Frequency tables, class boundaries, histograms and frequency polygons

Last time

Before we start: last time, in Stem-and-leaf diagrams, box-and-whisker diagrams, two-way tables and Venn diagrams, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Grouping data

A list of hundreds of values is hard to read, so statisticians group them into a frequency table. This lesson covers frequency tables, class boundaries, histograms and frequency polygons, including classes of unequal width, which the exam often uses.

Frequency tables

An ungrouped frequency table lists each value with its frequency, the number of times it occurs. A grouped table puts values into classes, such as 10 to 19. It is easier to read, but the original values are lost. A grouped two-way frequency table counts two variables at once, such as visits to a dentist and visits to a doctor.

Class boundaries

For data rounded to the nearest whole number, the class 10 to 19 has boundaries 9.5 and 19.5. For ages in completed years, the class 10 to 19 runs from 10 up to 20, because someone aged 19 and 11 months is still 19. The class width is the upper boundary minus the lower boundary: 10 in both cases. The mid-point is halfway between the boundaries.

Histograms

In a histogram, the area of each bar is proportional to its frequency, and there are no gaps between the bars. The height of each bar is the frequency density: frequency divided by class width. The class 10 to 20 with frequency 30 has density 3. The class 20 to 40, also with frequency 30, has density 1.5: it is wider, so it is lower.

Reading histograms

To find a frequency from a histogram, multiply the frequency density by the class width. A bar of density 2.4 over a class 5 wide holds 2.4 times 5, which is 12 items. To estimate the number in part of a class, take that fraction of the bar's area.

Frequency polygons

A frequency polygon plots each class's frequency at its mid-point and joins the points with straight lines. For classes of unequal width, plot the frequency density instead. Two polygons on one grid make it easy to compare two distributions, such as the lengths of two kinds of leaf.

A histogram table

Frequency densities for classes of unequal width.

Comparing distributions

Compare where the data are centred: which group has the larger values on the whole. Compare the spread: which group's values are more varied. Use numbers from the diagram, and write in the context of the question.

Mistakes that lose marks

Mistake one: drawing a histogram of unequal classes with frequency, not frequency density, as the height. Mistake two: using class limits, such as 10 and 19, instead of boundaries, to find widths. Mistake three: plotting a frequency polygon at the ends of the classes instead of their mid-points. The fix: boundaries first, then widths and densities.

The full-marks tip

The full-marks tip: add columns to the table for the boundaries, the width and the frequency density before you draw. Label the vertical axis frequency density, and choose a scale that fills the grid.

On one card

So, on one card: frequency distributions. Keep it for revision.