An investigator collecting data generally has two goals: to obtain descriptive information about the population from which the sample was drawn and to test hypotheses about that population. We focus here on the first goal: to summarize data collected on a single variable in a way that best describes the larger, unobserved population.
When the value of the variable associated with any given individual is more likely to fall near the mean (average) value for all individuals in the population under study than far from it and equally likely to be above the mean and below it, the mean and standard deviation for the sample observations describe the location and amount of variability among members of the population. When the value of the variable is more likely than not to fall below (or above) the mean, one should report the median and values of at least two other percentiles.
To understand these rules, assume that we observe all members of the population, not only a limited (ideally representative) sample as in an experiment
For example, suppose we wish to study the height of Martians and, to avoid any guesswork, we visit Mars and measure the entire population—all 200 of them. Figure 2-1 shows the resulting data with each Martian's height rounded to the nearest centimeter and represented by a circle. There is a distribution of heights of the Martian population. Most Martians are between about 35 and 45 cm tall, and only a few (10 out of 200) are 30 cm or shorter, or 50 cm or taller.
Distribution of heights of 200 Martians, with each Martian's height represented by a single point. Notice that any individual Martian is more likely to have a height near the mean height of the population (40 cm) than far from it and is equally likely to be shorter or taller than average.
Having successfully completed this project and demonstrated the methodology, we submit a proposal to measure the height of Venusians. Our record of good work assures funding, and we proceed to make the measurements. Following the same conservative approach, we measure the heights of all 150 Venusians. Figure 2-2 shows the measured heights for the entire population of Venus, using the same presentation as Figure 2-1. As on Mars, there is a distribution of heights among members of the population, and all Venusians are around 15 cm tall, almost all of them being taller than 10 cm and shorter than 20 cm.
Distribution of heights of 150 Venusians. Notice that although the average height and dispersion of heights about the mean differ from those of Martians (Fig. 2-1), they both have a similar bell-shaped appearance.