1The idea in one paragraph

A frequency table counts how often each value, or each range of values, occurs. From it you draw a histogram, which shows the shape of the data: where it piles up, and whether it trails off to one side. Add the frequencies up as you go and you get cumulative frequency, the number of values up to each point; its graph lets you read off the median, quartiles and any percentile by going across and down. Those five numbers (minimum, lower quartile, median, upper quartile, maximum) give a box-and-whisker diagram, the fastest picture for comparing two data sets and for spotting skew, outliers, and data that might be normal.

2Frequency tables

Discrete data go in a table with one row per value. Thirty students were asked how many books they read last month (invented data):

The frequencies add to 30, which is the first thing to check in any table. Figure 1 draws it: separate bars with gaps, because 2.5 books is not a possible answer.

Continuous data are grouped into class intervals. The guide says these are written as inequalities without gaps, so every value belongs to exactly one class. Eighty runners' times at a park run (invented data):

A runner who took exactly 25.00 minutes goes in 25 ≤ t < 30, not in the class below, because the lower class has "< 25". The class width here is 5 minutes, and the lower and upper boundaries of the third class are 25 and 30. Grouping costs information: once a time is in the table you know only its class, not its value. That is why everything read from grouped data later on is an estimate.

3Histograms

A frequency histogram draws each class as a bar whose width spans the class and whose height is its frequency. Because the data are continuous, the bars touch: there is no gap between 24.99 minutes and 25.00. Figure 2 is the histogram of the park-run times.

Read it for shape. The times pile up in 25 ≤ t < 30, the modal class, and trail off further to the right than to the left: a few runners walked it. A longer tail on the right is positive skew; a longer tail on the left is negative skew; a histogram that is roughly a mirror image either side of its peak is symmetric.

For the histogram to be marked correctly: frequency on the vertical axis with a scale, the variable and its units on the horizontal axis, bars starting and ending at the class boundaries, and equal widths.

4Cumulative frequency

The cumulative frequency at a value is the number of data items less than or equal to it. You build it by adding each frequency to the running total.

The last entry must equal the total, 80. The cumulative frequencies belong to the upper boundaries: by 20 minutes, 4 runners had finished; by 25 minutes, 15 had. So the cumulative frequency graph plots (20, 4), (25, 15), (30, 37) and so on, and starts at (15, 0), because no one finished in under 15 minutes. Join the points with straight lines or a smooth curve; both are accepted, and readings will differ slightly between them. The graph always rises or stays level, because a running total can never go down. Figure 3 is the graph.

Plot cumulative frequency at the upper class boundary, and start the graph at the lower boundary of the first class with frequency 0.

5Reading median, quartiles and percentiles

To read a value off a cumulative frequency graph, go across from the cumulative frequency you want to the graph, then down to the horizontal axis.

• The median is the middle value: go across from n ÷ 2.
• The lower quartile Q₁ has a quarter of the data below it: go across from n ÷ 4.
• The upper quartile Q₃ has three quarters below it: go across from 3n ÷ 4.
• The kth percentile has k% of the data below it: go across from (k ÷ 100) × n. The median is the 50th percentile, Q₁ the 25th and Q₃ the 75th.

For the 80 runners (Figure 3):

With a large n, IB questions use n ÷ 2 for the median rather than fussing over (n + 1) ÷ 2; the graph is not accurate enough for the difference to matter. Give readings to the accuracy the scale allows, and expect the mark scheme to accept a small range.

The range from a grouped table is less certain. The graph runs from 15 to 50, so the range is at most 50 − 15 = 35 minutes, but the fastest runner might have taken 16.2 minutes and the slowest 48.9. Say "at most" when that is all you can know.

The graph also answers counting questions. Figure 4 reads two things off the same curve.

The trap in the second one is stopping at 64. The graph counts values below a point, so "more than" always needs the subtraction from the total.

6Box-and-whisker diagrams

A box-and-whisker diagram (box plot) draws five numbers on a scale: a box from Q₁ to Q₃ with a line at the median, and a whisker from each end of the box out to the most extreme value that is not an outlier. Any outlier is marked separately with a cross.

Nineteen apples from one orchard have these masses in grams (invented data), in order:

142, 148, 151, 153, 155, 156, 158, 160, 161, 163, 164, 166, 168, 170, 171, 174, 177, 181, 212

So the left whisker runs to the minimum, 142, and the right whisker stops at 181, the largest value inside the fences. The 212 gets a cross. Figure 5 shows the finished diagram.

To draw one in an exam: a labelled scale with units first, then the box and median line, then the whiskers, then the crosses. A box plot drawn without a scale scores almost nothing, because its whole value is where things sit on the scale.

Half the data lie inside the box, a quarter along each whisker (counting any outliers on that side). A long whisker does not mean many values; it means the values in that quarter are spread out.

7Comparing two distributions

Box plots are built for comparison: put them on the same scale, one above the other. Orchard B's apples have minimum 138, Q₁ 150, median 157, Q₃ 162 and maximum 175, with no outliers. Figure 6 draws both orchards.

A full comparison makes two points, each with numbers and each in context:

• Centre. Orchard A's median (163 g) is higher than Orchard B's (157 g), so A's apples are typically heavier.
• Spread. Orchard B's IQR (162 − 150 = 12 g) is smaller than A's (171 − 155 = 16 g), so the middle half of B's apples are more consistent in mass. The range tells the same story more dramatically (37 g against 70 g), but A's range is inflated by one outlier, which is why the IQR is the better measure to quote.

You may add a third point on symmetry: both boxes have the median near the centre, so neither orchard's middle half is strongly skewed. "A is bigger than B" scores nothing; "A has a higher median, 163 g compared with 157 g, so A's apples are typically heavier" scores the mark.

8Shape, and whether the data could be normal

The position of the median inside the box and the lengths of the whiskers tell you the shape, as Figure 7 shows.

• Symmetric: the median is near the middle of the box and the two whiskers are about equal in length.
• Positively skewed: the right whisker is longer and the median sits nearer the left of the box, because the values above the median are spread over a wider range.
• Negatively skewed: the mirror image.

The normal distribution, which you meet in 4.9, is symmetric and bell-shaped. So the guide asks you to judge from a box plot whether data may be normally distributed, by looking for symmetry: median near the centre of the box, whiskers roughly equal. A normal distribution also has most of its values packed near the middle, so each whisker is typically longer than half the box. A skewed box plot rules normality out; a symmetric one only makes it plausible, since many non-normal distributions are symmetric too. Say "may be", not "is".