Ogive

An ogive is a line plot of the cumulative frequency distribution against values of the random variable. Francis Galton coined the term ogive to describe the shape of the normal cumulative distribution function, as it has a form similar to the S-shaped Gothic ogival arch.

Table 1 Student Test Scores
Test Scores	0–19	20–39	40–59	60–79	80–100
Frequency	1	6	22	17	4
Relative frequency	0.02	0.12	0.44	0.34	0.08
Cumulative frequency	0.02	0.14	0.58	0.92	1.00

Note: Solid line is ogive plot of student test scores with frequency distribution, and dashed line is normal cumulative distribution curve

The ogive can display the population cumulative frequency distribution or an estimate from a sample. The random variable can be continuous or discrete (as long as there is a natural ordering of the outcomes). The range of possible outcomes is divided into classes. A line is drawn to connect points at the upper limit of each class interval and the cumulative frequency distribution. For a discrete random variable, each possible outcome can be a class, and therefore the upper limit of the class is the outcome itself.

The student test scores in Table 1 are split up into five class intervals. The ogive is often overlaid with the frequency distribution, as in Figure 1. The solid line is the empirical cumulative distribution, and the

Table 2 Frequency Distribution of a Discrete Data Set With Number of Minor Accidents in a Factory per Year
Number of Accidents	0	1	2	3	4	5	6
Frequency	3	5	6	3	2	1	0
Relative frequency	0.15	0.25	0.30	0.15	0.10	0.05	0
Cumulative frequency	0.15	0.40	0.70	0.85	0.95	1	1

[Page 711] dashed line is an estimate of the population distribution assuming normality (notice the distinctive S shape observed by Galton).

Table 2 summarizes the frequency distribution of a discrete data set consisting of the number of minor accidents recorded in a factory per year. The corresponding ogive is shown in Figure 2, which was created with Excel. Notice that the ogive points are now at the outcome values rather than at the upper limit, which was the case in the previous data set.