CBSE Class 11 Economics Notes — Organisation of Data

CBSE Class 11 Statistics for Economics Chapter 3 notes: raw data, classification, discrete vs continuous variables, frequency distributions and class intervals.

Chapter Summary

This chapter shows how raw data are brought into order for analysis. Using the kabadiwallah analogy, it explains classification, the problems of raw data, chronological, spatial, qualitative and quantitative classification, discrete and continuous variables, frequency distributions, class limits, class intervals and mid-points, the exclusive and inclusive methods, tally marking, and univariate versus bivariate distributions, with a worked example using marks of twenty students.

Key Concepts

Why Organise Data? The Kabadiwallah and Classification

Once data have been collected, the next job is to organise them so that they can be analysed easily. The chapter explains this through the local junk dealer, the kabadiwallah, who buys old newspapers, bottles, plastics and metals. With so much junk, his trade would be impossible if he did not classify it. He ties newspapers together with a rope, puts bottles in a sack, and heaps metals in a corner, sorting them further into iron, copper, aluminium and brass. When a buyer asks for something, he knows exactly where to look.

Your schoolbooks work the same way. Grouped by subject, each subject becomes a class, and finding a history book means searching only the History group. Classification saves time and effort, but it is never arbitrary. The dealer puts empty bottles, broken mirrors and windowpanes under Glass because of the market for reused goods, and you would not put a science book in the History group, or the point of grouping would be lost.

So classification is arranging or organising things into groups or classes based on some criteria. The same applies to statistics: raw numbers are grouped by a rule that suits the purpose of the study. A post office does this when it sorts letters by the pin-code, which is the activity suggested in the chapter.

Key Points

Examples

Raw Data and the Trouble With Them

Unclassified data are called raw data. Like the junk in a disorganised shop, they are usually large, cumbersome and untidy. It is very hard to draw meaningful conclusions from them because they do not yield to statistical methods easily, so they must be organised and presented before any systematic analysis.

The chapter offers two examples. Table 3.1 shows marks in mathematics of 100 students, and Table 3.2 shows the monthly expenditure on food of 50 households, with values such as Rs 1,904, Rs 2,041, Rs 5,090 and Rs 1,007. In both tables the numbers are not arranged in any order. To find the highest mark among the 100 students, you would first have to arrange all the marks in ascending or descending order, a tedious job that becomes far worse for 1,000 students. Likewise, finding the average monthly food expenditure of 50 households from the jumbled list is hard, and harder for 5,000 households.

The Census of India shows the point on a huge scale. About 20 crore persons were contacted in Census 2001, as the chapter states, and the raw responses are so large and fragmented that conclusions seem almost impossible. When the same data are classified by gender, education, marital status and occupation, the structure of India's population becomes clear. The way data are classified depends on the purpose, for instance a teacher wanting to see how many students failed would build a frequency distribution of marks.

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Types of Classification: Chronological, Spatial, Qualitative and Quantitative

Data can be grouped on different bases, just as books can be arranged by subject, by author or by year of publication. The chapter describes four common kinds of classification.

Chronological classification arranges data by time, in ascending or descending order, using years, quarters, months or weeks. A table of India's food grain output year by year, or of monthly rainfall, is chronological.

Spatial classification groups data by place, such as villages, cities, districts, states or countries. For instance, rice production listed state by state is spatial.

Qualitative classification groups data by attributes or qualities that cannot be measured numerically, such as gender, literacy, religion or employment status. Persons may be classed as male or female, or as literate or illiterate. You count how many fall in each class.

Quantitative classification groups data by characteristics that can be measured numerically, such as height, weight, income or marks. Here the values are put into classes according to size, for instance students grouped into marks ranges.

The choice depends on the question to be answered. To see how production changed, use time. To compare regions, use place. To study the make-up of the population by attribute, use qualitative classification. To study how a numerical variable is spread, use quantitative classification, which leads naturally to the frequency distribution.

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Variables: Discrete and Continuous, and the Frequency Array

A variable is a characteristic that varies from one observation to another, such as marks, income or number of children in a household. Quantitative classification uses two kinds of variable.

