Two routes to the same office each had a mean delay of 5 minutes over five constructed trips. Does that make them equally reliable for someone who cannot afford a long wait? A mean answers only one question about a distribution, and a graph can make a small difference look large or hide a large one. This lesson compares whole distributions on a common scale, separates absolute change from relative change, and shows how to recognize a display whose picture does not match its numbers.
Key terms
Common scale
Definition Displays of the same variable for several groups are drawn on a common scale when their axes use the same units, the same range and the same length per unit, so that equal distances stand for equal differences in every panel. Comparing distributions means comparing their center, spread, shape and unusual values, together with the number of observations behind each, and a common scale lets the eye make those comparisons directly. Side-by-side dot plots on one axis, parallel box plots and histograms with the same bins on the same axis are typical examples.
In plain words Put the groups on the same ruler before you compare them. If each panel is scaled to fit its own data, a narrow distribution is stretched across the whole panel and can look as wide as a broad one, and the numbers on the axes do not always undo that first impression.
Example The delays on route A were 4, 5, 5, 5 and 6 minutes, and on route B they were 1, 3, 5, 7 and 9 minutes. Both routes have mean delay minutes and median delay 5 minutes, so neither center separates them. On one axis from 0 to 10 minutes, the delays of A span 2 minutes and those of B span 8 minutes, and the sample standard deviations are and minutes. If a long delay means more than 6 minutes, A had no long delay in 5 trips and B had 2 in 5, which is 0 percent against 40 percent. With the cutoff “6 minutes or more”, the 6-minute delay of A would count too, so the definition matters. Five trips per route describe these trips and cannot settle the future rate of long delays on either route.
Baseline
Definition The baseline of a bar chart is the value from which the bars are drawn. A bar for a value above the baseline has length proportional to , so two bars with values and above the baseline have lengths in the ratio
For two different values, this equals the ratio of the values only when . For bars that show counts, amounts or rates, a zero baseline is therefore the natural choice. Another baseline suits bars only when it is a meaningful reference marked on the chart, such as a target, and then each bar shows a difference from that reference. Line charts and dot plots show values by position rather than by length, so an axis that starts above zero can be reasonable there when its range is clearly labeled.
In plain words Readers take the length of a bar as the size of the quantity. Cut off the bottom of every bar and the remaining lengths no longer compare the quantities, even when every number printed on the chart is correct.
Example Two bars show the values 50 and 55. From a zero baseline their lengths are in the ratio , so the second value is 10 percent larger. With the axis starting at 49, the visible lengths are 1 and 6, and the second bar looks six times as long. The same two values shown as points on a dot plot with a labeled axis from 48 to 56 would not mislead in this way, because position, not length, carries the value there.
Absolute change
Definition The absolute change from a starting value to a new value is , measured in the units of the quantity itself. It is positive for an increase and negative for a decrease. When the quantity is a rate written as a percentage, the unit of the absolute change is the percentage point.
In plain words Absolute change says how much more or how much less, in the same units as the data. It ignores where you started, so the same absolute change can be large next to one starting value and small next to another.
Example A constructed failure rate rises from 10 percent to 15 percent. The absolute change is percentage points. Out of 1000 opportunities each, that is 100 failures before and 150 after, or 50 more failures per 1000 opportunities. The counts can be compared this way only because both rates refer to the same number of opportunities.
Relative change
Definition The relative change from a positive starting value to a new value is
usually reported as a percentage. It is negative for a decrease. A relative change from a starting value of 0 is undefined, because the formula divides by the starting value. Relative change also needs a ratio scale, whose zero means none of the quantity, as for counts, amounts and rates, so it has no unit-free meaning for temperatures in degrees Celsius.
In plain words Relative change compares the change with where you started. A nonzero change and its reversal do not have the same relative size, because they start from different values.
Example For the failure rate that rises from 10 percent to 15 percent,
an increase of 50 percent. Going back from 15 percent to 10 percent is a relative change of , a decrease of about 33.3 percent, although the absolute change again has size 5 percentage points. A clear report names the starting value and the direction of the comparison.
