Grade by Grade
D. Data · Data Literacy and Probability

Histograms

D1.3 D1.6About 50 minutes
Learning goals
  • I can explain how a histogram is different from a bar graph.
  • I can make a histogram from a frequency table with equal intervals.
  • I can read a histogram and answer questions about the data.
  • I can describe what the shape of a histogram tells me.

Words to know

histogram
A graph that shows how many data values fall in each interval. The bars touch because the intervals are next to each other on a number line.
bar graph
A graph that compares categories. The bars have spaces between them.
frequency
How many data values are in a category or interval.
scale
The numbers along an axis, going up by equal steps.

Remember from Grade 5

You read and made bar graphs and stacked-bar graphs with titles, labels, scales and sources. Each bar was a category, like sliding or tag, with spaces between the bars. A histogram looks similar, but it shows numbers grouped in intervals.

What is a histogram?

In the last lesson you grouped long-jump distances into intervals. A histogram is the graph of a table like that.

  • The horizontal axis is a number line (here, distance in cm), not a list of categories.
  • Each bar covers one interval. The bar for 140 to less than 150 sits between the 140 and 150 marks.
  • The height of each bar is the frequency: how many values are in that interval.
  • The bars touch. There are no gaps, because the intervals have no gaps. A jump could be 139.9 cm or 140 cm.
  • All the bars are the same width, because the intervals are the same width.
Long Jumps of 20 Grade 6 Students02468Number of students15752120130140150160170Distance (cm)Source: Room 12 gym class
Each bar covers one 10 cm interval. Frequencies from left to right: 1, 5, 7, 5, 2. The numbers under the bars are the edges of the intervals.

Histogram or bar graph?

  • A bar graph compares categories (qualitative data, or a few separate values). The bars have spaces, and you could put them in any order.
  • A histogram shows quantitative data in intervals, usually continuous data or discrete data with many values. The bars touch, and they must stay in number order.

Why does order matter? Because the horizontal axis is a number line. Moving the 160 bar to the front would be like moving 160 before 120 on a ruler!

Worked example

Example 1: Make a histogram

Make a histogram of the pages-read data from the last lesson:

10–19: 3 students; 20–29: 4; 30–39: 5; 40–49: 4.

  1. Title: "Pages Read in One Week".
  2. Horizontal axis: label it "Pages read". Mark the interval edges 10, 20, 30, 40, 50, equally spaced.
  3. Vertical axis: label it "Number of students". The largest frequency is 5, so a scale from 0 to 6 by 1s fits.
  4. Draw each bar from one edge to the next, as high as its frequency. The 30–39 bar goes from the 30 mark to the 40 mark and is 5 high.
  5. Make sure the bars touch and are all the same width.

See the histogram below.

Pages Read in One Week0123456Number of students34541020304050Pages read
Pages read by 16 students. The bar from 10 to 20 shows 10 to 19 pages. Frequencies: 3, 4, 5, 4.

Reading the shape

The shape of a histogram tells a story.

  • In the long-jump histogram, the bars are tallest in the middle and shorter at both ends. Most students jumped between 130 and 160 cm, and only a few jumped very short or very long distances.
  • A histogram can also be lopsided, with most of the data bunched at one side.
  • A short bar far away from the others shows a few unusual values.

One thing a histogram can't tell you: the exact values. You know 2 students jumped from 160 to less than 170 cm, but not exactly how far.

Worked example

Example 2: Ask and answer questions

Use the long-jump histogram. How many students jumped at least 140 cm? What fraction of the class is that?

  1. "At least 140" means 140 or more. Find the bars that start at 140 or higher: 140–150, 150–160 and 160–170.
  2. Add their heights: 7 + 5 + 2 = 14.
  3. There are 20 students in all, so the fraction is 14/20.
  4. Simplify: 14/20 = 7/10 = 0.7.

14 students, which is 7/10 of the class.

