55-93.55 |
| 93.6-98.5 | 1 | 93.55-98.55 |
| 32 |
Q.6) a) Briefly explain the use of charts in presenting statistical data,
b) Explain the following charts:
i) Simple Bar Charts,
ii) Multiple Bar Charts,
iii) Pie Charts,
iv) Sub-divided Bar charts
v) Rectangles and sub-divided rectangles
Answer:
The most important function of the science of statistics is to present complexity of qualitative data into simple forms and to make them easily intelligible. Classification and tabulation of data are the attempts in this regard. There is yet another method by which data can be represented more interestingly and it to present the data through charts.
The special feature of charts is that they do away with figures all together and present dry and uninteresting statistical figures in the shape of attractive, to simple and appealing charts. Figures are usually avoided by common man but charts always attract and impress him.
Use of charts in presenting statistical data:
Main usefulness of charts are briefly described below:
Easy to understand:
Through charts it becomes easy to understand the presentation of data. Charts are not much difficult to understand and more over these are self-explaining.
Economy of time and labour:
Since no efforts are necessary in understanding charts they save time which is otherwise needed in drawing inferences from a set of figures. They provide conclusion at a glance of the data and no mental effect are required for this purpose.
Lasting impression:
The impression created by charts is lasting than the effect of a set of figures.
Helpful in comparison:
It becomes easy to compare the items through charts.
Useful to all:
Charts have universal usefulness. They are useful to social scientists, mathematicians, businessmen etc.
PART B
Simple bar chart:
The simple bar chart is particularly appropriate for a linear or one-dimensional comparison. The scale for construction of simple bar charts should be such as facilitates the representation of largest bar quite conveniently. There should be sufficient spacing on all sides of diagram for writing headings, scale, unit etc. in general, the base line is taken horizontally because vertical bars facilitate better comparison.Following is the specimen of simple bar chart:
Multiple Bar Chart:
In this type of diagram, representing 2 or more sets of inter-related data in one chart then more than one bar is used.e.g
Pie charts:
The reason for the popularity of pie chart is the easiness and convenience in its constructions. This is also known as a circle chart the procedure is very simple, take the total of these quantities equal to an angle of 360 degrees and then convert the quantities in terms of angles.e.g
Subdivided bar charts:
Subdivided bar charts are used to present which such data, which are to be shown in parts, or which are the totals of various subdivisions. The component parts are shaded or colored differently so as to distinguish different parts. E.g
Q.7) Show by a bar chart the production of a factory from the following information.
| Quarter | Finished article | In complete article | Total |
| First | 200 | 100 | 300 |
| Second | 250 | 100 | 350 |
| Third | 300 | 50 | 350 |
| Fourth | 400 | 30 |
Answer:
Q.21) Write a note on the graphic representation of statistical data.
Answer:
A graph is a pictorial presentation of the relationship between the variables. Moreover the graphs give more accurate and precise results. They provide a very good method of showing fluctuations and trends in statistical data
Rules of drawing graphs:
1. Select a suitable scale
2. Clear and comprehensive title or heading
3. Plot independent variable on x-axis and dependent variable on y-axis, if the first item of the data is quite large, using false base line. In such a situation the vertical scale is broken and the space between the origin ‘O’ and the minimum value of the dependent variable is committed by drawing two zigzag horizontal lines above the base line. The scale along y-axis is then framed accordingly.
1. If more than one curves must be differentiated by different lines i.e.
i) Simple line (_______________ )
ii) Broken line (———————- )
iii) Dotted line ( …………………….. )
iv) Dot-dash line ( – – – – – – )
Graphic presentation:
In order to represent a frequency distribution graphically, the following methods or in common use:
i) Histogram, ii) Frequency polygon,
iii) Frequency curve, iv) Cumulative frequency curve
