EconomicsCBSE Class 11 · Presentation of Data

Histogram, Frequency Polygon, Frequency Curve, Ogive and Line Graph

Updated 17 October 2026 · Fact-checked

A histogram shows a continuous frequency distribution as adjacent rectangles whose areas are proportional to frequencies; for unequal classes, frequencies are adjusted to a common width. A frequency polygon joins class marks, an ogive plots cumulative frequencies against class limits, and its intersection with N/2 gives the median. A line graph shows a time series.

Understand Histogram, Frequency Polygon and Ogive

Histogram. Used for continuous frequency distributions with exclusive classes. Class limits on the horizontal axis, frequencies on the vertical axis; rectangles are adjacent (no gaps), width = class interval, height = frequency (equal classes). Area is proportional to frequency.

Unequal class intervals: heights must show frequency density. Adjusted frequency = (frequency ÷ class width) × smallest class width, so that every bar corresponds to the standard width. Example: classes 0-10 (f = 8), 10-20 (12), 20-40 (30), 40-60 (16). With standard width 10, adjusted frequencies = 8, 12, 30 ÷ 2 = 15 and 16 ÷ 2 = 8.
The mode can be located from the tallest bar.

Frequency polygon. Join the mid-points of the tops of the rectangles (or plot frequencies against class marks) with straight lines, extending to the horizontal axis at the class marks of an extra empty class at each end. For unequal classes use adjusted frequencies. Class marks for the example: 5, 15, 30, 50.

Frequency curve. A smooth curve through the polygon's points.

Ogive (cumulative frequency curve).

Less than ogive: plot cumulative frequencies against upper limits; rising curve. Example: (10, 8), (20, 20), (40, 50), (60, 66).
More than ogive: plot against lower limits; falling curve.
Median from ogives: the point where the two ogives intersect, or read on the less-than ogive the value corresponding to N/2. Here N = 66 and N/2 = 33 falls in 20-40: median ≈ 20 + (33 − 20)/30 × 20 = 28.67.
Quartiles from N/4 and 3N/4.

Arithmetic line graph (time series graph). Time on the horizontal axis, the variable on the vertical axis; points joined in time order; shows trends, rises and falls (population, prices, GDP). Use a broken axis only with a mark showing it.

Choosing. Histogram: shape of a distribution; polygon: compare two distributions on the same axes; ogive: medians, quartiles, 'how many below'; line graph: change over time.

Draw and label: a histogram with a superimposed frequency polygon; a less-than ogive with the median marked.

Histogram, Frequency Polygon and Ogive: key facts

Adjusted frequency: (Frequency / class width) x smallest class width; Unequal classes
Polygon: Join class marks of the top of bars; Add empty end classes
Ogive: Cumulative frequency against upper (less than) or lower (more than) limits; Cumulative curves
Median from ogive: Value at N/2 on the less-than ogive; Interpolation

How to answer Histogram, Frequency Polygon and Ogive questions

Questions give a frequency table and ask for a graph, adjusted frequencies or a median from an ogive.

1Check class widths; if unequal, adjust frequencies to the smallest width.
2Compute cumulative frequencies for the ogive and plot against limits.
3Locate N/2 and read across to the curve, then down to the horizontal axis.
4Use linear interpolation within the class for exact answers.
5Label axes, scale and title.

Histogram, Frequency Polygon and Ogive 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.

Wages of workers

Suppose a factory table gives weekly wages in classes 0-10, 10-20, 20-40, 40-60 (Rs hundred) with workers 8, 12, 30, 16.

Why must the frequencies be adjusted for the histogram, and what are the adjusted values?

Try it first, then show the solutionSolution

Classes 20-40 and 40-60 are twice as wide as the others.
Unadjusted bars would look too tall and mislead.
Adjusted frequencies for standard width 10: 8, 12, 15, 8.

Answer: Adjusted frequencies are 8, 12, 15 and 8.

Memory hook: Wider class, shorter bar.

Tricks to remember Histogram, Frequency Polygon and Ogive

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

Adjust = f over width times base
Adjusted f = f / width x smallest width.
Use it when: Unequal classes
Less-than uses upper
Less than ogive: upper limits; more than ogive: lower limits.
Use it when: Ogive
Median at N over 2
Read N/2 on the ogive.
Use it when: Median

Histogram, Frequency Polygon and Ogive: solved examples

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

Example 1: Histogram vs bar · 3 marks · Short answerBoard level

Explain why a histogram of wage classes has no gaps between bars while a bar diagram of five products has gaps, and what the width of a histogram bar stands for.

Try it first, then show the solutionSolution

Wage classes form a continuous scale, so one class runs straight into the next with no gap (1 mark).
Products are separate items with no scale between them, so gaps are used (1 mark).
Histogram bar width equals the class interval and its area is proportional to frequency; in a bar diagram width has no meaning (1 mark).

Answer: Continuous scale versus separate items.

