Mathematics Year 6: The modal class and choosing a summary measure
Mathematics, Year 6 (Portugal): When data are grouped in classes the mode becomes the modal class, and the nature of the data decides which summary measure is honest.
From the mode to the modal class
When data are separate numbers, the mode is the value that appears most often. But when data are continuous (heights, weights, times) and come grouped in classes, you no longer know each exact value. So you say which class is the modal class: the class with the greatest frequency. There can be two or more if they tie. Each kind of data has its own measure: categories only have a mode; numbers have a mode and a mean; data in classes have a modal class.
Choosing is deciding what to show
Summarising data means choosing what to show and what to hide, so the choice must be thought through. The mean puts all the values into one, but it can be pulled by values far from the others. The mode and the modal class show where most data lie, but say nothing about those outside. So ask: what kind of data are they? Are there atypical values? What do I want to know? Often two measures together say more than one.
The trees of a park
In a table, the heights of 30 trees in a park are grouped: [2, 4[ has 3; [4, 6[ has 5; [6, 8[ has 12; [8, 10[ has 7; [10, 12[ has 3. The symbol [6, 8[ means “from 6 m, included, up to 8 m, excluded”. Step 1: check the total: 3 + 5 + 12 + 7 + 3 = 30. Step 2: the greatest frequency is 12, so the modal class is [6, 8[. Step 3: interpret: no other class has as many trees; 12 out of 30 is 40%. Since there are only classes, each height is unknown: the measure you can give is the modal class.
The modal class is not “most of the data”
In the trees table, the modal class is [6, 8[ with 12 trees. Mistake 1: saying the modal class is 12. But 12 is the frequency; the modal class is [6, 8[. Mistake 2: saying “most trees are between 6 and 8 m tall”. Most means more than half, and half of 30 is 15; 12 is only 40%. The modal class is the most frequent one; it need not hold more than half of the data. Say “it is the class with the most trees”, and check with the percentage.
Asking “which measure?”
When you read “the average waiting time is 15 minutes”, ask which data it refers to and whether some values lie far from the rest. Whoever decides where to put more tables in a canteen, or how many seats a bus needs, wants to know where most cases are (mode or modal class), not only the mean. And in a letter to the school board, you choose the measures that support your argument without hiding the rest of the data.
Test what you learned
A table shows the heights of 30 trees grouped in classes, with the frequencies. Which summary measure can be read directly from it?
- a) The mean of the heights
- b) The modal class
- c) The exact height that appears most often
- d) The greatest frequency, as if it were the mode
Correct answer: b) The modal class — Yes: the modal class is the class with the greatest frequency, and it is read straight off the table.
The table shows the daily reading time of 25 pupils: [0, 10[ min has 4; [10, 20[ has 9; [20, 30[ has 9; [30, 40[ has 3. What is the modal class?
- a) [10, 20[ and [20, 30[
- b) Only [10, 20[, because it comes first
- c) 9
- d) There is no modal class, because two classes tie
Correct answer: a) [10, 20[ and [20, 30[ — Both classes have the greatest frequency, 9, and they tie: there are two modal classes.
Fifty pupils gave their travel time to school: [0, 10[ min has 18; [10, 20[ has 15; [20, 30[ has 11; [30, 40[ has 6. Which statement is correct?
- a) Most pupils take less than 10 minutes
- b) The modal class is 18
- c) The modal class is [10, 20[, the one in the middle
- d) The modal class is [0, 10[, but 18 out of 50 (36%) is less than half the pupils
Correct answer: d) The modal class is [0, 10[, but 18 out of 50 (36%) is less than half the pupils — Right: 18 is the greatest frequency, so [0, 10[ is modal, and 18 ÷ 50 = 36% does not reach half.
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