Quantitative Aptitude · Statistical Description of Data

Graphical Presentation: Histogram, Polygon and Ogive

Updated 1 October 2026 · Fact-checked

A histogram shows a continuous frequency distribution as adjacent rectangles. A frequency polygon joins the class mid-points of the tops of those bars. An ogive plots cumulative frequencies against class limits. The x-value of the point where the less-than and more-than ogives cross gives the median.

Understand Graphical Presentation: Histogram, Polygon and Ogive

Raw data is hard to read. Once you group it into a frequency distribution, you can draw it. Graphs let you see the shape of the data at a glance: where it bunches up, how spread out it is, and whether it leans to one side.

A histogram is used for a continuous frequency distribution. The x-axis shows the class intervals on a true scale. Each class becomes a rectangle with no gaps. The area of each rectangle is proportional to the class frequency. With equal class widths, height is proportional to frequency. With unequal widths, you must plot frequency density instead. A bar diagram is different: its bars have gaps, only the height matters, and it is used for discrete or categorical data.

A frequency polygon is made by plotting each class mid-point against its frequency and joining the points with straight lines. Close it at both ends by adding an empty class before the first and after the last, each with frequency 0. You can draw it on top of a histogram by joining the mid-points of the tops of the bars. A frequency curve is the polygon smoothed out. Common shapes are symmetrical (bell-shaped), positively skewed (long right tail), negatively skewed (long left tail), J-shaped, reverse J-shaped and U-shaped.

An ogive (cumulative frequency curve) is a graph of cumulative frequencies. A less-than ogive plots cumulative frequency against the upper limit of each class and rises from left to right. A more-than ogive plots cumulative frequency against the lower limit and falls from left to right. Start the less-than ogive at the lower limit of the first class with value 0, and start the more-than ogive at the lower limit of the first class with value N.

The two ogives cross at one point. Its x-value is the median. Its y-value is N ÷ 2. You can also read the median from a single ogive: find N ÷ 2 on the y-axis, go across to the curve, then drop down to the x-axis. The same method gives quartiles (N ÷ 4, 3N ÷ 4), deciles and percentiles.

Key formulas to remember

Frequency density (unequal classes): Frequency density = class frequency ÷ class width; Use this as the bar height when class widths differ. With equal widths, plain frequency can be used.
Adjusted frequency (alternative for unequal classes): Adjusted height = (frequency ÷ class width) × standard width; Standard width is usually the smallest class width. This is the frequency density scaled to a chosen width.
Mid-point of a class: Mid-point = (lower limit + upper limit) ÷ 2; Used as the x-value for the frequency polygon. For inclusive classes, convert to true class boundaries first.
Median from ogive: Median = x-value at cumulative frequency N ÷ 2; N is the total frequency. The same point is where the less-than and more-than ogives intersect.
Quartile from ogive: Q1 at N ÷ 4, Q3 at 3N ÷ 4; Read the x-value of the curve at that cumulative frequency. Deciles use kN ÷ 10 and percentiles use kN ÷ 100.
Area rule for histogram: Area of rectangle ∝ class frequency; Total area of the histogram represents total frequency.

How to solve Graphical Presentation: Histogram, Polygon and Ogive questions

Use this order for any question on histograms, polygons or ogives. Most MCQs test one of these steps directly.

1Check whether the classes are inclusive (10-19, 20-29) or exclusive (10-20, 20-30). Convert inclusive classes to true boundaries before drawing a histogram.
2Check class widths. If they are unequal, calculate frequency density for the histogram heights.
3For a histogram, mark class boundaries on the x-axis and draw touching rectangles of the required heights.
4For a polygon, find each class mid-point, plot it against frequency, join the points, and close the ends at zero using the imaginary classes on both sides.
5For an ogive, build the cumulative frequency column. Use upper limits for less-than and lower limits for more-than. Plot and join the points with a smooth curve.
6For the median, calculate N ÷ 2, locate it on the y-axis, move across to the curve and read the x-value. Or use the intersection of the two ogives.
7Check that your answer lies inside the median class and fits the shape of the data.

Quickest way: Shortcut for graph and ogive MCQs

When to use it: Use this when options give shapes, heights or a median value and you have under a minute per question.

For unequal classes, compute frequency ÷ width for each class. The tallest bar is the one with the highest density, not the highest frequency.
For median from ogive, compute N ÷ 2 first, then find the class where the cumulative frequency first reaches it. Reject any option outside that class.
Interpolate inside that class by proportion: median ≈ lower limit + (N ÷ 2 − previous cumulative) ÷ class frequency × width. This matches the reading from a straight-line ogive.
If a question asks what the intersection point of the two ogives gives, answer median without calculation.
If a question asks the starting point of an ogive, remember: less-than starts at 0 on the lower limit of the first class, more-than starts at N.

