For the class 60–80, the class mark and the class interval are respectively:

Answer: C. The class mark is the average of the limits, $(60 + 80)/2 = 70$, and the class interval is the difference, $80 - 60 = 20$. Option A swaps the two. Options B and D use the sum of the limits, 140, which is neither quantity.

The smallest and largest values in a data set are 9 and 58. If classes of width 10 are made from 0 onwards by the exclusive method, how many classes are needed?

Answer: B. The classes must run from 0 to cover 58, so they are 0–10, 10–20, 20–30, 30–40, 40–50 and 50–60. That is 6 classes. Option A comes from $49/10 \approx 5$, which ignores that the first class starts at 0 and not at 9. Option D is the range, not a count of classes.

Consider the statements. Statement I: A discrete variable can take fractional values such as 7.5 if it jumps between separate values. Statement II: A continuous variable can take any value, including fractional and irrational ones, within a range. Which is correct?

Answer: C. Discreteness depends on gaps between the allowed values, not on whether they are integers, so shoe sizes 7, 7.5, 8 are discrete. A continuous variable such as height or time can take any value in a range. Both statements are correct.

Organisation of Data — CUET Economics Notes & MCQs

📌 Snapshot

Turn raw marks into a frequency table, with class limits, class marks and exclusive classes made clear.
The four ways to classify data, and how to tell a discrete variable from a continuous one without guessing.
Worked examples that build a frequency distribution, convert inclusive classes to exclusive ones and handle open classes.
Frequency array, bivariate distribution and the exam traps on range, class interval and the 0.5 adjustment.

📖 Detailed Notes

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Based on NCERT Economics Class XI (Statistics for Economics), Chapter 3.

Can you read 30 marks at a glance?

Here are the marks of 30 students: 12, 45, 27, 33, 20, 38, 41, 9, 56, 24, 30, 47, 18, 36, 52, 29, 10, 43, 35, 22, 58, 31, 26, 40, 15, 49, 33, 28, 20, 44. How many scored between 30 and 40? Which stretch of marks holds the most students? You would have to count one number at a time, and with 3,000 students you would give up.

Raw data are like a pile of unsorted laundry: everything is there, but nothing can be found. Classification is the folding and sorting. You arrange the observations into groups (classes) by some criterion, and suddenly the pattern is visible. Every later chapter, from averages to correlation, starts from data that have been organised this way.

Four ways to classify data

The criterion you choose decides the type of classification. Learn the quick test in the last column, because CUET often gives a short description and asks you to name the type.

Type	Classified by	Example	Quick test
Chronological	Time (years, months, weeks)	India's population in each census year from 1951 to 2011	Rows are time periods
Spatial	Place (country, state, district)	Wheat yield in different countries in one year	Rows are places, time is fixed
Qualitative	An attribute that cannot be measured (gender, religion, literacy)	Population split into male and female, then married and unmarried	Counts of "has it or not"
Quantitative	A measurable characteristic (height, income, marks)	Students grouped by marks into 0–10, 10–20 and so on	Rows are value ranges

Data arranged by time form a time series: many time points for the same unit. Spatial data are the opposite, many places at one moment. A variable is a characteristic you can measure, such as income or weight. An attribute is a quality you can only record as present or absent, such as literacy. That is why attributes are classified by counting, while variables can be classified by counting and by magnitude.

Discrete or continuous?

Quantitative data come from a variable, and the variable decides how you will group it.

	Continuous variable	Discrete variable
What values are possible	Any value in a range: whole, fractional or irrational	Only certain separate values, with jumps between them
Test	Can you always find a value between two values?	Is there a gap with nothing allowed in between?
Examples	Height, weight, time, distance	Number of students in a class, number of cars, shoe size
Organised as	Frequency distribution with class intervals	Frequency array

The trap is thinking that discrete means "whole numbers". It does not. Shoe sizes run 7, 7.5, 8, 8.5, so they take fractional values, yet there is nothing between 7 and 7.5. What defines a discrete variable is the jump, not whether the number looks like an integer.

Pause and predict: is the number of children in a family a discrete or a continuous variable? And what about the time a student takes to finish a test?

Check your answer: the number of children is discrete (you cannot have 2.4 children in one family). Time is continuous, since 45 minutes, 45.3 minutes and 45.31 minutes are all possible.

