Frequency tables and histograms

From a column of numbers to a shape

A quantitative variable with dozens of values is unreadable as a list. The first move is to bin it: choose intervals of equal width, count how many values fall in each, and turn the counts into shares.

• Frequency $f$ — the count in a bin.
• Relative frequency — $f/n$, the bin's share of the data. The column sums to 1.
• Cumulative relative frequency — the running total of the relative-frequency column: the share of values at or below the top of that bin. It climbs from the first bin's share to exactly 1.

A histogram draws the bins as touching bars whose heights are the frequencies (or relative frequencies). It is the shape of that bar profile that the paper asks you to name.

Worked example — Maple Ledger's processing times

Maple Ledger Bookkeeping in Kitchener timed 40 invoice-processing jobs, in minutes, and binned them in widths of 5.

Check the two columns: the relative frequencies add to 1.000, and the cumulative column ends at exactly 1.000. If either fails, a count is wrong.

Read the table as a histogram. The tallest bar is 5–10; the bars fall away to the right in a long, thin tail (5, then 2). One peak, tail to the right: unimodal, right-skewed. A cumulative value answers a different question — "what share of jobs took under 15 minutes?" — 0.825, read straight off the third row.

The five shape words

• Symmetric — the left and right halves mirror each other; mean ≈ median.
• Right-skewed (positively skewed) — a long tail toward the large values; most observations are small, a few are large; the mean is pulled above the median. Invoice amounts, days overdue, incomes.
• Left-skewed (negatively skewed) — the long tail is toward the small values; the mean is pulled below the median. Exam marks with a hard ceiling; ages at retirement.
• Unimodal — one peak. Bimodal — two clear peaks, which usually means two groups have been pooled (day-shift and night-shift processing times) and should be split before any average is quoted.

The direction of a skew is the direction of the tail, not of the peak. A histogram whose hump sits on the left and trails off to the right is right-skewed — the most common mistake is to name it by where the bulk sits.

Choosing bins

Bins are of equal width, they cover every value, and their boundaries are chosen so that no data value can sit on a boundary — write "5 to under 10" or set the boundaries at 4.5, 9.5, … when the data are whole numbers. Between 5 and 15 bins is the usual range; too few hides the shape, too many turns it into a comb.