1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Four Measures of Central Tendency: The Mean, Median and Mode

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 2 New Statistical Notation An important symbol is , it is the Greek letter  and is called sigma The symbol  X means to sum (add) the X scores The symbol  X is pronounced “sum of X ”

3. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 3 What Is Central Tendency?

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 4 What is Central Tendency? A measure of central tendency is a score that summarizes the location of a distribution on a variable It is a score that indicates where the center of the distribution tends to be located

5. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 5 The Mode

6. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 6 The Mode The most frequently occurring score is called the mode The mode is typically used to describe central tendency when the scores reflect a nominal scale of measurement

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 7 Unimodal Distributions When a polygon has one hump (such as on the normal curve) the distribution is called unimodal.

8. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 8 Bimodal Distributions When a distribution has two scores that are tied for the most frequently occurring score, it is called bimodal.

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 9 The Median

10. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 10 The Median The median (Mdn) is the score at the 50th percentile The median is used to summarize ordinal or highly skewed interval or ratio scores

11. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 11 Determining the Median When data are normally distributed, the median is the same score as the mode. When data are not normally distributed, follow the following procedure: –Arrange the scores from lowest to highest –If there are an odd number of scores, the approximate median is the score in the middle position –If there are an even number of scores, the approximate median is the average of the two scores in the middle

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 12 The Mean

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 13 The Mean The mean is the score located at the mathematical center of a distribution The mean is used to summarize interval or ratio data in situations when the distribution is symmetrical and unimodal

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 14 Determining the Mean The formula for the sample mean is

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 15 On a perfect normal distribution all three measures of central tendency are located at the same score. Central Tendency and Normal Distributions

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 16 Central Tendency and Skewed Distributions

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 17 Deviations Around the Mean

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 18 A score’s deviation is the distance separate the score from the mean The formula for computing a score’s deviation is The sum of the deviations around the mean always equals 0 In symbols, this is Deviations

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 19 When using the mean to predict scores, a deviation indicates our error in prediction A deviation score indicates a raw score’s location and frequency relative to the rest of the distribution More About Deviations

20. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Population Mean The symbol for a population mean is  The formula for determining  is Chapter 4 - 20

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 21 Summarizing Research

22. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Summarizing an Experiment Summarize experiments by computing the mean of the dependent scores in each condition A relationship is present if the means from two or more conditions are different Chapter 4 - 22

23. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Line Graphs Create a line graph when the independent variable is an interval or a ratio variable. Chapter 4 - 23

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Bar Graphs Chapter 4 - 24 Create a bar graph when the independent variable is a nominal or an ordinal variable.

25. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Inferring the Relationship in the Population 1.Compute each sample mean to summarize the scores and the relationship found in the experiment 2.Perform the appropriate inferential procedure 3.Determine the location of the population of score by estimating  for each condition Chapter 4 - 25

26. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 26 14 13151115 131012131413 14151714 15 Example For the following data set, find the mode, the median, and the mean

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 27 Example Mode The mode is the most frequently occurring score In this data set, the mode is 14 with a simple frequency of 6

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 28 10111213 14 15 17 Example Median The median is the score at the 50th percentile. To find it, we must first place the scores in order from smallest to largest.

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 29 Since this data set has 18 observations, the median will be half-way between the 9th and 10th score in the ordered dataset. The 9th score is 14 and the 10th score also is 14. To find the midpoint, we use the following formula. The median, then is 14. Example Median

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 4 - 30 Example Mean For the mean, we need  X and N. We know that N = 18.

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms bar graph bimodal distribution deviation line graph mean measure of central tendency Chapter 4 - 31 median mode sum of the deviations around the mean sum of X unimodal distribution