Fundamentals of Business Mathematics and Statistics · Statistical Representation of Data

Histogram and Frequency Polygon: How to Draw and Solve

Updated 10 October 2026 · Fact-checked

A histogram shows a continuous frequency distribution as adjacent rectangles whose areas are proportional to class frequencies. A frequency polygon joins the mid-points of the tops of those rectangles with straight lines. For unequal classes, plot frequency density instead of frequency. A frequency curve is the smoothed polygon.

Understand Graphical Representation: Histogram and Frequency Polygon

A frequency distribution is a table of class intervals and frequencies. A graph makes the same data easy to see. The histogram, frequency polygon and frequency curve are the three standard graphs for a continuous distribution.

A histogram is a set of rectangles with no gaps between them. The class intervals go on the x-axis and the frequencies on the y-axis. Each rectangle's width is the class width and its height is the frequency. Because the classes must touch, the classes need to be continuous. If they are inclusive (like 10-19, 20-29), convert them to exclusive (9.5-19.5, 19.5-29.5) first.

A frequency polygon is made from the histogram. Mark the mid-point of the top of each rectangle, join these points with straight lines, and also join the ends to the x-axis at the mid-points of one empty class before the first and one after the last class. You can also draw it directly: plot class mid-values against frequencies and join them.

A frequency curve is the polygon smoothed into a curve. It keeps roughly the same area and shows the shape of the distribution, such as symmetric or skewed.

A histogram is not a bar diagram. A bar diagram uses separate bars with gaps, where only the height matters and the width is arbitrary. It is used for categories or discrete data. In a histogram the area matters, so width counts. With unequal class widths you must adjust the heights, using frequency density.

Key formulas to remember

Class width: Class width = Upper limit − Lower limit (exclusive classes); For inclusive classes, first convert to true limits.
Mid-value of a class: Mid-value = (Lower limit + Upper limit) ÷ 2; Used as the x-coordinate of each point on the frequency polygon.
Adjustment for inclusive classes: True lower limit = Lower limit − d ÷ 2; True upper limit = Upper limit + d ÷ 2; d is the gap between the upper limit of one class and the lower limit of the next. For 10-19, 20-29, d = 1.
Frequency density: Frequency density = Class frequency ÷ Class width; Use as the height of each rectangle when class widths are unequal. Some books first scale to a standard width; the shape is the same.

How to solve Graphical Representation: Histogram and Frequency Polygon questions

Use this order for any question that asks you to draw, read or identify a histogram, frequency polygon or frequency curve.

1Check whether the classes are exclusive or inclusive. If inclusive, convert to true (exclusive) limits.
2Find the class width of every class. If they are all equal, the height is the frequency. If they differ, compute frequency density for each class.
3Draw the axes. Put classes on the x-axis (use a kink if the scale starts far from zero) and frequency or frequency density on the y-axis.
4Draw adjacent rectangles with no gaps, with width equal to class width and height from step 2.
5For a polygon, find the mid-value of each class and mark the point at the top of each rectangle at its mid-value.
6Add one empty class at each end with zero frequency and join the end points to the x-axis. Join all points with straight lines.
7For a frequency curve, smooth the polygon freehand without a sharp corner.
8In a MCQ, check the answer against the definitions: no gaps means histogram, mid-points joined means polygon, area proportional to frequency.

Quickest way: Quick check for MCQs on histogram and polygon

When to use it: Use this when the question is a theory or identification MCQ, or asks for a value you can read without drawing the full graph.

Look for the key word: adjacent rectangles means histogram; mid-points joined means polygon; smooth line means frequency curve.
If the question gives unequal classes, divide each frequency by its class width before comparing heights.
To count points of a polygon, number of classes plus 2 (the two empty end classes).
Find the mid-value of the first class by adding its two limits and halving. If the widths are equal, add the class width to the previous mid-value to get each subsequent mid-value.
Eliminate any option that has gaps between rectangles or that uses frequency for height with unequal classes.

Common mistakes in Graphical Representation: Histogram and Frequency Polygon

Leaving gaps between rectangles when the classes are inclusive (10-19, 20-29).
Students draw directly from the table limits and do not notice the gap of 1.
Fix: Subtract d ÷ 2 from each lower limit and add d ÷ 2 to each upper limit first, so classes touch. The adjustment is 0.5 when the gap d is 1.
Using frequency as height with unequal class widths.
Equal-width examples are more common, so students apply the same habit.
Fix: Compute frequency density = frequency ÷ class width and use it as the height.
Joining the polygon points at the class limits instead of the mid-values.
Students confuse the polygon with an ogive, which uses limits.
Fix: Always use mid-values for a polygon. Ogives use upper (or lower) limits.
Not closing the polygon at the ends.
Students stop at the first and last plotted points.
Fix: Add an empty class before the first and after the last, and bring the line down to the x-axis.
Treating a histogram and a bar diagram as the same.
Both use rectangles.
Fix: Remember: bars have gaps and only height matters; histogram rectangles touch and area matters, so width counts.

Worked examples

Example 1

Draw a histogram for the data below and state the coordinates of the points of the frequency polygon. Marks of 50 students: 10-20: 4, 20-30: 8, 30-40: 15, 40-50: 13, 50-60: 10.

Show the solution

The classes are exclusive and each has width 10, so the height of each rectangle equals the frequency.
Rectangles: 10-20 height 4, 20-30 height 8, 30-40 height 15, 40-50 height 13, 50-60 height 10, all touching.
Mid-values: 15, 25, 35, 45, 55.
Add empty classes 0-10 (mid-value 5) and 60-70 (mid-value 65) with frequency 0.
Polygon points: (5, 0), (15, 4), (25, 8), (35, 15), (45, 13), (55, 10), (65, 0).

Answer: Polygon points are (5, 0), (15, 4), (25, 8), (35, 15), (45, 13), (55, 10), (65, 0); total of 7 points.

Example 2

Find the heights of the rectangles of a histogram for the data: 0-10: 5, 10-20: 12, 20-40: 18, 40-80: 20.

Show the solution

The widths are 10, 10, 20 and 40, which are unequal, so use frequency density.
0-10: 5 ÷ 10 = 0.5.
10-20: 12 ÷ 10 = 1.2.
20-40: 18 ÷ 20 = 0.9.
40-80: 20 ÷ 40 = 0.5.

Answer: Heights (frequency densities) are 0.5, 1.2, 0.9 and 0.5.

Exam tips

Questions are often definition based: know the one-line difference between a histogram, polygon, frequency curve and bar diagram.
With unequal classes, the answer is nearly always frequency density. Divide first, then compare.
If a question gives inclusive classes, expect a conversion to true limits as the catch.
A polygon uses mid-values; an ogive uses limits. Keep these two apart.
Histogram can also be used to find the mode graphically, so read the tallest rectangle carefully.

Practice questions from Statistical Representation of Data

Graphical Representation: Histogram and Frequency Polygon in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Graphical Representation: Histogram and Frequency Polygon: frequently asked questions

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

A histogram has touching rectangles for a continuous distribution, and the area shows frequency, so width matters. A bar diagram has separate bars with gaps, used for categories or discrete data, and only the height matters.

How do you draw a histogram with unequal class intervals?

Divide each class frequency by its class width to get frequency density. Draw rectangles with the class width as the base and the frequency density as the height.

How is a frequency polygon drawn from a histogram?

Mark the mid-point of the top of each rectangle and join them with straight lines. Close the figure by joining the ends to the x-axis at the mid-points of an empty class on each side.

What is a frequency curve?

It is a smooth curve drawn through the points of a frequency polygon. It shows the overall shape of the distribution, such as symmetric or skewed.