02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 39 PART II Σigma Freud and Descriptive Statistics 39 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 40 40—— Statistics for People Who (Think They) Hate Statistics ne of the things that Sigmund Freud, the founder of psychoanalysis, did quite well was to observe and describe Othe nature of his patients’ conditions. He was an astute observer and used his skills to develop what was the first systematic and comprehensive theory of personality. Regardless of what you may think about the validity of his ideas, he was a good scientist. Back in the early 20th century, courses in statistics (like the one you are taking) were not offered as part of undergraduate or graduate curricula. The field was relatively new, and the nature of scientific explorations did not demand the precision that this set of tools brings to the scientific arena. But things have changed. Now, in almost any endeavor, numbers count (as Francis Galton, the inventor of correlation and a first cousin to Charles Darwin said as well). This section of Statistics for People Who (Think They) Hate Statistics . The Excel Edition is devoted to how we can use Excel’s most basic statistical functions to describe an outcome and better understand it. Chapter 2 discusses measures of central tendency and how com- puting one of several different types of averages gives you the one best data point that represents a set of scores. Chapter 3 completes the coverage of tools we need to fully describe a set of data points in its discussion of variability, including the standard deviation and variance. When you get to Chapter 4, you will be ready to learn how distributions, or sets of scores, differ from one another and what this difference means. Chapter 5 deals with the nature of relationships between variables, namely, correlations. When you finish Part II, you’ll be in excellent shape to start understanding the role that probability and inference play in the social and behavioral sciences. 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 41 2 Computing and Understanding Averages Means to an End Difficulty Scale ☺☺☺☺ (moderately easy) How much Excel? (a ton) What you’ll learn about in this chapter • Understanding measures of central tendency • Computing the mean for a set of scores using the AVERAGE function • Computing the mode using the MODE function • Computing the median for a set of scores using the MEDIAN function • Using the Data Analysis ToolPak to compute descriptive statistics • Selecting a measure of central tendency ou’ve been very patient, and now it’s finally time to get started working with some real, live data. That’s exactly Ywhat you’ll do in this chapter. Once data are collected, a usual first step is to organize the information using simple indexes to describe the data. The easiest way to do this is through comput- ing an average, of which there are several different types. An average is the one value that best represents an entire group of scores. It doesn’t matter whether the group of scores is the number correct on a spelling test for 30 fifth graders or the bat- ting percentage of each of the New York Yankees or the number of people who registered as Democrats or Republicans in the most recent election. In all of these examples, groups of data can be 41 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 42 42—— Part II ♦Σigma Freud and Descriptive Statistics summarized using an average. Averages, also called measures of central tendency, come in three flavors: the mean, the median, and the mode. Each provides you with a different type of information about a distribution of scores and is simple to compute and interpret. COMPUTING THE MEAN The mean is the most common type of average that is computed. It is simply the sum of all the values in a group, divided by the number of values in that group. So if you had the spelling scores for 30 fifth graders, you would simply add up all the scores and get a total and then divide by the number of students, which is 30. The formula for computing the mean is shown in Formula 2.1. X X--- = (2.1) n where • The letter X with a line above it (also called “X bar”) is the mean value of the group of scores or the mean. • The Σ, or the Greek letter sigma, is the summation sign, which tells you to add together whatever follows it. • The X is each individual score in the group of scores. • Finally, the n is the size of the sample from which you are computing the mean. To compute the mean, follow these steps: 1. List the entire set of values in one or more columns. These are all the Xs. 2. Compute the sum or total of all the values. 3. Divide the total or sum by the number of values. For example, if you needed to compute the average number of shoppers at three different locations, you would compute a mean for that value. 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 43 Chapter 2 ♦ Computing and Understanding Averages 43 Location Number of Annual Customers Lanham Park store 2,150 Williamsburg store 1,534 Downtown store 3,564 The mean or average number of shoppers in each store is 2,416. Formula 2.2 shows how it was computed using the formula you saw in Formula 2.1: --- X 2,150 + 1,534 + 3,564 7,248 X = = = = 2,416 (2.2) n 3 3 See, we told you it was easy. No big deal. And Now . Using Excel’s AVERAGE Function To compute the mean of a set of numbers using Excel, follow these steps. For some reason, the people who name functions would rather call the one that computes the mean AVERAGE, rather than MEAN. Yikes—these same folks used the name MEDIAN to name the function that computes the median and they assigned the name MODE to the function that computes the mode, so why not make everyone’s life easier and assign the name MEAN to the function that computes the average? If you find out, let us know. 1. Enter the individual scores into one column in a worksheet, such as you see in Figure 2.1. Figure 2.1 Data That Will Be Used to Compute an Average Score 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 44 44—— Part II ♦Σigma Freud and Descriptive Statistics 2. Select the cell (an intersection of a row and a column in a workbook) into which you want to enter the AVERAGE function. In this example, we are going to compute the mean in Cell A5. 3. Now, create a formula in any cell that would average the three values. The formula would look like this: = (A1 +A2 + A3)/3 or click on Cell A5 and type the Average function (which we did) as follows: = AVERAGE (A1:A3) and press the Enter key or use the Insert Function menu option and the “Inserting a Function” technique we talked about on page 26 in Chapter 1 to enter the AVERAGE function in Cell A5. Whether you type in a function or enter it using the Insert Function menu option, it looks the same and no one will ever, ever, ever know how you did it. Once it’s there, whether typed or inserted, it does exactly the same thing. As you can see in Figure 2.2, the mean was computed and the value returned to Cell A5. Notice that in the formula bar (where you can see the contents of a cell) in Figure 2.2, you can see the AVERAGE function fully expressed and the value computed as 2,416, just as we did manually earlier in the chapter. Figure 2.2 Using the AVERAGE Function to Compute the Mean of a Set of Numbers 02-Salkind-45022.qxd 6/21/2006 6:02 PM Page 45 Chapter 2 ♦ Computing and Understanding Averages 45 More Excel You may also want to explore the geometric mean (the function is GEOMEAN). The geometric mean uses multiplication rather than addition to summarize data values. It’s used when one expects that changes occur in a relative fashion in the data. THINGS TO REMEMBER The mean is sometimes represented by the letter M and is also called the typical, average, or most central score. If you are reading another statistics book or a research report, and you see something like M = 45.87, it probably means that the mean is equal to 45.87. • In the formula, a small n represents the sample size for which the mean is being computed. A large N (like this) would rep- resent the population size. In some books and in some journal articles, no distinction is made between the two. • The sample mean is the measure of central tendency that most accurately reflects the population mean. • The mean is like the fulcrum on a seesaw. It’s the centermost point where all the values on one side of the mean are equal in weight to all the values on the other side of the mean. • Finally, for better or worse, the mean is very sensitive to extreme scores. An extreme score can pull the mean in one direction or another and make it less representative of the set of scores and less useful as a measure of central tendency.