1. Anthony J Greene1 Central Tendency 1.Mean Population Vs. Sample Mean 2.Median 3.Mode 1.Describing a Distribution in Terms of Central Tendency 2.Differences Between Group Means as the Foundation of Research

2. Anthony J Greene2 Mean, Median and Mode The mean of a data set is the sum of the observations divided by the number of observations. The arithmetic average The median is the middle point of a distribution. The 50 th %ile The mode is the most frequently occurring score

3. Anthony J Greene3 Mean, Median and Mode Nominal or Categorical Variables. One cannot average categories or find the midpoint among them. Since categorical variables do not allow mathematical operations, only the mode can be used as a central tendency for categorical variables.

4. Anthony J Greene4 Mean, Median and Mode Ordinal Variables Since ordinal variables are ordered a midpoint or median may be obtained, but because the intervals are not even, the arithmetic average cannot be used. A mode may also be meaningfully obtained.

5. Anthony J Greene5 Mean, Median and Mode Interval and Ratio Variables. Because interval and ratio scales are evenly distributed, a mean may be obtained. Median and Mode may also be obtained. Median may be preferable when the distribution is skewed.

6. Anthony J Greene6 Mean, Median and Mode Nominal / Categorical: Mode only Ordinal: Median and Mode Interval and Ratio: Mean, Median and Mode

7. Anthony J Greene7 Mean The mean of the population of a discrete random variable X is denoted by  x or, when no confusion will arise, simply by . It is defined by Where N is the population size The terms expected value and expectation are commonly used in place of mean.

8. Anthony J Greene8 A population of N = 6 scores with a mean of  = 4. The mean does not necessarily divide the scores into two equal groups. In this example, 5 out of the 6 scores have values less than the mean.

9. Anthony J Greene9 The mean as the balance point A distribution of n = 5 scores with a mean of µ = 7.

10. Anthony J Greene10 The mean as the balance point

11. Anthony J Greene11 Mean -6 -15 -17 x 2 -20 -22 -23 -27 +1 +2 +4 +6 +9 x 3 +11 +14 x 2 +15 +16 +18 +20 μ = 33

12. Anthony J Greene12 Some New Notation Statistics quiz scores for a section of n = 8 students.

13. Anthony J Greene13 Some New Notation or f 10 + 9 + 9 + 8 + 8 + 8 + 8 + 6 = 66 Or 10 + 18 + 32 + 0 + 6 = 66

14. Anthony J Greene14 Population Versus Sample Mean

15. Anthony J Greene15 Sample Mean For a variable x, the mean of the observations for a sample is called a sample mean and is denoted M. Symbolically, we have where n is the sample size.

16. Anthony J Greene16 Samples and Populations Parameter: A descriptive measure for a population. Statistic: A descriptive measure for a sample

17. Anthony J Greene17 Samples and Populations Statistics Parameters M

18. Anthony J Greene18 M

19. Anthony J Greene19 Data Transformations Measurement of five pieces of wood.

20. Data Transformations Whether in Fahrenheit or Celsius, the information is identical. For this transformation C = (F-32)(5/9) In General: With ratio and interval scales you can 1.Add or subtract a constant 2.Multiply or divide by a constant DayF°F°C° Mon5814.4 Tues6216.7 Wed6820 Thurs7523.9 Fri5613.3 Sat5110.6 Sun6317.2 Average61.916.6

21. Anthony J Greene21 Median The median is the middle score or the 50 th %ile. Thus half the scores occur above the mean, and half occur below the mean. Could the mean and the median be different? If so, why?

22. Anthony J Greene22 The median divides the area in the graph in half

23. Anthony J Greene23 Median Arrange the data in increasing order. If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list. If the number of observations is even, then the median is the mean of the two middle observations in the ordered list. In both cases, if we let n denote the number of observations, then the median is at position (n + 1)/2 in the ordered list.

24. Anthony J Greene24 The median divides the area in the graph exactly in half.

25. Anthony J Greene25 The median divides the area in the graph exactly in half.

26. Anthony J Greene26 The First Trick About Medians: Dealing With an Even Number of Scores 121124126129135191 In this Simple Case, simply take the mean of the two middle scores, 127.5

27. Anthony J Greene27 The Second Trick About Medians: What Happens When There Are Several Instances of the Middle Score The most basic rule is that there have to be as many above the median as below, in this case 5

28. Anthony J Greene28 A Direct Comparison of Mean and Median Consider a sample of three scores: 5, 7, 9 Mean and Median are identical Consider a second sample: 5, 7, 28 Mean is affected, median is not Median is insensitive to extreme scores. When the mean and median differ, the distributions is skewed.

29. Anthony J Greene29 Mode Obtain the frequency of occurrence of each value and note the greatest frequency. If the greatest frequency is 1 (i.e., no value occurs more than once), then the data set has no mode. If the greatest frequency is 2 or greater, then any value that occurs with that greatest frequency is called a mode of the data set.

30. Anthony J Greene30 Mode: Favorite restaurants

31. Anthony J Greene31 Describing Distributions by Central Tendency Mean, Median and Mode are identical Not a naturally occurring distribution

32. Anthony J Greene32 Describing Distributions by Central Tendency No Mode

33. Anthony J Greene33 Describing Distributions by Central Tendency Mode is the lowest or highest score

34. Anthony J Greene34 Describing Distributions by Central Tendency Median and Mean are different: For right skewed the median is lower than the mean, for left skewed, the median is higher than the mean.

35. Anthony J Greene35 Describing Distributions by Central Tendency More than one Mode.

36. Anthony J Greene36 Describing Distributions by Central Tendency

37. Anthony J Greene37 The Basic Idea of Experimental Design Are two (or more) means different from one another? e.g., experimental vs. control group.

38. Anthony J Greene38 Differences Between Means: Maze Learning

39. Anthony J Greene39 The mean number of errors made on the task for treatment and control groups according to gender.

40. Anthony J Greene40 Amount of food (in grams) consumed before and after diet drug injections.

41. Anthony J Greene41 The relationship between an independent variable (drug dose) and a dependent variable (food consumption). Because drug dose is a continuous variable, a continuous line is used to connect the different dose levels.

42. Anthony J Greene42 The Basic Idea of Experimental Design Are two (or more) means different from one another: e.g., experimental vs. control group. Remember that the means will always differ somewhat by chance factors alone. In the next chapter we will explore how to measure the “spread” of a variable which, ultimately, will be the basis for understanding how far apart means must be to not be attributable to chance factors

43. Anthony J Greene43 Significant Differences? μ 1 = 40 μ 2 =60

44. Anthony J Greene44 Significant Differences? μ 1 = 40 μ 2 =60