Statistics - Week 3 Learning Objectives Colour Code: Definitions Descriptions Explanations 1. Define variability and explain its use and importance as a statistical measure. Variability has the same meaning in statistics as it has in everyday language; to say that things are variable means that they are not all the same. In statistics, our goal is to measure the amount of variability for a particular set of scores. In simple terms, if the scores in a distribution are all the same, then there is no variability. If there are small differences between scores, then the variability is small, and if there are large differences between scores, then the variability is large. In general, a good measure of variability serves two purposes: 1. Variability describes the distribution. Specifically, it tells whether the scores are clustered close together or are spread out over a large distance. Usually, variability is defined in terms of distance. It tells how much distance to expect between one score and another, or how much distance to expect between an individual score and the mean. For example, we know that the heights for most adult males are clustered close together, within 5 or 6 inches of the average. Although more extreme heights exist, they are relatively rare. 2. Variability measures how well an individual score (or group of scores) represents the entire distribution. This aspect of variability is very important for inferential statistics, in which relatively small samples are used to answer questions about populations. For example, suppose that you selected a sample of one person to represent the entire population. Because most adult males have heights that are within a few inches of the population average (the distances are small), there is a very good chance that you would select someone whose height is within 6 inches of the population mean. On the other hand, the scores are much more spread out (greater distances) in the distribution of weights. In this case, you probably would not obtain someone whose weight was within 6 pounds of the population mean. Thus, variability provides information about how much error to expect if you are using a sample to represent a population.
Population distribution of male heights and weights. (a) (b) Ă— 58 64 70 76 82 Adult heights (in inches) 110 140 170 x 200 230 Adult weights (in pounds) 2. Define and calculate the range as a simple measure of variability and explain its limitations. The first step toward defining and measuring variability is the range, which is the distance covered by the scores in a distribution, from the smallest to the largest score. Range as a discrete variable: When the scores are measurements of a discrete variable, the range simply measures the difference between the largest score (X max) and the smallest score (X min). range = X max - X min Therefore, scores that have values from 1 to 5 cover a range of 4 points. This definition works well for discrete variables that have defined upper