Grade by Grade
D. Data · Data Literacy and Probability

Collecting Data and Organizing It in Intervals

D1.2 D1.1 A1About 50 minutes
Learning goals
  • I can plan how to collect qualitative, discrete or continuous data to answer a question about a population.
  • I can organize data in a frequency table.
  • I can group data into equal intervals with no gaps and no overlaps.
  • I can choose a good interval size for a data set.

Words to know

population
The whole group you want to learn about.
sample
The part of the population you actually collect data from.
frequency table
A table that shows how many times each value, category or interval appears.
interval
A range of values grouped together, like 130 to less than 140. All intervals in a table should be the same width.
interval width
The size of each interval. From 130 to less than 140 is a width of 10.

Remember from Grade 5

You chose a fair sample from a population and made relative-frequency tables, where each count is divided by the total. Today you will collect three kinds of data and learn a new way to organize numbers when there are lots of different values.

Plan before you collect

Good data starts with a good plan. Ask yourself:

  1. What is my question? Make it clear. "How far can Grade 6 students jump?"
  2. Who is my population? All Grade 6 students at my school.
  3. Who will be in my sample? A fair (random) group, or everyone if the group is small.
  4. What kind of data will I get? Qualitative (words), discrete (counted) or continuous (measured)?
  5. How will I collect it? A survey, an experiment, measuring, or a secondary source (data someone else collected)?
  6. How will I record it? A tally chart, a list, a spreadsheet.

Organizing different kinds of data

Qualitative data goes in a frequency table with one row for each category.

Discrete data with only a few values (like the number of pets: 0, 1, 2, 3) also gets one row for each value.

But what about continuous data, or discrete data with lots of different values? Here are the standing long-jump distances, in centimetres, of 20 Grade 6 students:

132, 145, 151, 128, 160, 139, 147, 155, 142, 136, 149, 158, 141, 134, 150, 163, 146, 138, 152, 144

Every value is different! A row for each value would be a very long table that tells us very little. Instead, we group the data into intervals.

20 long jumps (cm), sorted into intervals of 10 cm1 jump5 jumps7 jumps5 jumps2 jumps120130140150160170Each dot is one student. A dot on a dashed line counts in the interval to its right.
The 20 jumps placed on a number line. The dashed lines split it into intervals 10 cm wide, and the counts 1, 5, 7, 5, 2 become the frequency table.
Distance (cm)Frequency
120 to less than 1301
130 to less than 1405
140 to less than 1507
150 to less than 1605
160 to less than 1702
Total20
Long-jump distances of 20 Grade 6 students, grouped in intervals of width 10 cm.

Rules for good intervals

  • Same width. Every interval here is 10 cm wide.
  • No overlaps. Each value fits in exactly one interval. A jump of exactly 140 cm goes in "140 to less than 150", not in "130 to less than 140".
  • No gaps. Every possible value has a home, even 139.6 cm.
  • Cover all the data. Start at or below the smallest value (128) and end above the largest (163).
  • About 4 to 8 intervals is usually a good number. Too few hides the pattern. Too many gives a long, spread-out table.

For discrete data, you can write intervals like 10–19, 20–29, because no value can fall between 19 and 20. For continuous data, write "10 to less than 20" so values like 19.7 have a home.

Where does a jump of exactly 140 cm go?130140150160130 to less than 140140 to less than 150150 to less than 160140 lives herefilled dot: includedopen dot: not included
A filled dot means the value is included; an open dot means it is not. So 140 goes in "140 to less than 150", not in "130 to less than 140". No overlaps, no gaps.
Worked example

Example 1: Group discrete data

Sixteen students recorded how many pages they read during one week:

12, 35, 48, 22, 30, 41, 18, 27, 36, 44, 25, 33, 15, 29, 38, 47

Organize the data in a frequency table with intervals.

  1. The smallest value is 12 and the largest is 48. The values go from 12 to 48, a spread of 36 pages.
  2. Try an interval width of 10: 10–19, 20–29, 30–39, 40–49. That is 4 intervals, and it covers 12 to 48.
  3. Go through the list once, making a tally mark for each value. Cross off each value as you go so you don't miss one.
  4. 10–19: 12, 18, 15 → 3. 20–29: 22, 27, 25, 29 → 4. 30–39: 35, 30, 36, 33, 38 → 5. 40–49: 48, 41, 44, 47 → 4.
  5. Check: 3 + 4 + 5 + 4 = 16. That matches the number of students.

Pages read: 10–19: 3; 20–29: 4; 30–39: 5; 40–49: 4. Total 16.

Worked example

Example 2: Plan a data collection

Ravi's question is: "How long can Grade 6 students at my school hold a plank?" Plan his data collection.

