Mathematics Year 7: Grouping counted data in classes: the frequency table
Mathematics, Year 7 (Portugal): When counted values are very spread out and almost none repeats, group them in classes of equal width and organise them in a titled frequency table.
When one line per value is too many
For a variable with few values, such as pens in a pencil case (0 to 8), a table with one line per value works. But if each of 30 students says how many messages they sent in a week, the values can go from 12 to 870 and almost none repeats. Then you group the values in classes of equal width, such as [0 ; 100[, [100 ; 200[, [200 ; 300[… and count how many fall in each. For whole numbers, [0 ; 100[ holds 0, 1, …, 99.
You lose exact values and gain the shape
A table with 30 lines all with frequency 1 shows no pattern; a table with a few classes shows where the values pile up. The price is that inside a class you no longer see the exact values. Equal width is what lets you compare the counts of two classes fairly. Try two or three widths, make sure each value belongs to exactly one class and that the counts add up to the total, and give the table a title that says what, who and the unit.
From 30 different values to 9 classes
30 students wrote how many messages they sent in a week: all different, from 12 to 870, so a table without classes has 30 lines of frequency 1. Step 1: width 100, from 0 to 900: (900 − 0) ÷ 100 = 9 classes. Step 2: counts: [0 ; 100[ 10, [100 ; 200[ 8, [200 ; 300[ 6, [300 ; 400[ 3, [400 ; 500[ 2, then three classes with 0, and [800 ; 900[ 1. Step 3: 10 + 8 + 6 + 3 + 2 + 0 + 0 + 0 + 1 = 30. Reading: 10 + 8 = 18 of 30, that is 60%, sent fewer than 200. Title: “Messages sent in a week by the 30 students of the class”.
Do not widen a class just to avoid empty ones
The last classes of the table, [500 ; 600[, [600 ; 700[ and [700 ; 800[, are empty, and it is tempting to merge everything from 400 up into one class, [400 ; 900[, with 2 + 0 + 0 + 0 + 1 = 3 students. But that class is five times wider than the others, so its 3 cannot be compared with the 3 of [300 ; 400[, and it hides that one student is far above everyone else. Empty classes are information: they show a gap. Keep equal width.
Steps, followers, posts and scores
Classes appear whenever counted data are very spread out: steps per day, followers of an account, posts of each student on a social network, points in a game, vehicles passing a street each day. A spreadsheet can group the values for you, but you decide where the classes start and how wide they are, and that choice changes what the table shows. When you read such a table, look for the title, the class limits and the total.
Test what you learned
When is it useful to group counted (discrete) data in classes?
- a) When the values are very spread out and almost none repeats
- b) Always, for any counted data
- c) When the variable is qualitative
- d) When there are fewer than 10 data
Correct answer: a) When the values are very spread out and almost none repeats — Right: a line per value would give many lines of frequency 1 and no pattern. Classes show where the values pile up.
Points scored by 20 students in a game, sorted: 3, 18, 27, 35, 44, 50, 52, 58, 61, 66, 74, 83, 91, 97, 100, 108, 115, 126, 133, 149. With classes of width 50 starting at 0, how many values are in [50 ; 100[?
- a) 5
- b) 8
- c) 9
- d) 10
Correct answer: c) 9 — Right: 50, 52, 58, 61, 66, 74, 83, 91 and 97 are 9 values. The 100 already belongs to [100 ; 150[.
A table of followers of 20 accounts has these classes: [0 ; 50[ 7, [50 ; 100[ 6, [100 ; 150[ 4, [150 ; 400[ 3. What is wrong with it?
- a) Nothing: fewer classes are always clearer
- b) The counts should add up to 100
- c) The classes should overlap so that no value is left out
- d) The last class is 5 times wider than the others, so its count is not comparable
Correct answer: d) The last class is 5 times wider than the others, so its count is not comparable — Right: 400 − 150 = 250 and 250 ÷ 50 = 5. Classes must have equal width so their counts can be compared.
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