Key takeaways
- Raw data are unordered; an array orders them and a frequency distribution groups them.
- Data are classified by time, place, attribute or value.
- Class mark = (upper + lower) ÷ 2; class interval = upper − lower.
- Inclusive classes are converted to exclusive by adjusting half the gap.
- A bivariate table classifies units by two variables together.
This is Chapter 3 of the Statistics for Economics chapter list for Class 11 Economics. It takes the data collected in Chapter 2 and prepares them for presentation.
3.1Raw Data and Why It Needs Organising
Raw data are figures exactly as collected, with no order or grouping, such as the study hours of 30 students in the order they were surveyed. They are hard to read and impossible to compare at a glance. Organising them, by arrangement and classification, is the first step before any average or graph.
Here is raw data on weekly study hours for 30 students:
4, 12, 7, 15, 9, 6, 11, 3, 14, 8, 10, 5, 13, 7, 9, 16, 6, 11, 8, 12, 5, 10, 9, 14, 7, 13, 4, 8, 10, 12
Even this small set tells little. We cannot say how many study fewer than 8 hours, or where most students fall. An array arranges the values in ascending or descending order, which shows the smallest (3) and largest (16) at once, but 30 numbers are still too many to read. Grouping is the next step.
Why arranging the data in order helps
Arranging data in ascending or descending order is the first step in organising them. In the array, the lowest and highest values can be seen at once, and the range follows by subtraction. Repeated values show up together, and the middle value can be located. This is also the starting point for finding the median and quartiles.
An array becomes unwieldy when the data are large, which is why classification and frequency tables are used. Even so, an array of a small data set, with a few dozen values, is often the quickest way to see the shape of the data.
3.2Classification of Data
Classification groups data by a common feature. Chronological classification arranges data by time, spatial by place, qualitative by an attribute such as gender and quantitative by a measurable value such as marks. It removes needless detail, brings out similarities and differences, and makes comparison and further analysis possible.
| Basis | Grouped by | Example |
|---|---|---|
| Chronological | Time | Foodgrain output year by year |
| Spatial (geographical) | Place | Literacy rate by state |
| Qualitative | An attribute | Workers by sex or religion |
| Quantitative | A measured value | Students by marks scored |
Variables: continuous and discrete
A variable is a characteristic that varies. A discrete variable takes only separate values, usually whole numbers, such as the number of children in a family. A continuous variable can take any value in a range, such as height or time.
Choosing a basis for classification
The basis chosen should suit the purpose. To study how crop output has changed, classify by time. To compare states, classify by place. To study the workforce, classify by an attribute such as occupation or sex. To study the spread of marks, classify by the measured value. The same data can be classified in more than one way and each way answers a different question.
A good classification is complete, so that every item has a place, and exclusive, so that no item falls in two groups. It should also be stable, with the same categories used over time, so that comparisons across years are valid.
3.3Frequency Distribution
A frequency distribution shows how many observations fall in each class of a variable. The classes have a lower and upper limit, a width called the class interval, and a mid-value called the class mark. Grouping 30 values into five classes of width 4 turns an unreadable list into a table that can be read at once.
Using classes of width 4, with the lower limit included and the upper excluded (the exclusive method), the study hours become the following. The value 16 falls in 16–20, not in 12–16.
| Hours (class) | Frequency (f) | Class mark | Cumulative f |
|---|---|---|---|
| 0–4 | 1 | 2 | 1 |
| 4–8 | 9 | 6 | 10 |
| 8–12 | 11 | 10 | 21 |
| 12–16 | 8 | 14 | 29 |
| 16–20 | 1 | 18 | 30 |
| Total | 30 |
The class mark is the mid-value: (upper limit + lower limit) ÷ 2. The class 0–4 has mark 2. The class interval is upper limit − lower limit, here 4. Tables usually have 6 to 15 classes; this small set uses five. Too few classes hide the pattern, too many bring back the clutter.