  1. Population: all Grade 6 students at his school (say 90 students).
  2. Sample: choose 30 students at random, for example by drawing names from a container.
  3. Kind of data: time, measured with a stopwatch, so it is continuous.
  4. Recording: write each time to the nearest second in a list.
  5. Organizing: the times will all be different, so group them in equal intervals, like 0 to less than 30 s, 30 to less than 60 s, and so on.

Collect continuous data from a random sample of 30, then organize it in a frequency table with equal intervals.

Plan, check, persevere

Grouping data takes patience. Go through the list once, in order, and cross off each value as you tally it. Then check that your total matches the number of values. If it doesn't, that isn't a failure. It's a clue! Recount one interval at a time.

Just for fun

What did 139.9 cm say to 140 cm?

"We're so close... but we live in different intervals!"

Hands-on activity

Leaf (or crayon) lengths

You need: a ruler, paper, and 20 leaves from outside (if it's winter, use 20 crayons, pencils, spoons or socks instead).

  1. Write your question: "How long are the leaves under our maple tree?" (or "How long are the crayons in our box?").
  2. Measure each object to the nearest millimetre and write it in centimetres (for example 7.4 cm). What kind of data is this? (Continuous.)
  3. Find the smallest and largest values.
  4. Choose an interval width so you get 4 to 7 intervals. Write intervals like "5 to less than 6 cm".
  5. Make a tally and frequency table. Check that your total is 20.
  6. Write one sentence about what your table shows.

Keep your table. You will graph it in the next lesson!

Your turn

1

Look at the long-jump table. Which interval has the most jumps?

2

How many students jumped 150 cm or more?

3

In which interval does a jump of exactly 150 cm go?

4

What fraction of the 20 students jumped less than 140 cm?

5

Why do these intervals start at 120 and not at 0?

6

Ravi records each student's favourite winter sport. Should he use intervals? Why or why not?

7

A student wants to group the page data as 10–20, 20–30, 30–40, 40–50. What is the problem?

More practice: warm-up

1

What is an interval?

2

Is the interval 30–39 the same width as 40–49?

3

The data is 4, 9, 11, 17, 18, 22. How many values are in the interval 10–19?

4

Is how far you can throw a ball qualitative, discrete or continuous?

5

A frequency table has frequencies 3, 6, 8 and 3. How many data values are there?

More practice: on your own

1

Group these ages of people at a skating party into intervals 0–9, 10–19, 20–29, 30–39, 40–49: 8, 12, 35, 11, 41, 9, 13, 38, 6, 12, 44, 10.

2

In the skating-party table, the 20–29 interval has a frequency of 0. Should you leave it out of the table? Explain.

3

Some times (in seconds) are 42.3, 57.9, 61.0 and 75.5. Choose intervals of width 10 that would work for continuous data.

4

Write a clear survey question to collect discrete data about screen time, and one to collect continuous data.

5

Why should all intervals be the same width?

Challenge

1

Masses of 12 pumpkins (kg): 3.2, 5.8, 4.1, 7.6, 2.9, 6.3, 4.8, 5.5, 3.7, 6.9, 4.4, 5.1. Make a frequency table with intervals of width 1 kg, starting at 2.

2

Use the pumpkin table from the last question. Explain why intervals of width 5 kg (0 to less than 5, 5 to less than 10) would be a poor choice.

3

A school has 120 Grade 6 students: 60 in French Immersion and 60 in English. Mia wants to know how long Grade 6 students take to get to school. Describe a good plan: question, population, sample, kind of data and how she will organize it.

Check your understanding

Quiz

1.

Which intervals follow the rules for discrete data?

2.

Which data set is best organized with intervals?

3.

Data: 23, 27, 31, 35, 38, 41. How many values are in 30–39?

4.

A jump of exactly 140 cm goes in which interval?

5.

A frequency table has frequencies 4, 9, 7 and 5, but there are 26 values. What should you do?

6.

Why do we group continuous data into intervals?

For parents and teachers

What this lesson teaches: planning how to collect qualitative, discrete and continuous data to answer a question about a population, and organizing data in frequency tables, including equal intervals (D1.2, D1.1). Checking totals and recounting practises perseverance (A1).

How to help at home:

  • When your child tallies, suggest crossing off each value in the list as it is counted.
  • Ask: "Where would 140 go? Where would 139.5 go?" to check there are no overlaps or gaps.
  • Use real measurements (heights of family members, lengths of leaves or socks) to make the data meaningful.

Common difficulties:

  • Writing overlapping intervals (10–20, 20–30).
  • Unequal interval widths.
  • Leaving out intervals with a frequency of 0. They should stay in the table.
  • Using intervals for qualitative data.