A discrete variable takes only certain isolated values, usually whole numbers, with gaps between them. The number of children in a family can be 0, 1, 2 or 3 but not 2.4. The number of students in a class or of rooms in a house is also discrete.

A continuous variable can take any value within a range, including fractions, because it is measured and not counted. Height, weight, time and income are continuous. A person may be 150.3 cm tall, and 150.37 if measured more finely.

This distinction decides how the data are presented. For a discrete variable we can list each value with the number of times it occurs. Such a simple listing is called a frequency array. For example, if 30 households are surveyed for the number of children, the distribution might be as follows: 0 children in 6 households, 1 child in 10, 2 children in 8, 3 children in 4 and 4 children in 2. The frequencies add up to 6 + 10 + 8 + 4 + 2 = 30, which must equal the number of households. For a continuous variable, listing every value is not useful, so values are grouped into classes.

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Frequency Distribution: Classes, Limits, Width and Mid-point

A frequency distribution is a table that shows the different values or classes of a variable and the number of observations, called the frequency, in each. To make one from a large set of continuous data, the numbers are grouped into class intervals.

Some terms you must know. The lower limit and upper limit are the smallest and largest values of a class. The class interval or class width is the difference between the upper and lower limits, when classes are formed on the exclusive pattern. The class mid-point or class mark is the middle value, calculated as the average of the lower and upper limits. For a class of 20-30, the width is 10 and the mid-point is (20 + 30) divided by 2, which is 25.

The chapter's technique for forming classes starts with the range, the largest value minus the smallest. Decide the number of classes, often between 5 and 15, choose a convenient width, and prepare class limits so that every observation falls in exactly one class. Classes of equal width are easy to compare.

For example, if the lowest mark among 100 students is 0 and the highest is 100, the range is 100. Choosing a width of 10 gives 10 classes from 0-10 up to 90-100. Choosing a width of 20 would give only 5 classes, hiding more detail. There is no single correct answer, but the classes must suit the purpose and not leave out or double count any observation.

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Exclusive and Inclusive Methods

The upper limit of one class and the lower limit of the next can be handled in two ways.

In the exclusive method, a class includes its lower limit but excludes its upper limit. In classes 10-20, 20-30 and 30-40, an observation of exactly 20 goes into 20-30, not 10-20. This method has no gaps between classes, which makes it suitable for continuous data. It is the method in which the width is simply upper minus lower.

In the inclusive method, both limits belong to the class. The classes are written as 10-19, 20-29 and 30-39, so a value of 19 is in the first class and 20 is in the second. There are apparent gaps between 19 and 20, which is acceptable when data are whole numbers, as with marks or the number of persons.

If inclusive classes are used for continuous data, the gaps must be closed by an adjustment. Take the gap between the upper limit of one class and the lower limit of the next, here 20 minus 19, which is 1, halve it to 0.5, and subtract it from every lower limit and add it to every upper limit. Class 10-19 becomes 9.5-19.5, and 20-29 becomes 19.5-29.5. The widths remain 10.

In exam questions, always state which method you have used, and make sure the highest observation fits inside the last class.

Key Points

Examples

Worked Example: Making a Frequency Distribution With Tally Marks

Tally marking is the technique of counting observations by making a stroke for each one. Four strokes are drawn vertically and the fifth is a diagonal line across them, so that every bundle of five is quick to count. Tally marking prevents errors of missing or double counting.

Let us organise the marks of 20 students out of 100: 12, 25, 33, 47, 38, 52, 45, 61, 29, 41, 55, 48, 36, 72, 64, 18, 44, 50, 58, 39. The smallest is 12 and the largest is 72, so the range is 60. Choose exclusive classes of width 10 from 10 to 80, which gives seven classes. Go through the list once and put a tally in the right class for each mark.

The frequencies are: 10-20 has 12 and 18, so 2; 20-30 has 25 and 29, so 2; 30-40 has 33, 38, 36 and 39, so 4; 40-50 has 47, 45, 41, 48 and 44, so 5; 50-60 has 52, 55, 50 and 58, so 4; 60-70 has 61 and 64, so 2; 70-80 has 72, so 1. The total is 2 + 2 + 4 + 5 + 4 + 2 + 1, which is 20, matching the number of students. The mid-points are 15, 25, 35, 45, 55, 65 and 75.

