Representing and interpreting data

Histograms: area is frequency

A bar chart has equal-width bars, so height alone carries the information. A histogram copes with unequal class widths, and the price of that is recalculating every height as frequency density.

Now area equals frequency, and every histogram question becomes an area question. Given a bar, multiply height by width to recover its frequency. Given a frequency, divide by the width to set the height. A histogram with a key, saying one square represents 5 observations, is the same idea with a scale factor bolted on.

A frequency polygon joins the midpoints of the tops of the bars and is the usual way to lay two distributions over each other. It shows shape well and reads frequencies badly, so pick it when you are comparing and the histogram when you are counting.

Box plots and outliers

A box plot displays five numbers. The extremes, the quartiles and the median. The box spans the middle half of the data, the median line splits it, and the whiskers reach the most extreme values that are not outliers.

The 1.5 × IQR fences are the usual rule, though a question may instead define an outlier as more than two standard deviations from the mean. Use whichever definition the question gives you, even when the other one is more familiar. An outlier gets plotted as a separate cross, and the whisker stops at the last value inside the fence, never at the fence itself.

Comparing two box plots is a two-sentence answer with a fixed shape. One sentence on location, using the medians, in context. One on spread, using the IQRs, in context. “The median mass is higher for brand A, and brand A's masses are more consistent since its IQR is smaller.” Marks vanish when the context does.

Shape and skew

A distribution is symmetric when the median sits midway between the quartiles, so Q₂ − Q₁ = Q₃ − Q₂. When the upper gap is the bigger one, the long tail points right and the data has positive skew. The reverse gives negative skew.

The mean and median tell the same story. A long right tail drags the mean above the median, so mean > median indicates positive skew. Either test earns the mark provided you quote the comparison you used, and quoting both when they disagree is a good way to start a sentence about which measure the data deserves.

Cumulative frequency

Plot cumulative frequency against the upper class boundary and join the points with a smooth increasing curve. Reading across from n/2, n/4 and 3n/4 gives the median and quartiles. Reading up from two values and subtracting counts the data in any interval. Every reading is an estimate, because the grouping already blurred the exact values, and no examiner expects three significant figures off a graph.

Cleaning the data

Real data arrives dirty, and the large data set is deliberately so. Entries are missing, rainfall reads tr, wind direction is a compass point instead of a number. Decide what to do with each before any calculation, and say what you decided. Removing a genuinely impossible value is cleaning. Removing a merely inconvenient one is bias with better manners, so an anomaly should only go when you can state a reason to doubt the measurement itself.