Class mark = (upper limit + lower limit) ÷ 2 · Class interval = upper limit − lower limit
Exclusive and inclusive classes
In the exclusive method (0–4, 4–8) a value equal to the upper limit goes into the next class. In the inclusive method (10–19, 20–29) both limits belong to the class. Inclusive classes leave a gap between classes, so for continuous variables they are converted by subtracting half the gap from each lower limit and adding it to each upper limit.
Discrete series and frequency arrays
For a discrete variable no classes are needed: list each value with its frequency, which gives a frequency array. For example, if 30 families have 0, 1, 2, 3 or 4 children, the table has five rows, one for each value. Grouping always costs something: once values are in a class, we can no longer see the individual figures.
Choosing class intervals
Start by finding the range, the difference between the largest and smallest values. Decide how many classes are wanted, usually between six and fifteen, and divide the range by that number. Round the result to a convenient width such as 5, 10 or 20. Begin the first class at a convenient value at or below the smallest observation.
For the 30 study-hour figures, the range is 16 − 3 = 13; with five classes the width is about 3, and rounding to 4 gives classes of 0–4 up to 16–20. Equal widths make frequencies comparable. When most of the data are concentrated, open-ended or unequal classes may be used, but they need adjustment before graphs are drawn.
Cumulative frequencies
The cumulative frequency of a class is the total of its frequency and those of all earlier classes. In the study-hours table, the cumulative frequencies are 1, 10, 21, 29 and 30. A less-than cumulative frequency says how many observations are below a given upper limit, and a more-than cumulative frequency says how many are above a given lower limit.
These totals are used to find the median and quartiles and to draw ogives. The last cumulative frequency always equals the total number of observations, which gives a quick check on the table.
3.4Bivariate Frequency Distribution
A bivariate frequency distribution classifies observations by two variables at the same time in a two-way table. The rows show classes of one variable and the columns classes of the other, and each cell counts the units with that pair of values. It is used to examine how two variables are related.
Suppose 20 students are classified by hours studied and by marks scored.
| Study hours | Marks 0–50 | Marks 50–100 | Total |
|---|---|---|---|
| 0–8 | 7 | 3 | 10 |
| 8–16 | 2 | 8 | 10 |
| Total | 9 | 11 | 20 |
The row and column totals (10, 10 and 9, 11) are the marginal distributions of each variable. Reading across, students who study more are more likely to land in the higher marks column; this pattern is studied properly in correlation.
Reading marginal and conditional distributions
In a bivariate table, the totals at the right and bottom edges give the marginal distributions of each variable on its own. To see how one variable behaves for a given value of the other, read along a single row or column. In the study-hours table, among students who study 8–16 hours, 8 of 10 score 50 or more, against 3 of 10 among those who study less.
Such comparisons suggest an association, which correlation measures. A table does not prove cause, but it often points to relationships worth investigating.
Building a bivariate table step by step
Choose the two variables and decide the classes for each, for example study hours in two classes and marks in two classes. Draw a grid with one variable along the rows and the other along the columns. Take each student in turn and place a tally mark in the cell that matches both of his or her values. Count the tallies in each cell and then add along rows and columns.
Check that the grand total equals the number of students, and that the row totals and column totals both give the same grand total. This step-by-step method prevents omissions and double counting.
Worked Examples
Example 1: Class mark and width
- Given
- Classes of weekly wages: 100–200, 200–300, 300–400.
- Formula
- Class mark = (upper + lower) ÷ 2; width = upper − lower
- Steps
- Marks: 150, 250, 350. Width: 100 in every class.
- Answer
- Class marks are 150, 250 and 350; the width is 100.
Example 2: Inclusive to exclusive
- Given
- Classes of marks: 40–49, 50–59, 60–69.
- Formula
- Adjustment = (lower limit of next class − upper limit of this class) ÷ 2
- Steps
- Gap = 50 − 49 = 1, so adjustment = 0.5. Classes become 39.5–49.5, 49.5–59.5, 59.5–69.5.
- Answer
- The exclusive classes are 39.5–49.5, 49.5–59.5 and 59.5–69.5.
Example 3: Building a frequency table
- Given
- The 30 study-hour values of section 3.1, with classes 0–4, 4–8, 8–12, 12–16, 16–20 (upper limit excluded).