Ask good questions

When you meet a histogram, ask questions at three levels:

  • Read the data: How many students are in the 140 to 150 bar?
  • Read between the data: How many jumped less than 140?
  • Read beyond the data: If a new student joins, which interval will their jump probably be in?
Just for fun

Why are histogram bars such good friends?

They always stick together. No gaps allowed!

Hands-on activity

Graph your leaves

You need: the frequency table you made in the last lesson (or measure 20 objects now), grid paper, a ruler and a pencil crayon.

  1. On grid paper, draw the horizontal axis. Mark your interval edges, using the same number of squares for each interval.
  2. Draw the vertical axis. Choose a scale that fits your largest frequency.
  3. Draw a bar for each interval. Make the bars touch.
  4. Add a title, axis labels, and a source ("Leaves from our backyard, October").
  5. Describe the shape: Where is the tallest bar? Is it in the middle or at one side?
Time Spent Outside on a Snowy Saturday0246810Number of students2597430306090120150180Time (minutes)Source: survey of 30 Grade 6 students
Use this histogram for the Your turn questions. Frequencies from left to right: 2, 5, 9, 7, 4, 3. The scale counts by 2s.

Your turn

1

How many students were surveyed?

2

Which interval has the most students?

3

How many students spent 2 hours (120 minutes) or more outside?

4

How many students spent less than 1 hour outside?

5

What fraction of the students spent 60 to less than 90 minutes outside?

6

Can you tell from the histogram exactly how long the student who stayed outside the longest was outside? Explain.

7

Why do the bars in this graph touch?

More practice: warm-up

1

Do the bars of a histogram touch?

2

Long-jump histogram: how many students jumped from 130 to less than 140 cm?

3

Pages histogram: which interval is the tallest bar?

4

What does the height of a histogram bar show?

5

Would you use a histogram or a bar graph for favourite season?

More practice: on your own

1

Long-jump histogram: how many students jumped less than 150 cm? What fraction is that?

2

Pages histogram: how many students read at least 30 pages?

3

Describe the shape of the time-outside histogram.

4

Arjun drew a histogram with the bars in this order: 40–49, 10–19, 30–39, 20–29. What did he do wrong?

5

A histogram has bars of heights 4, 8, 6 and 2. The scale on the vertical axis goes 0 to 10. Is that a good scale? Would 0 to 100 be good?

Challenge

1

Time-outside histogram: in which interval is the median time? (Hint: line up all 30 students in order. Where are the 15th and 16th students?)

2

A new student jumps 171 cm. What must you change in the long-jump histogram to include them?

3

Make up a set of 10 values whose histogram (intervals 0–9, 10–19, 20–29, 30–39) has bars of heights 1, 2, 3, 4.

Check your understanding

Quiz

1.

How is a histogram different from a bar graph?

2.

Which data is best shown in a histogram?

3.

Long-jump histogram: how many students jumped from 150 to less than 160 cm?

4.

Time-outside histogram: how many students were outside 90 minutes or more?

5.

In a histogram, where does a value of exactly 30 go when the bars are 20 to less than 30 and 30 to less than 40?

6.

What can't you learn from a histogram?

For parents and teachers

What this lesson teaches: making and reading histograms (new in Grade 6), and how they differ from bar graphs (D1.3). Your child also analyses the data by asking and answering questions and describing the shape of the graph (D1.6).

How to help at home:

  • Compare a histogram to a ruler: the bottom axis is a number line, so the bars must be in order and must touch.
  • Ask "read between" questions: "How many are less than 60 minutes?" means adding several bars.
  • Remind your child that the numbers under a histogram are the edges of intervals, not labels for single values.

Common difficulties:

  • Leaving gaps between bars, or putting bars out of order.
  • Placing labels in the middle of bars and then misreading which interval a bar covers.
  • Forgetting that "at least 140" includes the 140 interval.
  • Thinking a histogram shows exact values.