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D1.6 · Interpret and critique histograms and misleading graphs

Learn to interpret and critique histograms and misleading graphs through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Data

Ontario Grade 6 Mathematics — D1.6

Graphs help us see information quickly, but a graph can also give the wrong impression if it is unclear or poorly designed. In this lesson, you will read histograms and check graphs carefully. All example data are hypothetical. They are made for learning and are not reports of real observations.

What you will learn

Read a histogram and describe how values are grouped.
Compare frequencies and identify the interval with the most values.
Check a graph’s title, labels, scale, and bars.
Explain how a graph’s design might give a misleading impression.

Start with groups of numbers

Imagine sorting hypothetical numbers of pages read into groups. One group could be from 0 to 4 pages. The next could be from 5 to 9 pages. These groups are called intervals. An interval is a range of numbers grouped together. Here, the intervals are consecutive: one ends just before the next begins. They do not overlap.

A frequency is the number of values in a group. If 6 values fall in the interval from 0 to 4, that interval has a frequency of 6. Frequency is a count.

A histogram shows numerical data grouped into intervals. The interval labels run along the bottom. The frequencies run up the side. In a histogram, the bars touch because the intervals follow one another without gaps. The height of each bar shows the frequency for its interval.

Read the interval labels along the bottom.
Read the frequency scale on the side.
The tallest bar shows the interval with the greatest frequency.

Read the whole picture

Before comparing bars, check the graph’s title and labels. They tell you what the data are about and what each axis shows. An axis is a line on a graph that carries labels or numbers.

Next, check the scale. A scale is the number marks on an axis and the steps between them. Look to see whether the steps are equal. For example, marks at 0, 2, 4, and 6 increase by equal steps of 2.

Then compare the bar heights. Find which interval has the most values and which has the fewest. Describe any pattern you can see, and use the labels or frequencies as evidence.

A histogram groups values, so it may not show the exact values of each item. A bar for the interval from 10 to 14 tells you how many values are in that interval. It does not tell you how many values were exactly 12.

Check the title and axis labels before interpreting the bars.
Use the scale marks to read frequencies.
Describe what the grouped data show without guessing exact values inside an interval.

Check whether a graph is fair

A misleading graph is a graph whose design may make the data seem different from what the numbers show. A graph might mislead by mistake, or a design choice might give one comparison too much attention.

Check that the title matches the data and that the labels explain what the numbers mean. Look closely at the scale. Do its steps stay equal? Does the axis begin at a value that could make small differences look very large? In a histogram, are the intervals in order and the bars touching?

Compare the visual impression with the numbers shown. If two counts are close but their bars look very different, check the scale before deciding what the graph means. Explain which feature could mislead a reader and why. If information is missing or unclear, say what the reader needs to know.

Check the title, labels, scale, and intervals.
Compare the graph’s appearance with the values it shows.
Name the feature that may mislead and explain its effect.

Practise explaining your thinking

A clear response answers three questions: What does the graph show? What evidence supports that reading? Is anything unclear or unfair? Point to a bar, label, or scale mark as evidence.

For example: “The interval from 10 to 14 has the most values because its bar reaches 8. The graph labels the interval and frequency axes. I would also check that the frequency scale uses equal steps.” This response gives a claim, evidence, and a check.

Try this practice prompt: A histogram has a tallest bar for the interval from 15 to 19, with a frequency of 7. State what the graph shows. Then name one graph feature you would check before trusting the display. A good response says that this interval has the most values and checks the title, labels, scale, or bar arrangement.

Make a claim about what the graph shows.
Support the claim with a label, count, or scale mark.
Mention a possible problem when you see one.

Worked example

Find the interval with the greatest frequency

This hypothetical histogram groups the number of pages read. Which interval has the greatest frequency? How many values are in that interval?

Hypothetical pages read

The touching bars show counts for consecutive, non-overlapping intervals.

Compare the bars
Each bar’s height gives the number of values in its interval. Compare the four heights.
Read the highest bar
The bar for the interval from 10 to 14 reaches 8. That is higher than the other bars, so this interval has the greatest frequency.