Example 2: Adjusted frequencies · 3 marks · Short answerBoard level

Classes 0-10, 10-20, 20-40, 40-60 have frequencies 8, 12, 30, 16. Find the adjusted frequencies for a histogram.

Try it first, then show the solutionSolution

Smallest width is 10 (1 mark).
20-40: 30 / 20 x 10 = 15 (1 mark).
40-60: 16 / 20 x 10 = 8; others stay 8 and 12 (1 mark).

Answer: 8, 12, 15, 8.

Example 3: Median by ogive · 4 marks · Short answerHarder board level

For the same data find the cumulative frequencies, N/2, the median class and the median.

Try it first, then show the solutionSolution

Cumulative: 8, 20, 50, 66 (N = 66) (1 mark).
N/2 = 33, so the median class is 20-40 (1 mark).
Median = 20 + (33 - 20)/30 x 20 (1 mark).
= 20 + 8.67 = 28.67 (1 mark).

Answer: Median 28.67.

Example 4: Plotting an ogive · 3 marks · Short answerHarder board level

State the points to plot for a less-than ogive for the above data and explain how the more-than ogive differs.

Try it first, then show the solutionSolution

Less than: (10, 8), (20, 20), (40, 50), (60, 66) (1 mark).
Starts with (0, 0) at the first lower limit (1 mark).
More than ogive plots against lower limits with decreasing cumulative frequencies and slopes downward (1 mark).

Answer: Upper-limit points; falling curve for more than.

Example 5: Reading a time series · 4 marks · Short answerHOTS / competency

Sales (Rs lakh) were 300 in year 1 and 330 in year 2. Find the percentage change, state which graph suits, and explain how to avoid a misleading slope.

Try it first, then show the solutionSolution

Change = 30/300 x 100 = 10 per cent (1 mark).
A line graph (time series graph) suits (1 mark).
Use a suitable scale on both axes and start the vertical axis at zero or show the break clearly (1 mark).
A stretched axis would exaggerate a small change (1 mark).

Answer: 10 per cent; line graph; honest scale.

Where students lose marks in Histogram, Frequency Polygon and Ogive

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

Mistake: Gaps in a histogram
Bar diagram mix.
Wrong working
Leave gaps between bars.
Right working
Histogram bars touch.
Do this: No gaps.
Usually costs: 1 mark.
Mistake: Ogive plotted at class marks
Wrong axis.
Wrong working
Plot ogive against mid-points.
Right working
Plot against limits.
Do this: Limits.
Usually costs: 1 mark.
Mistake: No adjustment for unequal widths
Height misleading.
Wrong working
Use raw frequencies as heights.
Right working
Adjust to the smallest width.
Do this: Adjust.
Usually costs: 1 mark.

Exam tips

Show adjusted frequencies for unequal classes.
Plot ogives against limits.
Interpolate for the median.
Label axes with units.

Practice questions on Histogram, Frequency Polygon and Ogive

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

Q1 · 1 mark · MCQBoard level
A graph used to find the median of a distribution is
(a) histogram (b) ogive (c) pie (d) simple bar
Hint
Cumulative.
Answer
(b).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The frequency polygon joins the mid-points of the tops of the histogram bars.
Reason (R): Class marks represent the classes.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Mid-points.
Answer
(a).
Q3 · 2 marks · Very short answerBoard level
Why are frequencies adjusted when class intervals are unequal in a histogram?
Hint
Area ~ frequency.
Answer
Because the area of each rectangle must be proportional to frequency; otherwise wider classes would look larger than they are.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose a less-than ogive passes through (10, 8), (20, 20), (40, 50), (60, 66).
(a) How many items are below 40? (1)
(b) How many are 20 or more? (1)
(c) Find the approximate value below which 25 per cent of items fall (1st quartile). (2)
Hint
N/4.
Answer
(a) 50. (b) 66 - 20 = 46. (c) N/4 = 16.5 lies in 10-20: 10 + (16.5 - 8)/12 x 10 = 17.08 (about 17.1).
Q5 · 3 marks · Short answerHOTS / competency
Suppose two firms' wage distributions are to be compared on one diagram. Explain whether to use two histograms or two polygons, and why unequal total frequencies matter.
Hint
Overlay.
Answer
Frequency polygons can be drawn on the same axes and compared easily, whereas overlapping histograms hide each other. If the totals differ, use relative frequencies (percentages) so shapes can be compared fairly.

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.Where does the polygon join?Show answer
Class marks.
2.Less than ogive uses?Show answer
Upper limits.
3.Median from?Show answer
N/2 on the ogive.
4.Histogram for?Show answer
Continuous distributions.
5.Time series graph?Show answer
Line graph.
6.Frequency curve?Show answer
Smoothed polygon.

Histogram, Frequency Polygon and Ogive: frequently asked questions

What is a histogram?

Adjacent rectangles for a continuous frequency distribution.

What is an ogive?

A cumulative frequency curve.

How to find the median from an ogive?

Read the value at N/2.

What is a frequency polygon?

A line joining class marks.

What is a time series graph?

A line graph of values over time.