Anatomy of a frequency distribution

A frequency distribution shows how the values of a quantitative variable are spread over classes, together with the class frequency, the number of observations in each class. Take the class 40–50 and name its parts.

Term	Meaning	For the class 40–50
Class limits	The two ends of the class	Lower limit 40, upper limit 50
Class interval (class width)	Upper limit minus lower limit	$50 - 40 = 10$
Class mark (mid-point)	The average of the two limits	$(40 + 50)/2 = 45$
Range	Largest value minus smallest value of the whole data set	Not a property of one class
Class frequency	Number of observations in the class	Counted from the data

$$m = \dfrac{\text{LCL} + \text{UCL}}{2}, \qquad h = \text{UCL} - \text{LCL}, \qquad R = X_{\max} - X_{\min}, \qquad k \approx \dfrac{R}{h}$$

Once data are grouped, the individual values disappear, so one number must stand in for the whole class. That number is the class mark, and every later calculation of a mean or a similar measure uses it. The usual number of classes is between 6 and 15. Too few classes hide the shape of the data, and too many bring back the clutter you were trying to escape.

Worked example: building a frequency distribution. Use the 30 marks from the start of this section. Build a frequency distribution by the exclusive method with class width 10.

Step	What we do	Result
1	Find the range: largest minus smallest	$58 - 9 = 49$
2	Divide by the width to estimate the number of classes	$49 / 10 \approx 5$, but we start at the round number 0 and 58 needs the class 50–60, so use 6 classes
3	Fix one rule for boundary values. In the exclusive method a value equal to the upper limit goes to the next class	10 goes to 10–20, 20 goes to 20–30
4	Place each mark in its class and count	See the table below
5	Check that the frequencies add up to the number of observations	$1 + 4 + 8 + 7 + 7 + 3 = 30$

Class	Marks that fall in it	Frequency $f$	Class mark $m$
0–10	9	1	5
10–20	10, 12, 15, 18	4	15
20–30	20, 20, 22, 24, 26, 27, 28, 29	8	25
30–40	30, 31, 33, 33, 35, 36, 38	7	35
40–50	40, 41, 43, 44, 45, 47, 49	7	45
50–60	52, 56, 58	3	55

Common wrong step. Putting the two 20s in 10–20 because "20 is the upper limit of that class". That is the inclusive habit leaking into an exclusive table. The frequencies become 6 and 6 instead of 4 and 8, and the total still comes to 30, so the check in step 5 does not catch it. Choose one rule and apply it to every boundary value.

Exclusive and inclusive classes

Boundary values like 20 need a rule. The two methods give two different rules.

	Exclusive method	Inclusive method
Classes look like	0–10, 10–20, 20–30	0–9, 10–19, 20–29
A value equal to a limit	The lower limit belongs to this class, the upper limit goes to the next	Both limits belong to this class
Gap between classes	None, the classes are continuous	A gap (for example between 9 and 10)
Typical use	Continuous variables	Variables recorded in whole numbers, such as income in rupees

An inclusive table has a problem: the value 9.5 has no home, because it falls in the gap between 9 and 10. To restore continuity, subtract 0.5 from every lower limit and add 0.5 to every upper limit. The 0.5 is not magic. It is half of the gap between the upper limit of one class and the lower limit of the next, so each class stretches halfway across the gap and the two meet in the middle.

Worked example: inclusive to exclusive. Monthly incomes of 55 employees (in ₹) are grouped in the inclusive method. Convert to continuous classes and find the class marks.

Step	What we do	Result
1	Find the gap between the upper limit of one class and the lower limit of the next	$900 - 899 = 1$
2	Take half of the gap as the adjustment	$0.5$
3	Subtract 0.5 from each lower limit and add 0.5 to each upper limit	See the table below
4	Recompute the class marks and compare	$(799.5 + 899.5)/2 = 849.5$, the same as $(800 + 899)/2$

Inclusive class	Exclusive class	Frequency	Class mark
800–899	799.5–899.5	12	849.5
900–999	899.5–999.5	20	949.5
1000–1099	999.5–1099.5	15	1049.5
1100–1199	1099.5–1199.5	8	1149.5

The class marks do not change, because both limits moved by the same amount in opposite directions. Every class now has width 100 and there are no gaps.