Answer: The interval from 10 to 14 has the greatest frequency. It contains 8 values.

Check: The answer identifies the tallest bar and reads its count from the frequency scale.

Worked example

Know what grouped data cannot tell you

Use the same hypothetical histogram. Can you tell exactly how many values were equal to 12? Explain your answer.

Counts for grouped values

The interval from 10 to 14 contains 8 values in all; the histogram does not separate the exact values.

Find the interval
The value 12 belongs in the interval from 10 to 14. The bar shows 8 values in that interval altogether.
State what is not shown
The histogram does not split the interval’s count into separate exact values. It does not tell us how many values were exactly 12.

Answer: No. The histogram shows 8 values in the interval from 10 to 14, but it does not show how many were exactly 12.

Check: The count of 8 applies to the whole interval, not to one exact value inside it.

Worked example

Spot a scale that could mislead

A hypothetical graph compares two groups. It shows counts of 20 and 22, but its vertical axis begins at 19 and ends at 22. The bars look very different in height. What could mislead a reader?

Hypothetical group counts

The count axis is shortened from 19 to 22, which can make the difference appear unusually large.

Compare the counts
The difference between the counts is 2. The graph should show the values clearly so a reader can judge that difference.
$22-20=2$
Check the axis
The axis begins at 19, so the bars show only the part of each count above 19. This makes the difference take up much of the displayed scale and can make it look unusually large.
Suggest a clearer display
A clearer graph could show the count scale starting at 0, with clear labels and equal steps. Then readers could compare the counts more fairly.

Answer: The axis beginning at 19 can make the difference of 2 look much larger than it is. Readers should check the scale and compare the actual counts.

Check: The displayed counts differ by 2, but the shortened scale can exaggerate how large the difference looks.

Common mistakes and how to avoid them

Saying the tallest bar tells the exact value that appeared most often.

Correction: A histogram bar shows how many values fall in an interval. It does not show the exact values inside that interval.

Comparing bar heights without checking the vertical scale.

Correction: Read the scale marks and check where the axis begins. The scale affects how large a difference looks.

Thinking histogram bars should have gaps.

Correction: Histogram intervals follow one another, so the bars touch.

Calling a graph misleading without giving a reason.

Correction: Point to a title, label, scale, or bar that could give the wrong impression. Explain why.

Lesson summary

A histogram groups numerical values into intervals and shows the frequency in each interval.
Use the title, labels, scale, and bar heights to interpret a histogram.
A grouped interval does not reveal every exact value inside it.
To critique a graph, compare its visual message with its labels, scale, and values.

Check your understanding

Question 1

A histogram has bars with frequencies 2, 7, and 5. Which interval has the greatest frequency?

The interval with frequency 2
The interval with frequency 7
The interval with frequency 5
The interval with frequency 0

Show answer and explanation

The interval with frequency 7

The greatest frequency is 7, so the interval with that bar has the most values.

Question 2

A histogram shows 9 values in the interval from 20 to 24. What can you conclude?

Exactly 9 values were equal to 22.
There were 9 values from 20 to 24 together.
There were 24 values in all.
There were 9 intervals.

Show answer and explanation

There were 9 values from 20 to 24 together.

The bar gives the count for the whole interval, not the count for one exact value.

Question 3

A graph’s count axis starts at 49. Its bars show counts of 50 and 52. What should you check?

Whether the axis start makes the difference look larger than it is
Whether the interval labels are in alphabetical order
Whether the bars show exact values inside each interval
Whether both counts are the same

Show answer and explanation

Whether the axis start makes the difference look larger than it is

An axis beginning at 49 may make the difference between 50 and 52 look unusually large. Compare the scale with the counts.

Key terms

Histogram: A graph that groups numerical data into intervals and shows how many values are in each interval.
Interval: A range of numbers grouped together.
Frequency: The number of values in a group or interval.
Scale: The number marks on a graph axis and the steps between them.
Misleading graph: A graph whose design may make the data seem different from what the numbers show.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation D1.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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