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• Measures of central tendency – We like to boil things down. It’s easier to report a single number than many – In the behavioral and natural sciences, the frequency of many data sets are symmetrical For a normal distribution, one should be able to imagine the frequency distribution histogram if given the mean and standard deviation. For a normal distribution, the mean median and mode are the same number. X f 2 3 4 5 6 7 8 X= Calculate the mean (a.k.a. M or X for a sample and μ for a population). More terminology for populations vs. samples • Population mean μ = ΣX N • Sample mean M = ΣX n X 1 2 3 4 5 6 7 8 9 X= f X 1 2 3 4 5 6 7 8 9 X= f The mean is the arithmetic average, but it is also the “balance point.” The total distance from the mean to all points above is equal to the total distance from the mean to all points below. The distance between the mean and all points determines variance. More on this later… Sometimes the means from two or more samples need to be combined. The importance of each mean is determined by the size of the respective sample, so you shouldn’t simply combine means and divide by two. • Example – Two classes take an exam • Class 1: n = 4 and M = 65% • Class 2: n = 45 and M = 89% – Calculating overall mean “unfairly” leads to 65% + 89% = 77% 2 The two courses should not be treated equally since one has many more students than the other. The larger class has more weight and should have more influence in calculating the overall mean. • Example – Two classes take an exam • Class 1: n = 4 and M = 65% – This is like having 4 scores of 65% (ΣX1 = 260%) • Class 2: n = 45 and M = 89% – This is like having 45 scores of 89% (ΣX2 = 4005%) – Weighted mean = ΣX1 + ΣX2 = 256 + 4005 n1 + n 2 4 + 45 = 87% • The weighted mean is the same number you would get if you had totaled all of the original scores and divided by N. • The weighted mean calculation is a shortcut. You can also calculate it by multiplying each sample mean by the proportion of total participants found in each sample. The two courses should not be treated equally since one has many more students than the other. The larger class has more weight and should have more influence in calculating the overall mean. • Example – Two classes take an exam • Class 1: n = 4 and M = 65% – This is like having 4 scores of 65% (ΣX1 = 256%) • Class 2: n = 45 and M = 89% – This is like having 45 scores of 89% (ΣX2 = 4005%) Weighted mean = ΣX1p1 + ΣX2p2 = 65% (4/49) + 89% (45/49) = 87% • Influencing the mean: – It should be obvious to you that changing a score, adding a score, or removing a score will change the mean – Adding or subtracting a constant from each score will change the mean by the same amount – Multiplying or dividing each score by a constant will change the mean in the same way • Think about converting meters to yards or ounces to grams Figure 3.3 (p. 63) - Consider the changes just described as applied to the scale model Table 3.2 (p. 64) The weight of a freight truck with and without a 200 lb. passenger. Truck A B C D E F Passenger 1200 1600 1400 2100 2500 1400 X=1700 No Passenger 1000 1400 1200 1900 2300 1200 X= Table 3.3 (p. 65) • The median is the score that divides the distribution in half so that 50% are above and 50% are below. – If number of scores is odd, it is the middle score. Otherwise it is the average of the two middle scores Figure 3.4 (p. 66) - The median divides the area of a frequency distribution histogram in half. (Blue = White) Figure 3.5 (p. 67) 2.25 Figure 3.6 (p. 67) Sometimes the result looks strange, but the rule still applies. Though you may round numbers off to suit your study. It may be awkward to report some variables, such as children in a household, as non-discrete. Median = 2.5 Figure 3.7 (p. 68) mean = 4, median = 2.5 For a skewed distribution, the mean is pulled away from the median in the direction of the skew. Be careful!! The mode is Luigi’s, NOT 42 Table 3.4 (p. 70) The mode is a score or a category that has the greatest frequency. Figure 3.8 (p. 70) The term “bi-modal” (or multi-modal) is casually used to describe a distribution with more than one clear peak. Figure 3.9 (p. 72) Use the mode when describing distributions with few unusually high or low scores. What’s the mean? Table 3.5 (p. 74) Use it in cases in which there is no data available for a participant. Table 3.6 (p. 75) Illustrating means in discrete groups. Figure 3.10 (p. 76) Illustrating means in continuous groups. Figure 3.11 (p. 76) Figure 3.12 (p. 77) Figure 3.13 (p. 78) Using a Box-and-Whisker Plot to Illustrate skewed distributions. • Two classes have the following quiz 6 f1 f2 1 5 1 2 8 0 3 4 0 4 3 1 5 2 3 6 1 5 7 0 8 8 1 2 9 0 2 10 0 1 4 3 2 1 0 1 2 3 4 5 Scores for Class 1 Frequency x Frequency 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 1 2 3 Scores for Class 2 4 5 These scores cannot simply be reported with a mean and standard deviation Basics of SPSS (Appendix D in your book) • Start SPSS • Start by defining variables by clicking on the “Variable View” Tab Basics of SPSS • We have two variables to consider – Class – Score • Define these variables and return to “Data View” •Generating Descriptive Statistics Choose the variable of interest and move it to the “variable “ window” Select the “Charts” button to display a frequencydistribution histogram •Generating Descriptive Statistics Select “Explore” to Create Box-and-Whisker Plots The Box has the mean going through it as well as the 25th and 75th percentiles (1st and 3 quartiles) •The plot splits scores into four equal groups by defining the first, second, and third quartiles 3 2 1 6 Frequency 5 4 3 2 1 0 1 2 3 4 5 Scores for Class 1 Frequency •Visualizing the distribution with Boxand-Whisker Plots 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 1 2 3 Scores for Class 2 4 5