- Formula
- Frequency = number of values in the class
- Steps
- Count the values: 1, 9, 11, 8 and 1. Check: 1 + 9 + 11 + 8 + 1 = 30.
- Answer
- Frequencies are 1, 9, 11, 8 and 1; the most common class is 8–12 hours.
Common Mistakes
- Placing a value equal to the upper limit in the same class under the exclusive method. It belongs in the next class.
- Using inclusive classes for a continuous variable without adjusting them to true boundaries.
- Forgetting to check that the frequencies add up to the number of observations.
- Choosing unequal or too few classes without a reason, which hides the pattern.
10 Important Questions for Class 11 Board and School Examinations
Questions are written in original words. Marks are shown beside each question.
Q1The class mark of the class 20–30 is (a) 5 (b) 10 (c) 25 (d) 50 1 mark · MCQ
The correct option is (c) 25, since (20 + 30) ÷ 2 = 25.
Q2The number of children in a family is (a) a continuous variable (b) a discrete variable (c) an attribute (d) a constant 1 mark · MCQ
The correct option is (b) a discrete variable; it takes only whole-number values.
Q3Assertion (A): Classification makes data easier to compare. Reason (R): It groups items with common features. Which is correct? 1 mark · Assertion-reason
Both A and R are true, and R explains A.
Q4What is a frequency distribution? Name its parts. 3 marks
A frequency distribution is a table showing how many observations fall in each class or value of a variable.
- Class limits: the lower and upper values of a class.
- Class interval: upper limit minus lower limit.
- Class mark: the mid-value of the class.
- Frequency: the number of items in the class.
The frequencies add up to the total number of observations.
Q5Distinguish between exclusive and inclusive classes. 3 marks
In exclusive classes the upper limit is not part of the class, while in inclusive classes both limits belong to it.
For example, 0–10, 10–20 is exclusive, so a value of 10 goes into the second class. By contrast, 0–9, 10–19 is inclusive, which leaves a gap between classes. For a continuous variable the inclusive classes are converted by subtracting half the gap from each lower limit and adding it to each upper limit.
Q6Explain the four bases of classification with one example each. 4 marks
Data can be classified by time, place, attribute or measured value.
- Chronological: by time, such as year-wise foodgrain output.
- Spatial: by place, such as literacy rates by state.
- Qualitative: by an attribute, such as workers by sex or occupation.
- Quantitative: by a measured value, such as students by marks.
Classification removes needless detail, brings out similarities and differences, and makes comparison easier.
Which basis is used depends on the purpose of the study, and the same data can be classified in more than one way. A classification must be complete and clear, with no overlap between categories.
Q7Convert the inclusive classes 15–24, 25–34, 35–44 into exclusive classes and find their class marks. 4 marks
The exclusive classes are 14.5–24.5, 24.5–34.5 and 34.5–44.5, with class marks 19.5, 29.5 and 39.5.
- Given: inclusive classes 15–24, 25–34, 35–44.
- Gap between classes = 25 − 24 = 1, so half the gap is 0.5.
- Subtract 0.5 from each lower limit and add 0.5 to each upper limit.
- Class mark = (lower + upper) ÷ 2, for example (14.5 + 24.5) ÷ 2 = 19.5.
The class width is 10 in each class.
All three classes have the same width, so the frequencies can be compared directly.
Q8Explain the steps in preparing a frequency distribution from raw data. 6 marks
To prepare a frequency distribution, find the range, decide the classes, tally the values and total the frequencies.
- Find the smallest and largest values to get the range.
- Decide the number of classes, usually 6 to 15, and choose a convenient equal width.
- Write class limits so that they do not overlap and every value fits in one class.
- Go through the data and tally each value against its class.
- Count the tallies to get the frequency of each class.
- Check that the total of frequencies equals the number of observations.
The resulting table shows where values cluster and can be used to find averages and draw graphs, though some detail within each class is lost.
A properly built table makes the shape of the data visible.
Q9What is a bivariate frequency distribution? How does it help in studying two variables? 6 marks
A bivariate frequency distribution is a two-way table that classifies items by two variables together.