Now the table answers questions quickly. Five students scored in the 40-50 band, the most common one, and 4 + 2 + 1 = 7 students scored 50 or more.

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Examples

Univariate and Bivariate Frequency Distributions

A frequency distribution may deal with one variable or with two together.

A univariate frequency distribution describes one variable only. The table of marks of 20 students above is univariate, as is the distribution of households by number of children. It answers questions about a single characteristic.

A bivariate frequency distribution describes two variables at the same time, with the values of one variable arranged along rows and the other along columns. Each cell shows how many observations have a particular combination. It helps to examine whether the two variables are connected.

Take the same 20 students and also record gender. Suppose 12 are boys and 8 are girls. Among the boys, 7 scored below 50 and 5 scored 50 or above. Among the girls, 6 scored below 50 and 2 scored 50 or above. The bivariate table has the marks groups as rows and gender as columns. Below 50: 7 boys, 6 girls, a total of 13. 50 and above: 5 boys, 2 girls, a total of 7. The column totals are 12 boys and 8 girls and the grand total is 20, matching the univariate table (13 students below 50 and 7 at 50 or above).

The row and column totals of a bivariate table give the univariate distributions of each variable separately.

Key Points

Examples

Frequently Asked Questions

What is classification of data and why is it needed?

Classification is arranging raw data into groups or classes on the basis of some criterion. It is needed because raw data are large and disorganised, making it hard to locate, compare or summarise information. Once classified, similar items are placed together and statistical analysis becomes easier, as with the kabadiwallah's sorted junk.

What are raw data?

Raw data are unclassified data, kept in the order in which they were collected, with no grouping or arrangement. Marks of 100 students listed randomly are an example. They are bulky and difficult to analyse, so they must be classified and presented before systematic statistical work.

Distinguish between chronological and spatial classification.

In chronological classification, data are grouped by time, such as years, months or weeks, in ascending or descending order. In spatial classification, they are grouped by place, such as districts, states or countries. Yearly food grain output is chronological, while state-wise rice production is spatial.

What is the difference between qualitative and quantitative classification?

Qualitative classification groups data by attributes that cannot be measured numerically, such as gender or literacy. Quantitative classification groups data by characteristics measured in numbers, such as height, income or marks. In the first we count units in each attribute class, while in the second we group values into numerical classes.

Distinguish between a discrete and a continuous variable.

A discrete variable takes only isolated values, usually whole numbers, such as the number of children in a family. A continuous variable can take any value in a range, including fractions, such as height or weight. Discrete data suit a frequency array, and continuous data are grouped in class intervals.

Define class limits, class interval and mid-point with an example.

Class limits are the lowest and highest values of a class, called the lower and upper limits. The class interval is the width, the difference between the limits. The mid-point is their average. For the class 20-30 the limits are 20 and 30, the interval is 10 and the mid-point is 25.

Explain the exclusive and inclusive methods of forming classes (4 marks).

In the exclusive method the upper limit of a class is not included in it, so classes like 10-20 and 20-30 leave no gaps and a value of 20 goes to 20-30. In the inclusive method both limits are included, as in 10-19 and 20-29. For continuous data, inclusive classes are adjusted by half the gap, giving 9.5-19.5.

What is tally marking and how is it used?

Tally marking is a counting technique in which one stroke is made for every observation in a class. Four vertical strokes are crossed by a fifth diagonal stroke, forming a bundle of five. It helps count frequencies quickly and avoids missing or double counting, and the totals are written as frequencies.

Differentiate between univariate and bivariate frequency distributions.

A univariate frequency distribution shows the frequencies of one variable only, such as marks of students. A bivariate frequency distribution shows two variables together in a two-way table, such as marks and gender. Each cell gives the number of observations with a particular pair of values, and the totals give the single-variable distributions.

How will you prepare a frequency distribution from raw data (6 marks)?

Find the range by subtracting the smallest value from the largest. Decide the number of classes and a convenient width, then write class limits using the exclusive or inclusive method. Take each observation and put a tally mark in its class, then count the tallies to get frequencies. Check that the frequencies add up to the number of observations, and compute mid-points if needed.

CBSE Class 11 Economics Notes — Organisation of Data