Frequency tables

Of 50 students in an OpenStax example, 16 bought three books, so the relative frequency of "three books" is 16 ÷ 50 = 0.32.

A distribution tells you two things about a variable: which values it takes and how often each one turns up. The simplest way to write a distribution down is a frequency table. One column lists the values or the classes, and the next counts how many times each one occurs. Everything else in this course, from histograms to the normal curve, is a way of drawing or summarising a table like that.

Frequency, relative frequency, cumulative frequency

OpenStax asked 50 part-time students how many books they bought. Eleven bought one book, 10 bought two, 16 bought three, 6 bought four, 5 bought five and 2 bought six. Those counts are the frequencies, and they add to 50. Dividing each by 50 gives the relative frequencies 0.22, 0.20, 0.32, 0.12, 0.10 and 0.04, which add to 1. Running totals give the cumulative frequencies 11, 21, 37, 43, 48 and 50. Cumulative columns answer "how many at most?" questions at a glance: 37 of the 50 students, or 74%, bought three books or fewer.

Grouped data

When the values are many and varied, a table lists classes instead of single values. OpenStax gives the scores of 100 students on a calculus final test grouped by tens: 49.5 to 59.5 has 5 students, 59.5 to 69.5 has 10, 69.5 to 79.5 has 30, 79.5 to 89.5 has 40 and 89.5 to 99.5 has 15. The cumulative frequencies are 5, 15, 45, 85 and 100, so 85 students scored below 89.5. Grouping loses detail: from the table alone you cannot tell whether the five lowest scores were 50 or 59. That is the price of a short summary.

Frequency polygons

A frequency polygon draws the same table as dots joined by lines. Each dot sits above the middle of a class at the height of its frequency, and an empty class is added at each end so the line comes down to the axis. OpenStax uses polygons to lay two distributions on one picture, such as the calculus test scores and the students' final grades, because two outlines overlap more clearly than two sets of bars.

One more table: ages of presidents

OpenStax also groups the ages of US presidents at inauguration into six classes five years wide: 41.5 to 46.5 has 4 presidents, 46.5 to 51.5 has 11, 51.5 to 56.5 has 14, 56.5 to 61.5 has 9, 61.5 to 66.5 has 4 and 66.5 to 71.5 has 2, for 44 in all. The largest class, 51.5 to 56.5, holds 14 ÷ 44 ≈ 0.318 of them. Notice the half-year boundaries: ages are whole numbers, so no president can land on a boundary.

Comparing groups of different sizes

Relative frequencies earn their keep when groups differ in size. Suppose, as a made-up example, that 12 of 40 students in one class and 15 of 60 in another scored above 90. The second class has more high scorers, 15 against 12, but a smaller share of them: 15 ÷ 60 = 0.25 against 12 ÷ 40 = 0.30. Counts answer "how many?", relative frequencies answer "how common?". Before comparing two tables, decide which question you are asking, and check that each table's counts add to its stated total.