Skip to content

Frequency Distribution and Diagrams, ISC Class 11

questionbuddy.in/isc/class-11/economics/collection-organisation-presentation-data/frequency-distributions-tables-and-diagrams

EconomicsISC Class 11 · Collection, Organisation and Presentation of Data

Frequency Distributions, Tabulation and Diagrams (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

Raw data are organised by grouping them into class intervals and counting the frequency of each class. Inclusive classes (10-19, 20-29) are converted to exclusive form (9.5-19.5) by subtracting and adding half the gap. Cumulative frequencies add up the frequencies; data are then shown in tables and diagrams: bar, pie and line diagrams, and the histogram, frequency polygon and ogive for continuous series.

Bars touch because classes are continuous; the polygon joins the mid-point of each bar top and is closed to the axis using empty classes at both ends.

Illustration drawn for QuestionBuddy.

Understand Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams

Variables and series. A variable is a measurable characteristic. *Discrete* variable: takes separate values (number of children). *Continuous*: any value in a range (height). An *individual series* lists values; a *discrete series* gives each value with its frequency; a *continuous series* groups values into classes.

Constructing a frequency distribution. (1) Find the range = highest - lowest. (2) Decide the number of classes (usually 5 to 15) and the class width = range / number of classes (rounded). (3) Write the classes. (4) Use tally marks to count frequencies. (5) Total to check the sum equals the number of observations.

Class terms. *Class limits* (lower and upper), *class width* = upper - lower, *mid-point* = (lower + upper) / 2.

  • Exclusive method: the upper limit of a class is the lower limit of the next and is not included in it (10-20: 10 is included, 20 goes to the next class).
  • Inclusive method: both limits are included (10-19, 20-29). To convert to exclusive form, find the gap between the upper limit of one class and the lower limit of the next (here 1), and subtract half of it from lower limits and add half of it to upper limits: 9.5-19.5, 19.5-29.5.

Cumulative frequency. *Less-than* cumulative frequency: frequencies added from the top, against upper limits. *More-than*: from the bottom, against lower limits.

Tabulation. A table has a number, title, stub (row heads), caption (column heads), body, headnotes and source/footnotes.

Diagrams. *Bar diagram:* compare values of items; *pie diagram:* parts of a whole (angle = value / total x 360); *line graph:* change over time. For continuous series: *histogram* (bars with no gaps; for unequal classes, plot frequency density or adjusted frequency), *frequency polygon* (join mid-points), *ogive* (plot cumulative frequencies against class limits; the median is read at N/2).

Adjusted frequency for unequal classes = Frequency / Class width x Smallest class width.

Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams: key facts

Class width and mid-point
Class width = Upper limit - Lower limit | Mid-point = (Lower + Upper) / 2
Used in grouped data.
Inclusive to exclusive
Adjustment = Half the gap between classes | Lower limit - adjustment, upper limit + adjustment
E.g. 0.5.
Pie angle
Angle = Value / Total x 360 degrees | Angles add to 360
Pie diagram.

How to answer Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams questions

Organise step by step and check totals.

  1. 1Find the range and choose class width.
  2. 2Tally and total the frequencies.
  3. 3Convert classes if inclusive, then find mid-points or cumulative frequencies.
  4. 4Choose a suitable diagram and compute the plotting values.

Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Cumulative frequency

Suppose a teacher groups the marks of 20 students and wants to know how many scored below 30.

Which cumulative frequency answers this?

Try it first, then show the solutionSolution
  1. Add frequencies of classes whose upper limit is up to 30.
  2. This is the less-than cumulative frequency at 30.

Answer: The less-than cumulative frequency at 30.

Memory hook: Less than a limit: add from the top.

Tricks to remember Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams

Short memory aids. Say each one aloud once, then use it on a question the same day.

  • Exclusive excludes the top

    In 10-20, the value 20 goes to the next class.

    Use it when: Class rule.

