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Stats+Primer - A critical reader's brief guide to...

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1 A critical reader's brief guide to statistics Gregory S. Gilbert Environmental Studies, UC Santa Cruz Statistics are simply formal, mathematical ways to (1) DESCRIBE observations about groups of things (e.g., to describe the size of a population of animals), and to (2) TEST for trends, patterns, or differences among groups. (e.g., to see if plants grow larger as soils become more fertile (a trend) or to see if more rats died after eating a particular chemical than after eating a placebo (a difference)). Let’s say we want to describe the frog population in Walden Pond. We go out and catch 60 frogs and weigh them (Table 1). Some frogs are bigger, some smaller. We can illustrate the variability in the population looking at the frequency DISTRIBUTION of the frog’s weights (Table 2) and making a graph (Figure 1). This is usually a frequency graph, or histogram, showing the number of frogs in each weight class (a weight class might be all the frogs from 0-0.4 g, or from 2.0- 2.4 g, and so on) distribution, that is, the population will tend to have most of its members clustered around some particular weight, with some a little heavier and some a little lighter, and even fewer much heavier or much lighter. We can describe this central tendency in several ways. One common statistic is the MEDIAN - this is the value where one half (50%) of the frogs are heavier and 50% of the frogs are lighter (median =1.8 g). Another common statistic is the MEAN (in common usage often called the average). The mean is calculated by adding up the weights of all the frogs and dividing by the number of frogs (mean = 1.75 g). This can be thought of as the weight of the “typical” frog. When the distribution of the population approximates a bell-shaped (“normal”) curve, the median and the mean are very similar. Sometimes the distribution has a longer tail in one direction than the other - in these cases the median and mean can be very different. But the weight of the “typical” frog is only part of the useful information. We also want to know how variable the population is. One measure of the spread of the distribution is the VARIANCE , which is a measure of how far away the members of the population are from the mean. The square Table 1. Weights of Table 2. Frequency frogs in Walden pond. of weights of frogs. Frog # Weight (g) Weight class (g) Num. of frogs 1 2.1 0-0.4 2 2 3.0 0.4-0.8 3 3 1.8 0.8-1.2 7 4 2.5 1.2-1.6 11 5 3.2 1.6-2.0 15 6 1.4 2.0-2.4 10 7 1.8 2.4-2.8 8 . . 2.8-3.2 4 . . . . 57 2.5 58 2.2 59 1.9 60 3.1
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2 Fig. 2. Number of frogs by weight class in Walden and Crystal Ponds ( N =60 in each). root of the variance is called the STANDARD DEVIATION , and is very often presented along with the mean (and the sample size, n , (number of frogs caught)) to describe the population (mean = 1.75 ± 0.7 g, n =60. Translation: Of the 60 frogs measured, the mean frog weighed 1.75 grams, with a variation (standard deviation) around that mean of 0.7 g).
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