Choosing and judging statistical representations
Choosing and judging statistical representations
Which display for which purpose
Each display keeps some information and loses some. Choose the display that shows what the question needs.
| Display | Shows well | Cannot show, or drawback |
|---|---|---|
| Stem-and-leaf diagram | every value, in order; the median and quartiles exactly | is hard to draw and read for a large data set |
| Dot diagram | each value as a dot; clusters, gaps and an isolated value | suits only small data sets |
| Histogram (equal class widths) | the frequency of each class and the shape of the distribution | the individual values; the median only to its class |
| Cumulative frequency curve | the number of values up to any value; estimates of the median, quartiles and percentiles | the individual values; class frequencies only by subtracting readings |
| Box-and-whisker plot | the five-number summary; a quick comparison of two groups on one scale | the number of values, the individual values, or how the values lie inside each quarter |
| Bar graph | counts of separate categories | continuous data such as heights or times |
| Pie chart | each category as a part of one whole | the total; two pie charts of different totals cannot compare counts |
Misleading displays
Both charts show the same sales: \(60\) and \(64\) units. On the left the vertical axis starts at \(56\), so the second bar looks twice as tall although sales rose by only \(\dfrac{4}{60}\times100\%\approx6.7\%\). On the right the axis starts at \(0\), so the bar heights keep the true ratio \(60:64\) and the two bars look almost the same height.
- A vertical axis that does not start at zero exaggerates differences between bar heights. Describe the effect, not just that the chart “looks wrong”.
- Percentages or pie charts without totals cannot show which group has more items.
- In a box plot each of the four sections holds about a quarter of the values. A longer whisker shows a wider spread of that quarter, not more values.
Choosing an average
The mean uses every value, so one extreme value pulls it towards itself. The median depends only on the middle of the ordered data. The mode is the most common value and can be used for categories. The waiting times \(4,5,5,6,7,30\) minutes have mean \(\dfrac{57}6=9.5\) minutes, larger than five of the six times; the median, \(\dfrac{5+6}2=5.5\) minutes, describes a typical wait better.
Checks
- Read the scale and where each axis starts before comparing heights.
- Compare groups of different sizes with percentages, not counts.
- Justify a choice of average with the data: name the extreme value or the reason the mean is pulled.
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