Common mistakes in Graphical Presentation: Histogram, Polygon and Ogive

Using frequency as bar height when class widths are unequal.
Students learn the equal-width case first and apply it everywhere.
Fix: Always check widths. If they differ, plot frequency density (frequency ÷ width).
Plotting the ogive against class mid-points.
Mid-points are used for the polygon, so they get mixed up.
Fix: Plot a less-than ogive against upper limits and a more-than ogive against lower limits.
Drawing a histogram directly on inclusive classes, leaving gaps.
Students ignore the gap between 19 and 20.
Fix: Subtract 0.5 from lower limits and add 0.5 to upper limits to get true boundaries. Then the bars touch.
Reading the median at N instead of N ÷ 2.
Students read the top of the cumulative frequency scale by habit.
Fix: Median sits at the middle, so go to N ÷ 2 on the y-axis.
Not closing the frequency polygon at the ends.
Students stop at the first and last mid-points.
Fix: Add one empty class on each side with frequency 0 and join to those mid-points.
Calling a bar diagram and a histogram the same thing.
Both use rectangles.
Fix: Histogram: continuous data, no gaps, area matters. Bar diagram: discrete or categorical data, gaps, only height matters.

Worked examples

Example 1

A histogram is drawn for these classes: 0-10 (frequency 20), 10-20 (frequency 30), 20-40 (frequency 40), 40-60 (frequency 20). Which class has the greatest bar height when frequency density is used? Options: (A) 0-10 (B) 10-20 (C) 20-40 (D) 40-60

Show the solution

Class widths are 10, 10, 20 and 20, so the widths are unequal.
Frequency density = frequency ÷ width.
0-10: 20 ÷ 10 = 2.
10-20: 30 ÷ 10 = 3.
20-40: 40 ÷ 20 = 2.
40-60: 20 ÷ 20 = 1.
The highest density is 3, in the class 10-20.

Answer: (B) 10-20

Example 2

For a distribution with total frequency 80, the less-than ogive and the more-than ogive intersect. At the point of intersection, the y-value is: Options: (A) 20 (B) 40 (C) 60 (D) 80

Show the solution

The two ogives intersect at the median.
The median corresponds to cumulative frequency N ÷ 2.
N = 80, so N ÷ 2 = 40.
At this point, 40 observations lie below and 40 lie above.

Answer: (B) 40

Example 3

Class intervals and frequencies: 0-10 (5), 10-20 (15), 20-30 (20), 30-40 (10). Using the less-than ogive, with a straight line between points, the median is: Options: (A) 22.5 (B) 25 (C) 20 (D) 27.5

Show the solution

N = 5 + 15 + 20 + 10 = 50, so N ÷ 2 = 25.
Less-than cumulative frequencies: below 10 is 5, below 20 is 20, below 30 is 40, below 40 is 50.
25 lies between 20 and 40, so the median class is 20-30.
Cumulative frequency rises from 20 to 40 across a width of 10, which is 2 per unit.
Needed rise from 20 to 25 is 5, so distance = 5 ÷ 2 = 2.5.
Median = 20 + 2.5 = 22.5.

Answer: (A) 22.5

Exam tips

Whenever a question shows unequal class widths, think frequency density first. This is the most common trap.
Memorise pairs: less-than ogive with upper limits, more-than ogive with lower limits, polygon with mid-points.
If an option says the ogive intersection gives the mode or mean, reject it. It gives the median.
Under negative marking, skip questions that need long cumulative tables unless the median class is easy to spot.
Know the frequency curve shapes by name: symmetrical, moderately skewed, J-shaped, reverse J-shaped and U-shaped.

Practice questions from Statistical Description of Data

Graphical Presentation: Histogram, Polygon and Ogive: frequently asked questions

What is the difference between a histogram and a bar diagram?

A histogram is for continuous data, its bars touch, and the area is proportional to frequency. A bar diagram is for discrete or categorical data, bars have gaps, and only the height carries meaning.

How do I draw a histogram with unequal class intervals?

Calculate frequency density for each class by dividing frequency by class width. Plot class boundaries on the x-axis and use the density as the height of each bar. This keeps area proportional to frequency.

How do I find the median graphically from an ogive?

Compute N ÷ 2 and mark it on the cumulative frequency axis. Draw a horizontal line to the ogive, then drop a vertical line to the x-axis. That value is the median. It is also the x-value where the less-than and more-than ogives cross.

What is the difference between a frequency polygon and a frequency curve?

A frequency polygon joins the plotted points with straight lines. A frequency curve is the smoothed version of the same shape. Both use class mid-points against frequencies.