  • Half the gap

    Subtract and add half the gap for inclusive classes.

    Use it when: Conversion.

  • Histogram has no gaps

    Continuous classes touch.

    Use it when: Histogram.

Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Mid-point · 1 mark · MCQBoard level

The mid-point of the class 30-40 is:
(a) 30 (b) 35 (c) 40 (d) 70

Try it first, then show the solutionSolution
  1. (30 + 40) / 2 = 35.

Answer: (b) 35

Example 2: Inclusive to exclusive · 3 marks · Short answerBoard level

Convert the classes 10-19, 20-29, 30-39, 40-49 to the exclusive form and find the mid-points. (3 marks)

Try it first, then show the solutionSolution
  1. Gap between 19 and 20 is 1; half is 0.5.
  2. Classes: 9.5-19.5, 19.5-29.5, 29.5-39.5, 39.5-49.5.
  3. Mid-points: 14.5, 24.5, 34.5, 44.5.

Answer: 9.5-19.5, 19.5-29.5, 29.5-39.5, 39.5-49.5; mid-points 14.5, 24.5, 34.5, 44.5.

Example 3: Construct a frequency distribution · 6 marks · Long answerHarder board level

The marks of 20 students are: 12, 25, 31, 18, 42, 37, 28, 15, 22, 34, 45, 29, 38, 19, 26, 33, 41, 23, 36, 14. Form a frequency distribution with classes 10-20, 20-30, 30-40, 40-50 (exclusive), and find the less-than cumulative frequencies, the mid-points and the number of students scoring 30 or more. (6 marks)

Try it first, then show the solutionSolution
  1. 10-20: 12, 18, 15, 19, 14 = 5.
  2. 20-30: 25, 28, 22, 29, 26, 23 = 6.
  3. 30-40: 31, 37, 34, 38, 33, 36 = 6.
  4. 40-50: 42, 45, 41 = 3. Total 20.
  5. Less-than cumulative at 20, 30, 40, 50: 5, 11, 17, 20. Mid-points 15, 25, 35, 45.
  6. 30 or more = 6 + 3 = 9 students (or 20 - 11).

Answer:

ClassfMid-pointLess-than cf
10-205155
20-3062511
30-4063517
40-5034520

9 students scored 30 or more.

Example 4: Histogram with unequal classes · 4 marks · Short answerHarder board level

Classes 0-10, 10-20, 20-40, 40-80 have frequencies 8, 12, 20, 16. Find the adjusted frequencies for a histogram (smallest width = 10) and the class that has the highest frequency density. (4 marks)

Try it first, then show the solutionSolution
  1. Adjusted f = f / width x 10.
  2. 0-10: 8; 10-20: 12; 20-40: 20 / 20 x 10 = 10; 40-80: 16 / 40 x 10 = 4.
  3. Highest is 10-20 with 12.

Answer: Adjusted frequencies 8, 12, 10, 4; the 10-20 class has the highest density although 20-40 has the largest frequency.

Example 5: Pie diagram interpretation · 5 marks · Long answerHOTS / competency

A government budget is Rs 800 crore: education Rs 240 crore, health Rs 160 crore, defence Rs 200 crore, others Rs 200 crore. Find the pie angles. If the next year's health spending rises to Rs 200 crore and the total to Rs 900 crore, find the new angle for health and comment on whether health's share rose. (5 marks)

Try it first, then show the solutionSolution
  1. Angles: education 240/800 x 360 = 108; health 72; defence 90; others 90; total 360.
  2. New health angle = 200 / 900 x 360 = 80 degrees.
  3. Share: 20% to 22.2%, so health's share rose though the total also rose.

Answer: Angles 108, 72, 90, 90; new health angle 80 degrees; the share rose from 20% to about 22.2%.

Where students lose marks in Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams

The slips examiners see most often, and what to write instead.

  • Mistake: Counting the upper limit in an exclusive class

    Inclusive habit.

