Histograms, cumulative frequency and box plots. Three ways to draw the same data, and three different impressions.
Here are 60 journey times to school, in minutes. The data never changes on this page. Only the width of the groups changes. Watch what that does.
At width 20 the data looks like one smooth lump. At width 5 you can see there are really two groups of travellers, and a gap between them. Same 60 numbers. If you choose the groups before you look, you can hide the most interesting thing in your own data.
The same sixty values, binned three ways, telling three different stories. The data did not change.
When data is grouped you lose the individual values. You can still estimate, but you must be honest that it is an estimate.
The modal class is the group with the highest frequency. It is a group, not a number, so "the mode is 23" is wrong when the data is grouped.
An estimated mean uses the midpoint of each class as if every value in that class sat exactly there. That is why it is an estimate and not the mean.
| Time, t minutes | 0 ≤ t < 10 | 10 ≤ t < 20 | 20 ≤ t < 30 | 30 ≤ t < 40 |
|---|---|---|---|---|
| Frequency | 6 | 14 | 16 | 4 |
| Midpoint | 5 | 15 | 25 | 35 |
A cumulative frequency curve answers "how many were less than this?". Read across at half the total to estimate the median, at a quarter for Q₁, at three quarters for Q₃.
Those three readings, with the smallest and largest values, are the five numbers a box plot draws. The box plot is the cumulative frequency curve with everything except the quartiles thrown away.
The five-number summary: minimum, Q₁, median, Q₃, maximum. The box spans Q₁ to Q₃, so the box itself is the interquartile range, and half the data sits inside it.
A long box does not mean more people. Every quarter of a box plot holds the same number of data values. A wide section means those values are spread out, not that there are more of them. This is the single most common misreading of a box plot.
1. Using the table above, which is the modal class?
2. A different group of 50 has Σfx = 1710. Estimate the mean, to 1 decimal place.
3. On a box plot, the section between Q₃ and the maximum is much longer than the section between Q₁ and the median. What does that tell you?
Estimated mean questions are marked on method. Show Σfx and Σf separately and you keep the marks even if the arithmetic slips.
Say "estimate" when the data is grouped. Writing "the mean is 19.5" when only grouped data was given loses a mark that costs nothing to keep.
For a modal class, give the class. Giving a single number is the most frequent error on this topic, and it is a whole mark.
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