Common wrong step. Adding 0.5 only to the upper limits, which gives 800–899.5 and 900–999.5. A gap of 0.5 is still left between 899.5 and 900. Both ends must move.

Pause and predict: a table has the inclusive classes 0–9 and 10–19, and the data are recorded to one decimal place. Which class does the value 9.6 belong to?

Check your answer: neither. It falls in the gap between 9 and 10. This is exactly why an inclusive table has to be adjusted before it can hold every possible value.

Beyond the textbook. For the exam: convert an inclusive table to an exclusive one by subtracting 0.5 from the lower limits and adding 0.5 to the upper limits. In reality: the correct boundaries depend on how the figures were recorded. Incomes rounded to the nearest rupee make the class 800–899 really cover 799.5 up to 899.5, so the 0.5 adjustment is right. Ages are usually recorded in completed years, so the class 10–19 really covers 10.0 up to just under 20.0, and the true boundaries are 10 and 20. Why it matters: the gap in an inclusive table comes from how the data were recorded, not from the variable itself.

Unequal classes and open-ended classes

Equal classes are the default, but they are a poor fit when the data are crowded in one part of the range or the range is very wide, as with incomes. A class width of 10 over incomes from 0 to 150 would need 15 classes, with the top few nearly empty. So wider classes are used where the data thin out.

Worked example: unequal classes. Monthly incomes (₹ thousand) of 100 households are grouped as below. Find the class widths and class marks.

Class	Frequency	Class width	Class mark
0–10	30	10	5
10–25	36	15	17.5
25–50	22	25	37.5
50–100	9	50	75
100 and above	3	not given	not defined

The class mark is still the mid-point of each class, but the marks (5, 17.5, 37.5, 75) are no longer evenly spaced. The last class is open-ended: it has no upper limit, so it has no mid-point. If a calculation forces you to give it one, the usual practice is to assume it has the same width as the class before it.

Common wrong step. Assuming the class mark of 10–25 is $10 + 5 = 15$ by copying the width of the first class. Always take the mid-point of the limits of that class: $(10 + 25)/2 = 17.5$.

Frequency array: when the variable is discrete

When a variable is discrete there is no need for class intervals, because each distinct value is its own category. The result is a frequency array.

Worked example: a frequency array. The household sizes of 25 families are: 3, 4, 2, 5, 3, 4, 1, 3, 6, 4, 2, 3, 5, 4, 3, 2, 4, 7, 3, 4, 5, 2, 3, 4, 3. Organise them.

Size of household	Frequency
1	1
2	4
3	8
4	7
5	3
6	1
7	1
Total	25

Counting each size once gives the array, and the total matches the 25 families. Reading it is immediate: the most common household has 3 members.

Common wrong step. Grouping the sizes into classes such as 1–2, 3–4 and 5–6. This throws away exact information for no reason, since a discrete variable with only seven values needs no grouping.

The three kinds of table you must tell apart:

Table	Variables	Used for	Example
Frequency array	One, discrete	Each value has its own frequency	Household size
Frequency distribution	One, continuous (or many discrete values)	Frequencies of class intervals	Marks of 30 students
Bivariate frequency distribution	Two together	Joint frequency in each cell	Advertising spend and sales

The price of grouping: loss of information

When you replace 9, 10, 12, 15 and 18 by "the class 10–20 has four students", you can never recover the individual marks. You cannot tell whether those four scored 10, 12, 15 and 18, or all clustered near 19. Statisticians accept this loss because a table of 6 numbers is far easier to grasp than a list of 30. But the loss is real, and it shows up in the calculations.

Worked example: the cost of grouping. For the 30 marks, compare the true average from the raw data with the average estimated from the grouped table (you will meet the mean fully in the chapter on averages; here we only need the idea).

Step	What we do	Result
1	Add all 30 raw marks and divide by 30	$971 / 30 = 32.37$
2	Multiply each class mark by its frequency and add	$5(1) + 15(4) + 25(8) + 35(7) + 45(7) + 55(3) = 990$
3	Divide by the total frequency	$990 / 30 = 33.0$

The grouped answer (33.0) is close to but not equal to the true one (32.37). Treating every student in a class as if they scored the class mark is an approximation, and that is the whole reason the answers differ.

Common wrong step. Believing that calculations on a grouped table use the actual observations. Once the data are grouped, every later calculation uses the class marks.

