Year 7 · Unit 6 Collecting and presenting data · Lesson 3 of 7

From a heap of numbers to a table

A list of 24 raw numbers hides its own pattern. Tally them in fives, read the frequencies, then group measurements into classes and watch the shape change with the class width.

Statistics and ProbabilityNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Unit 1, lesson 7) Round 17.8 to the nearest 10.
    Answer20. The ones digit is 7, which rounds the tens up.
  2. (Unit 2, lesson 1) What is 25% as a fraction in simplest form?
    Answer1/4. 25 out of 100 divides by 25 on the top and the bottom.
  3. (Lesson 1 of this unit) Heights are measured. Is height discrete or continuous?
    AnswerContinuous. A measurement can take any value in a range.
Words used in this lesson
tally
a stroke for each value, in groups of five. For example: four strokes and a diagonal across them.
frequency
how many times a value appears. For example: 2 books has frequency 8.
frequency table
a table of each value with its frequency. For example: books borrowed and frequency.
total frequency
the sum of all the frequencies, the number of values. For example: 3 + 5 + 4 = 12.
class
a range of values, used to group measurements. For example: 10 to under 20, written 10 ≤ h < 20.
class width
the size of each class. For example: 10 to under 20 has width 10.

A heap of numbers

A librarian has a slip for each of 24 students with one number on it: how many books that student borrowed in a week. 2, 0, 1, 3, 2, 1, 4… The pattern is in there but you cannot see it in a heap. Predict first: which number of books do you think will turn out to be the most common? Then tally the values one at a time, and watch the pattern appear.

12
12values tallied
2most frequent so far

12 values tallied. The value 2 has come up 4 times, more than any other.

Show me the steps
  1. After 5 values: 2 has come up twice, and 0, 1 and 3 once each. No value has a full group of five yet.
  2. After 12 values: 0 twice, 1 three times, 2 four times, 3 twice and 4 once. They add to 12.
  3. After 24 values: 0 four times, 1 six times, 2 eight times, 3 four times and 4 twice. The 2s have a full group of five and a group of three: 8.

A tally counts as you go, in groups of five (four strokes and a diagonal across them), so the count can be read quickly. The frequency of a value is how many times it appears. A frequency table lists each value with its frequency.

Why it works: the frequencies of all the values add up to the number of values in the list. If they do not, something was missed or counted twice, so the total is a check.

A common mistake: “The frequency is the value.” In the row for 2 books, the 2 is the value and the 8 is the frequency: 8 students borrowed 2 books. The books borrowed in total would be 2 × 8 = 16 from that row, which is a different number again.

When every value is different

Heights, times and masses are measured, so almost every value is different. A table with one row per value would be as long as the list. Instead, the values are put into classes, each a range of equal width, and the table counts how many values fall in each class. This is the fix for lesson 2’s “0 to 2, 2 to 4” boxes. Change the class width.

2 cm
8classes
4largest frequency

Classes 2 cm wide: 8 classes. The frequencies add to 20, and the largest is 4. These bars are part of the table; lesson 4 draws them as a histogram.

Show me the steps
  1. Classes 2 cm wide: 8 classes with frequencies 0, 2, 3, 4, 4, 4, 2 and 1. They add to 20.
  2. Classes 4 cm wide: 4 classes with frequencies 2, 7, 8 and 3. They add to 20.
  3. Classes 8 cm wide: 2 classes with frequencies 9 and 11. They add to 20.
  4. A height of exactly 12.0 cm is in “12 to under 14” (width 2), “12 to under 16” (width 4) and “12 to under 20” (width 8): always the class that starts at 12.

In this unit the classes have the same width, with no gaps and no overlaps, so the frequencies can be compared directly. Write them as “10 to under 20”, “20 to under 30”, so a value of exactly 20 has one place to go: the class that starts at 20. In symbols, “10 to under 20” is written 10 ≤ h < 20.

Grouping shows the shape but loses the exact values: a plant in “10 to under 12” could be 10.0 or 11.9, and the table does not say which.

Worked example: a list to a frequency table

Goals scored by a team in 15 matches: 2, 0, 1, 1, 3, 0, 2, 1, 2, 2, 0, 1, 4, 1, 2. Make a frequency table.

  1. Work through the list once and tally each value in its row: 0 has 3 strokes, 1 has 5 (one full group), 2 has 5, 3 has 1 and 4 has 1.
  2. Write the frequencies: 3, 5, 5, 1 and 1.
  3. Check the total: 3 + 5 + 5 + 1 + 1 = 15, the number of matches. 0: 3, 1: 5, 2: 5, 3: 1, 4: 1.

Your go. The first step is given. The times, in minutes, of 10 bus journeys: 12, 25, 18, 31, 22, 15, 28, 20, 34, 19. Group them into classes “10 to under 20”, “20 to under 30” and “30 to under 40”.

  1. The class “10 to under 20” holds 12, 18, 15 and 19: frequency 4.
  2. Now find the other two classes yourself and check the total, then check.
Check the answer

“20 to under 30” holds 25, 22, 28 and 20: 4. “30 to under 40” holds 31 and 34: 2. The total 4 + 4 + 2 = 10 matches the 10 journeys.

Check before you go on

1. In a frequency table of “pets owned”, the row for 2 pets has a frequency of 7. What does the 7 mean?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A tally for one value shows two full groups of five strokes, then three more single strokes. What is the frequency?

3. A table shows the stickers each student has: 0 stickers has frequency 3, 1 sticker has frequency 5, and 2 stickers has frequency 4. How many students are in the table?

Level 2 · Building

4. Parcel masses are grouped in classes (kg): 0 to under 10, 10 to under 20, 20 to under 30 and 30 to under 40. A parcel has a mass of exactly 20 kg. Which class does it go in?

5. Siblings per student: 0 siblings has frequency 5, 1 has 9, 2 has 7, 3 has 3 and 4 has 1. What percentage of the students have 2 or more siblings?

6. 40 values are grouped into four classes. The first three classes have frequencies 7, 12 and 15. What is the frequency of the fourth class?

Level 3 · Stretch

7. Heights are grouped in classes 5 cm wide starting at 140: 140 to under 145, 145 to under 150, and so on. How many classes are needed, counting from the first, to include a height of 163.2 cm?

8. The goals in 8 matches were 1, 3, 2, 1, 0, 2, 1, 3. Which frequency table is correct for 0, 1, 2 and 3 goals?

Something to think about

Two students group the same 20 plant heights. One uses classes 2 cm wide and the other 8 cm wide. Whose table would you use to say how many plants are shorter than 12 cm, and whose would you use to see the shape of the data? Is one of them wrong?

One way to think about it

Both tables can say how many plants are shorter than 12 cm, because 12 is a class boundary in each: 9 plants. The 2 cm classes show the shape much better, while the 8 cm classes only say 9 and 11. Neither is wrong: narrow classes give detail, wide classes give a quicker picture.

Exit ticket

9. 30 parcels have masses from 0.4 kg to 9.8 kg. Which set of classes (in kg) is suitable for a grouped frequency table?