ZOL 415, Spring 2006 Statistics pg 1 of 9 Some basic statistical concepts to help with reading journal articles. We will be reading scientific research reports that base their conclusions on statistical analyses and we need to be able to judge whether the conclusions are valid, or whether the data show something other than what the authors have inferred. I will introduce some of the basic ideas below, using some simple exercises we can do in class. We will learn more as we go along from examples in our readings. For a nice, humorous introduction, see: Gonick. G. & W. Smith. 1993. The cartoon guide to statistics. HarperPerenial. New York, NY. For a less humorous introduction to applications in Behavioral Ecology, see Brown. L. & J.F. Downhower. 1988. Analyses in behavioral ecology. Sinauer. Describing & comparing two things. In general, the way we understand things in nature is to describe them and try to discover how they relate to or interact with other things. Ideally, we can use this understanding to make useful predictions. The things might be a trait, like height, for two objects: Bob and Mary. In this case, Bob is a population of one and Mary is a population of one. Suppose Bob is 6 units tall and Mary is 5 units tall. In this case, 6 is also the average height of the population of one Bob: 6 = 6/1. There is no variation in the “variable” height in the population of one Bob. If we ask 'Is Bob tall?' it wouldn't be a well-defined question. If we ask 'Is Bob taller than Mary?' the answer is straightforward: Yes, 6>5. We didn't need statistics for this and we didn't need to make any statistical inferences because we could measure the entire population of one Bob and the entire population of one Mary and just compare the trait value of each. We have described the two populations and their relationship, but this has no generality or predictive value beyond the little universe of Bob and Mary. Describing & comparing two populations. Usually, the things we examine are classes of objects, or populations bigger than one. For instance, consider the population of all males in this class and the population of all females in this class. Both of these populations are small enough to measure. Suppose we want to describe and compare the heights of the men in the class and the women in the class. Height is a variable; it varies across individuals within each population. Some are relatively short, some relatively tall, and some about average. We call this a distribution of the variable height. There are various ways to describe a variable distribution. We’ll illustrate a few using the heights of students in class.ZOL 415, Spring 2006 Statistics pg 2 of 9 Describing & Comparing The Variable Trait 'Height' in Two Populations: Males & Females in This Class: In class, we will measure the heights of students (it is convenient to do this in units of cinderblocks or bricks if the walls are accommodating, otherwise, we can resort to some other arbitrary unit, like inches). Fill out the Name & Height columns. We'll return to the squared deviations and averages below. Female Students Male students Initials Height Squared deviation from mean Initials Height Squared deviation from mean Total = Total = # Students = # Students = Average = Average = Standard deviation = Standard deviation =ZOL 415, Spring 2006 Statistics pg 3 of 9 Next, we will describe the populations of males and females with two frequency histograms. A frequency histogram is a bar graph. The x-axis shows the range of trait values. The y-axis shows a bar at each trait value whose height is proportional the number of individuals having that trait value (you should be familiar with these graphs from 'class curves,' which are frequency distributions of scores within a class). In a frequency histogram, the y-axis should be converted from the number having a particular x-value to the proportion having value x. That way we can compare the male and female histograms on the same y-axis, regardless of whether there are the same number of males and females. You can convert the y-axis scale without re-drawing the graph; just make a new scale on the right. We can see a lot by just looking at frequency histograms. • Describe and compare the two histograms. Do they overlap much along the x-axis? Is the difference between groups large compared to the variation within groups (this is the essence of most statistical tests)? We would like to have some precise, quantitative ways to describe these variable distributions. You are already familiar with one of these descriptive statistics, the mean, or average trait value. • Compare the mean height for females and for males. 012345678910x = Female Height Trait Values012345678910x = Male Height Trait Valuesy = Number having value xZOL 415, Spring 2006 Statistics pg 4 of 9 Another important property is variation around the mean. A simple measure of this variation is the variance. The variance is the average squared deviation from the mean value. • Go back to the data table and calculate the average of the squared deviations. This is the variance. • The square root of the variance is known as the standard deviation. It is the average absolute deviation from the mean, ignoring the direction or sign. Calculate the standard deviation for both populations. When we finally get to the point where we want to make statistical inferences about populations from data on samples, we will find that the amount of variation within and between groups is crucial. Inferring population properties from sample properties. If the students in this class are an unbiased sample of students at the University, then we can use statistics to make inferences about the larger, unmeasured population 'students at the University' from the smaller, measured sample 'students in this class.' This could save us time and trouble but it would come at a price - uncertainty. When we make a statistical inference from a sample to a population, we can never be 100% confident of the result. As a matter of convention, statisticians have agreed that being right 95% of the time will just have to do. This is why you will often see the statement 'p < 0.05'. This means that we estimate that the probability