Mathematics Year 6: Classes of equal width and the frequency table
Mathematics, Year 6 (Portugal): When almost every measured value appears only once, you group them into classes [a, b) of equal width and organise them in a table of absolute and relative frequencies.
When every value appears only once
The 20 bag weights are all different: 2.9, 3.2, 3.4… A table with one row per value would have 20 rows, all with frequency 1, and would show no pattern at all. The solution is to group: split the number line into consecutive intervals of equal width, the classes, and count how many values fall into each one. You write [2, 3): it includes 2 and the values up to 3, but excludes 3. The class is closed on the left and open on the right, and 3 is the start of the next class.
How many classes? Try more than one
There is no official rule: you choose whole-number limits that include all the data and a number of classes that shows something. With very few classes, different values get mixed and you lose information. With many, you are back to about one value per class and nothing shows. So it is worth trying two or three options and seeing what each one shows. Once chosen, each class gets two counts: the absolute frequency (how many values) and the relative frequency (that count divided by the total, as a fraction or a percentage).
From 2.9 to 6.8: building the table
The 20 weights run from 2.9 kg to 6.8 kg. Step 1: whole-number limits that include everything: start at 2 (2.9 fits in [2, 3)) and end at 7 (6.8 fits in [6, 7)). Step 2: with width 1 kg that is 7 − 2 = 5 classes. Step 3: count each class: 1, 7, 4, 6 and 2 values, and 1 + 7 + 4 + 6 + 2 = 20. Step 4: relative frequency = count ÷ 20: 5%, 35%, 20%, 30% and 10%, which add up to 100%. Title: “Weight of the class's bags (kg)”. If someone typed 53 instead of 5.3, that value would fall outside the classes and the total would be 19: the sign to hunt for the typo.
Which class does 5.0 belong to?
The list has one bag of exactly 5.0 kg. Many people put it in [4, 5) because it “reaches 5”. But the class [4, 5) is open on the right: 5 is not in it. The 5.0 goes into [5, 6). It matters: with the mistake, [4, 5) would get 5 values instead of 4, and [5, 6) would get 5 instead of 6. A hint: the left-hand value belongs to the class, the right-hand one is the start of the next class. And always check that the counts add up to 20.
Heights, race times, rainfall
Whenever there are many different measured values, classes appear: the heights of the students in a school, the times in a race, the minutes waiting at a bus stop, the amount of rain each day. A spreadsheet groups for you, but you decide where the classes start and end, and that decision changes what the table shows. Anyone reading a table of classes should look for the title, the class limits and the total.
Test what you learned
The 20 bag weights are all different. Why is it useful to group them into classes of equal width?
- a) So you no longer need to measure with decimals.
- b) So the values become more exact.
- c) So that each value has frequency 1.
- d) Because, with every value appearing once, the table shows no pattern; a class gathers nearby values and you see where they concentrate.
Correct answer: d) Because, with every value appearing once, the table shows no pattern; a class gathers nearby values and you see where they concentrate. — Right: classes gather nearby values and the table now shows where the weights concentrate.
The 20 weights run from 2.9 kg to 6.8 kg. Which classes of width 1 kg, with whole-number limits, include all the values?
- a) [3, 4), [4, 5), [5, 6), [6, 7), [7, 8)
- b) [2, 3), [3, 4), [4, 5), [5, 6), [6, 7)
- c) [2, 3), [3, 4), [4, 5), [5, 6)
Correct answer: b) [2, 3), [3, 4), [4, 5), [5, 6), [6, 7) — Right: 2.9 fits in [2, 3) and 6.8 in [6, 7). That is 7 − 2 = 5 classes of 1 kg.
A bag weighs exactly 5.0 kg. Which class does it go in, if the classes are [4, 5) and [5, 6)?
- a) In [4, 5), because it “reaches 5”.
- b) In both, because it sits on the limit of both.
- c) In [5, 6), because the class includes its left-hand value.
Correct answer: c) In [5, 6), because the class includes its left-hand value. — Right: 5 is the left limit of [5, 6), which includes it, and the right limit of [4, 5), which excludes it.
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