    Wrong working

    20 counted in 10-20

    Right working

    20 counted in 20-30

    Do this: Upper limit goes to the next class.

    Usually costs: 2 marks.

  • Mistake: Adjusting inclusive classes by 1

    Using the gap not half.

    Wrong working

    9-20

    Right working

    9.5-19.5

    Do this: Half the gap.

    Usually costs: 1-2 marks.

  • Mistake: Plotting ogive against mid-points

    Polygon confusion.

    Wrong working

    cf against mid-points

    Right working

    cf against upper limits

    Do this: Ogive uses limits.

    Usually costs: 2 marks.

Exam tips

  • Show tallies or working for the frequencies.
  • State clearly whether classes are inclusive or exclusive.
  • Write the angle formula and show the angles sum to 360.

Practice questions on Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

  1. Q1 · 1 mark · MCQBoard level

    A histogram is suitable for:
    (a) discrete data only (b) a continuous frequency distribution (c) time series (d) pie data

    Hint

    Bars touch.

    Answer

    (b) a continuous frequency distribution

  2. Q2 · 3 marks · Short answerBoard level

    State the difference between the inclusive and exclusive methods of class intervals. (3 marks)

    Hint

    Limits included or not.

    Answer

    In the exclusive method the upper limit of a class is excluded and counted in the next class (10-20, 20-30). In the inclusive method both limits are included (10-19, 20-29), and a correction of half the gap is needed to treat the series as continuous.

  3. Q3 · 5 marks · Long answerHarder board level

    The daily wages (Rs) of 25 workers are: 210, 245, 260, 310, 225, 275, 290, 330, 255, 240, 300, 265, 285, 315, 235, 270, 250, 295, 320, 220, 280, 305, 260, 243, 288. Form a frequency distribution with classes 200-250, 250-300, 300-350, and find the cumulative frequencies. (5 marks)

    Hint

    Count each class.

    Answer

    200-250: 210, 245, 225, 240, 235, 220, 243 = 7. 250-300: 260, 275, 290, 255, 265, 285, 270, 250, 295, 280, 260, 288 = 12. 300-350: 310, 330, 300, 315, 320, 305 = 6. Total 25. Less-than cumulative frequencies: 7, 19, 25.

  4. Q4 · 4 marks · Short answerHarder board level

    Classes 10-19, 20-29, 30-39 have frequencies 6, 14, 10. Convert to exclusive classes, find the mid-points, the less-than cumulative frequencies, and the number of observations below 29.5. (4 marks)

    Hint

    Use 9.5.

    Answer

    Classes 9.5-19.5, 19.5-29.5, 29.5-39.5; mid-points 14.5, 24.5, 34.5; less-than cf 6, 20, 30. Below 29.5 there are 20 observations.

  5. Q5 · 3 marks · Short answerHOTS / competency

    A bar diagram of the population of five cities and a pie diagram of the same data are proposed. Evaluate which is better to compare the sizes of cities and which for the share of each city in the total of the five. (3 marks)

    Hint

    Compare vs share.

    Answer

    A bar diagram is better to compare sizes, because lengths are easy to compare. A pie diagram is better to show each city's share in the total, as angles show the parts of a whole.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

  1. 1.Mid-point formula?Show answer

    (Lower + Upper) / 2.

  2. 2.Exclusive class 10-20 includes?Show answer

    10 but not 20.

  3. 3.Pie angle formula?Show answer

    Value / Total x 360.

  4. 4.Ogive is plotted against?Show answer

    Class limits.

  5. 5.Histogram has?Show answer

    No gaps between bars.

Frequency Distributions, Class Intervals, Cumulative Frequency and Diagrams: frequently asked questions

How many classes?

Usually 5 to 15.

What is class width?

Upper limit minus lower limit.

Which diagram shows change over time?

Line graph.

What is a stub?

The row heading of a table.