Two variables at once: bivariate distribution

A bivariate frequency distribution classifies the same units by two variables together. One variable is laid along the rows and the other along the columns, and each cell gives the number of units with that pair of values.

Sales (₹ crore) / Advertising (₹ lakh)	0–10	10–20	20–30	Total
10–20	4	1	0	5
20–30	2	5	1	8
30–40	0	2	5	7
Total	6	8	6	20

The row totals and column totals are called marginal frequencies, and the grand total equals the number of units, here 20 firms. The cell in the row 20–30 and the column 10–20 says that 5 firms had sales of ₹20–30 crore and advertising spend of ₹10–20 lakh. Notice that the heavy cells run from top left to bottom right: firms that advertise more tend to sell more. Measuring how strong that tendency is is the job of the chapter on correlation.

Pause and predict: in the table above, how many firms spent ₹0–10 lakh on advertising? And what does the column total of 8 mean?

Check your answer: 6 firms (the first column total). The 8 is the number of firms whose advertising spend was ₹10–20 lakh, whatever their sales.

Beyond the textbook. For the exam: data are either a time series (chronological) or spatial. In reality: statisticians also work with panel data, which follow the same units over time, for example the same households surveyed every year. A panel combines a time series (each household over the years) with a spatial cross-section (all households in a year). Why it matters: the chapter's four types are a starting point, and real surveys often mix them.

Exam traps

Trap	Why it is tempting	The truth
"Range is the width of a class"	Both involve a difference of two numbers	Range is largest minus smallest of the whole data. Class interval is upper limit minus lower limit of one class
"Class mark is the difference of the limits"	The width is also found from the limits	Class mark is the average of the limits, $(\text{LCL} + \text{UCL})/2$
"A discrete variable takes only whole numbers"	Counts like 3 children feel like integers	Discrete means separate values with jumps. Shoe sizes 7.5 and 8.5 are discrete
"In the exclusive method 20 belongs to 10–20"	The class ends at 20	The upper limit is excluded, so 20 belongs to 20–30
"The adjustment is always 1"	The gap in an inclusive table is 1	The adjustment is half the gap, so 0.5 here
"Grouped data calculations use the real values"	The real values still exist on paper	They use class marks. The individual values are lost
"Frequencies add up to 100"	Percentages add to 100	Class frequencies add up to $N$, the number of observations
"Bivariate means two classes or two groups"	The prefix bi means two	It means two variables recorded together for each unit

Formula sheet

Quantity	Formula	Unit	When to use
Class mark	$m = \dfrac{\text{LCL} + \text{UCL}}{2}$	Same as the variable	To represent a class in later calculations
Class interval	$h = \text{UCL} - \text{LCL}$	Same as the variable	Class width in an exclusive table
Range	$R = X_{\max} - X_{\min}$	Same as the variable	Planning the classes
Number of classes	$k \approx \dfrac{R}{h}$	Number	Equal classes, usually 6 to 15
Continuity adjustment	$\pm \tfrac{1}{2}(\text{gap})$, which is $\pm 0.5$ for a gap of 1	Same as the variable	Inclusive to exclusive
Total frequency	$\sum f = N$	Number	Checking a table

60-second revision

Topic	Lock this in
Classification types	Chronological (time), spatial (place), qualitative (attribute), quantitative (measurable)
Variable types	Continuous: any value in a range. Discrete: jumps, may be fractional (shoe size)
Class terms	Class limits, class interval $h = \text{UCL} - \text{LCL}$, class mark = average of the limits
Range	Largest minus smallest of the whole data
Number of classes	$R / h$, normally 6 to 15
Exclusive	Upper limit goes to the next class
Inclusive	Both limits stay in the class. Subtract 0.5 from lower limits, add 0.5 to upper limits for continuity
Unequal and open classes	Use when data are crowded or the range is wide. Open class has no mid-point
Frequency array vs distribution	Array for discrete values, distribution for classes
Bivariate	Two variables, marginal frequencies, grand total is $N$
Cost of grouping	Individual values lost, class marks used instead

Where this chapter connects

Previous: Collection of Data explains where the raw observations come from before you organise them.
Next: Presentation of Data turns the frequency tables built here into histograms, polygons and ogives.
Later use: Measures of Central Tendency computes means, medians and modes from the class marks and cumulative frequencies defined here, and Correlation starts from the bivariate table.

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