AFCAT Numerical Ability · Statistics

Histograms, Frequency Polygons and Ogives

Know how a histogram, a frequency polygon, a frequency curve and an ogive are drawn from a frequency table, and what each one lets you read off.

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Why this matters

Most questions here are statements to judge true or false, so each mark goes to one exact fact about a graph. The usual losses are the height of a histogram bar, which follows the frequency only when the classes are equally wide, and the limit a less than ogive is plotted against.

Concept 1 of 2: The graphs of a frequency distribution

All of these graphs are drawn from the same frequency table, but each shows something different: a histogram shows the shape of the data, a frequency polygon shows the same shape as a line, and an ogive shows the running total.

Definition

A histogram's rectangles stand side by side with no gaps, because the classes are continuous: each one ends where the next begins. The area of each rectangle is proportional to the frequency of its class, as if the class's values were spread evenly across it. The height is proportional to the frequency only when every class has the same width.

Graph	How it is drawn	What you read from it
Histogram	Touching rectangles on the class intervals, each with area proportional to its frequency	The shape of the data, and the modal class (the tallest bar, when the widths are equal)
Frequency polygon	Drawn after the histogram: join the mid-points of the tops of its rectangles with straight lines	The same shape as a line, useful for comparing two distributions on one graph
Frequency curve	A smooth curve through the same mid-points	The overall pattern, such as whether the data are symmetric
Bar chart	Separate bars with gaps, one for each category or value	A comparison between separate categories
Ogive	Cumulative frequencies plotted at the class boundaries and joined	The median, read at half the total (the next concept)

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Histograms, Polygons and Ogives

Frequency polygon can be drawn after drawing

Area, not height, follows the frequency

When one class is twice as wide as the others, a rectangle of the same height stands for twice the frequency. Halve its height to keep the area right.

A histogram is not a bar chart

A bar chart's bars have gaps and stand for separate categories. A histogram's bars touch because its classes run on from one another.

Drill 3 more on the graphs of a frequency distribution

Concept 2 of 2: The ogive and reading the median

An ogive plots the running total of the frequencies. Go up the vertical axis to half the total frequency, across to the curve, and down: the value you land on is the median.

Definition

Less than ogive: the cumulative frequency up to each class is plotted against the class's upper boundary, because it counts the items below that point.
More than ogive: the count from each class upwards is plotted against the class's lower boundary.
The median is read at $\tfrac N2$ on the vertical axis, and the two ogives cross at the median.
A straight-line ogive means the running total grows by the same amount in every class: each of the $k$ classes has frequency $\tfrac Nk$ , and no class stands out as the modal class, so there is no mode.

Reading the median

\text{cumulative frequency at } M = \dfrac N2

Worked example

The classes 0-10, 10-20, 20-30 and 30-40 have frequencies 4, 10, 16 and 10. List the points of the less than ogive and read the median from it.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Statistics · Histograms, Polygons and Ogives

The cumulative frequency curve of a frequency distribution with 6 classes and total frequency 60 is a straight line. Consider the following statements I. The first and the last classes have a frequency of 10 each. II. Both the middle classes have a total frequency of 30. III. The frequency distribution does not have a mode. Which of the statements given above are correct?

Less than goes at the upper boundary

The number of items below 20 is plotted at 20, the top of the class 10-20, not at its start or its mid-point.

Half the total frequency, not half the axis

Read the median where the curve reaches

\tfrac N2

on the vertical axis. The middle of the horizontal axis is not the median unless the data are symmetric.

Drill 2 more on the ogive and reading the median

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The ogive and reading the median
Reading the median
$\text{cumulative frequency at } M = \dfrac N2$

Reference tables (1)

The graphs of a frequency distribution5 rows

Graph	How it is drawn	What you read from it
Histogram	Touching rectangles on the class intervals, each with area proportional to its frequency	The shape of the data, and the modal class (the tallest bar, when the widths are equal)
Frequency polygon	Drawn after the histogram: join the mid-points of the tops of its rectangles with straight lines	The same shape as a line, useful for comparing two distributions on one graph
Frequency curve	A smooth curve through the same mid-points	The overall pattern, such as whether the data are symmetric
Bar chart	Separate bars with gaps, one for each category or value	A comparison between separate categories
Ogive	Cumulative frequencies plotted at the class boundaries and joined	The median, read at half the total (the next concept)

Watch out for (4)

Area, not height, follows the frequency
→ The graphs of a frequency distribution
A histogram is not a bar chart
→ The graphs of a frequency distribution
Less than goes at the upper boundary
→ The ogive and reading the median
Half the total frequency, not half the axis
→ The ogive and reading the median

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