Percentage point
Definition A percentage point is the unit of an absolute difference between two percentages. A change from percent to percent is a change of percentage points, and for it is a relative change of percent.
In plain words Percentage points say how far the percentage moved. Percent, used for a relative change, says how large that move is compared with the starting percentage. A sentence saying that a rate “rose by 4 percent” is ambiguous until you know which of the two is meant.
Example An approval rate that moves from 40 percent to 44 percent rises by 4 percentage points, which is a relative increase of , or 10 percent. A relative increase of 4 percent from 40 percent would end at percent. A rate that goes from 2 percent to 4 percent has doubled, a relative increase of 100 percent, yet it has risen by only 2 percentage points.
Misleading graph
Definition A graph is misleading when its visual encoding suggests a different comparison from the one its numbers support, even if every label on it is correct. Common causes are bars drawn from a baseline that is not meaningful, panels on different scales placed side by side for comparison, histograms with unequal bin widths that use frequency rather than frequency density as height, uneven or missing periods on a time axis, icons enlarged in both height and width so that their area grows as the square of the quantity, counts plotted where a rate is claimed, percentages shown without the number of cases, and error bars whose meaning is not stated.
In plain words Ask what the picture makes you believe, then check whether the numbers say the same thing. Error bars are a good test. A bar of one standard deviation on each side of a mean describes how individual observations vary, while an interval for a mean describes uncertainty about the average, and the two are different objects, so a graph that does not say which one it shows is incomplete. Equal percentages need their counts as well, since 2 out of 10 and 20 out of 100 are both 20 percent but rest on very different amounts of data.
Example A caption says a rate doubled from 2 percent to 4 percent, beside bars of raw counts that rise from 20 to 80. The counts quadrupled, so the picture shows a fourfold rise. Both statements hold only if the number of opportunities doubled, from to , since
A repaired display plots the two rates as bars from a zero baseline, states both denominators, and describes the change as a rise of 2 percentage points, a relative increase of 100 percent.
Check yourself
- A failure rate falls from 12 percent to 9 percent. State the absolute change and the relative change, each with its correct unit.
Show the answer
The absolute change is percentage points, a decrease of 3 percentage points. The relative change is , a decrease of 25 percent. Calling this a drop of 3 percent would be wrong if read as a relative change, because a relative drop of 3 percent from 12 percent ends at percent.
- A bar chart compares the values 80 and 84, with the vertical axis starting at 78. How many times as long as the first bar does the second one look, how large is the real difference in relative terms, and how would you repair the chart?
Show the answer
The visible lengths are and , so the second bar looks three times as long as the first. The values are in the ratio , so the second value is only 5 percent larger. To repair the chart, draw the bars from zero so that their lengths are proportional to the values. If the difference of 4 is what matters, show it directly, either as one bar of height 4 from zero or as two points on a dot plot whose axis range is clearly labeled.
- Over eight constructed trips each, route A took 21, 22, 22, 23, 23, 24, 24 and 25 minutes, and route B took 18, 19, 19, 20, 20, 21, 22 and 41 minutes. A report says B is faster because its mean is lower. Write a fuller comparison covering center, spread, shape, unusual values and the limits of the data.
Show the answer
Plot both routes as dot plots on one axis, for example from 15 to 45 minutes. For the center, the means are 23 and 22.5 minutes, a difference of only half a minute, because one long trip pulls the mean of B upward. The medians are 23 and 20 minutes, so a typical trip on B was about 3 minutes faster.
For the spread, the trips on A span only 4 minutes, from 21 to 25. The trips on B span 23 minutes, from 18 to 41, although seven of its eight trips lie between 18 and 22 minutes. For the shape, A is roughly symmetric, while B is skewed to the right by its single 41-minute trip, which is an unusual value at about twice the median of B.
For reliability, all eight trips on A took at most 25 minutes, against seven of eight on B. Someone who must arrive within 25 minutes would have been late once on B and never on A in these records.
As for the limits, eight trips per route is very little data. We do not know how the trips were chosen, whether both routes were driven at similar times, or whether a delay like the 41-minute trip is rare or common, so these numbers describe these trips and are not estimates of future rates.