Collecting and presenting data
Statistics is the branch of mathematics that deals with collecting, organising, presenting and interpreting data.
Primary data is collected by the investigator directly, for example by surveying classmates.
Secondary data is collected by someone else and used by the investigator, such as census figures.
Data in its original form, as collected, is raw data.
Ungrouped frequency distribution
When the values repeat a lot, list each value with its frequency (how many times it occurs).
Example. Marks out of 10 of 20 students: 6, 7, 5, 8, 6, 9, 7, 6, 5, 10, 7, 8, 6, 7, 9, 6, 8, 7, 5, 6.
Marks | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|
Number of students | 3 | 6 | 5 | 3 | 2 | 1 |
Grouped frequency distribution
When there are many different values, group them into class intervals.
Example. Heights (in cm) of 30 students, grouped:
Height (cm) | Number of students |
|---|---|
140–145 | 4 |
145–150 | 7 |
150–155 | 10 |
155–160 | 6 |
160–165 | 3 |
Terms used:
In 145–150, 145 is the lower class limit and 150 the upper class limit.
The class size (width) is the difference between them: 5.
The class mark is the midpoint: (145 + 150) ÷ 2 = 147.5.
These intervals are exclusive (continuous): a height of exactly 150 cm is counted in 150–155, not in 145–150. The upper limit belongs to the next class.
If the classes are written as inclusive intervals such as 10–19, 20–29, there is a gap between 19 and 20. To make them continuous, subtract 0.5 from each lower limit and add 0.5 to each upper limit: 9.5–19.5, 19.5–29.5, and so on.
Bar graphs
A bar graph uses bars of equal width with equal gaps; the height shows the frequency. It suits data in separate categories, such as the number of students who like each sport.
Histograms
A histogram is used for continuous grouped data.
Class intervals go on the horizontal axis and frequencies on the vertical axis.
Bars are drawn next to each other with no gaps, since the classes are continuous.
The area of each bar is proportional to the frequency.
When all classes have equal width, the heights can simply be the frequencies.
Unequal widths. If some classes are wider, heights must be adjusted so that areas stay proportional to frequencies. Use the smallest class width as the standard and compute
adjusted frequency = (minimum class width ÷ this class's width) × frequency
Example. Classes 20–30 (frequency 8) and 30–50 (frequency 12). The minimum width is 10. The 30–50 class has width 20, so its bar's height is 10/20 × 12 = 6, not 12. Otherwise the wider bar would look misleadingly large.
If the first class doesn't start at 0, show a small zig-zag (kink) on the horizontal axis to indicate a break in the scale.
Frequency polygons
A frequency polygon shows the same information as a histogram with line segments.
Mark the class mark of each interval on the horizontal axis, and plot the point (class mark, frequency).
Join the points with straight lines.
To close the polygon, add an imaginary class with frequency 0 before the first and after the last class, and join to their class marks on the axis.
It can be drawn on top of a histogram by joining the midpoints of the tops of the bars. Frequency polygons are especially handy for comparing two data sets on the same axes, such as the marks of two sections of a class.
Summary
Raw data is organised into frequency tables, ungrouped or grouped.
Class mark = (lower limit + upper limit) ÷ 2; class size = upper limit − lower limit.
A histogram has no gaps between bars, and areas are proportional to frequencies; adjust heights for unequal widths.
A frequency polygon joins (class mark, frequency) points.