- Rows give classes of one variable and columns give classes of the other.
- Each cell shows how many units have that pair of values.
- Row and column totals give the marginal distributions of each variable.
- Patterns in the cells show whether the two variables move together.
For example, a table of study hours against marks may show more students in the high-marks column as study hours rise, which suggests a link that correlation then measures.
The marginal distributions are found by adding along rows and columns, and they show each variable on its own. A conditional reading, such as looking only at students who study long hours, shows how the second variable behaves for a given value of the first. This is the starting point for correlation analysis.
Q10From the values 3, 5, 8, 9, 12, 14, 15, 18, 21, 24, prepare a frequency distribution with classes 0–10, 10–20, 20–30 and find the class marks. 4 marks · Numerical
The frequencies are 4, 4 and 2, and the class marks are 5, 15 and 25.
Steps: 0–10 holds 3, 5, 8, 9 (4 values). 10–20 holds 12, 14, 15, 18 (4 values). 20–30 holds 21, 24 (2 values). Total = 10. Marks = (0 + 10) ÷ 2 = 5, 15 and 25.
10 Points for State PSC and UPSC Aspirants
- Data are classified chronologically, spatially, qualitatively or quantitatively; official tables follow these lines.Why it matters: the framework behind every statistical table.Source: NCERT; MoSPI publications
- The Census tabulates population by age group, sex and literacy in class intervals, a bivariate-style classification.Why it matters: real use of class intervals.Source: Census of India
- NSS consumption surveys classify households into monthly per capita expenditure classes, the basis of fractile class tables.Why it matters: distribution of consumption across classes.Source: MoSPI
- Monthly per capita expenditure classes are used to describe the distribution of household spending between poorer and richer groups.Why it matters: a standard way to read inequality from grouped data.Source: MoSPI
- The Economic Survey arranges macro data as time series, an example of chronological classification.Why it matters: most economic indicators are shown over time.Source: Economic Survey
- State-wise tables of literacy, poverty or per capita income are spatial classification.Why it matters: common in Indian economy sections of PSC papers.Source: Census; Economic Survey
- Discrete variables such as number of workers differ from continuous ones such as GDP growth rate.Why it matters: a basic definitional question.Source: NCERT
- Frequency distributions underlie the Lorenz curve, which plots cumulative shares of income.Why it matters: links grouped data with inequality measurement.Source: NCERT; standard economics textbooks
- Cumulative frequency is the base for ogives and for locating the median in the next chapters.Why it matters: connects Chapter 3 to Chapters 4 and 5.Source: NCERT
- Open-ended classes such as “above 50,000” appear in income tables and need special treatment when averages are found.Why it matters: a common difficulty with published tables.Source: NCERT
These are revision notes, not official predictions of exam content.
Quick Revision
- Raw data → array → frequency distribution.
- Classification: time, place, attribute, value.
- Discrete = separate values; continuous = any value in a range.
- Class mark = (upper + lower) ÷ 2.
- Exclusive: upper limit left out; inclusive: both limits in.
- Adjust inclusive classes by half the gap.
- Bivariate table classifies by two variables.
Related Glossary Terms and Chapters
Glossary:
Frequency distributionClass intervalClass markVariable
Related chapters: Collection of Data, where raw data come from; Presentation of Data, which turns the organised table into diagrams.
Sources
- NCERT textbooks: Statistics for Economics, Class 11, National Council of Educational Research and Training. Syllabus reference. Accessed 23 October 2026.
- Ministry of Statistics and Programme Implementation (MoSPI), Government of India. Accessed 23 October 2026.
- Office of the Registrar General and Census Commissioner, India, Ministry of Home Affairs. Accessed 23 October 2026.
- Press Information Bureau, Government of India. Accessed 23 October 2026.
- NITI Aayog, Government of India. Accessed 23 October 2026.
About this page
Drafted with AI assistance from the NCERT chapter structure, in original words. Content is checked and approved by Dr. Anamika Shrivastava, PhD Economics, before publication. Read how this site works or send a correction.
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