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Chapter Four

Chapter Four. Measures of Central Tendency: The Mean, Median and Mode. New Statistical Notation. An important symbol is S , it is the Greek letter S and is called sigma The symbol S X means to sum (add) the X scores The symbol S X is pronounced “sum of X ”. Chapter 4 - 2.

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Chapter Four

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  1. Chapter Four Measures of Central Tendency: The Mean, Median and Mode

  2. New Statistical Notation An important symbol is S, it is the Greek letter S and is called sigma The symbol SX means to sum (add) the X scores The symbol SX is pronounced “sum of X” Chapter 4 - 2

  3. What Is Central Tendency? Chapter 4 - 3

  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 Chapter 4 - 4

  5. The Mode Chapter 4 - 5

  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 Chapter 4 - 6

  7. Unimodal Distributions When a polygon has one hump (such as on the normal curve) the distribution is called unimodal. Chapter 4 - 7

  8. Bimodal Distributions When a distribution has two scores that are tied for the most frequently occurring score, it is called bimodal. Chapter 4 - 8

  9. The Median Chapter 4 - 9

  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 Chapter 4 - 10

  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 Chapter 4 - 11

  12. The Mean Chapter 4 - 12

  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 Chapter 4 - 13

  14. Determining the Mean The formula for the sample mean is Chapter 4 - 14

  15. Central Tendency and Normal Distributions On a perfect normal distribution all three measures of central tendency are located at the same score. Chapter 4 - 15

  16. Central Tendency andSkewed Distributions Chapter 4 - 16

  17. Deviations Aroundthe Mean Chapter 4 - 17

  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 Chapter 4 - 18

  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 Chapter 4 - 19

  20. Population Mean The symbol for a population mean is m The formula for determining m is Chapter 4 - 20

  21. Summarizing Research Chapter 4 - 21

  22. 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. Line Graphs Create a line graph when the independent variable is an interval or a ratio variable. Chapter 4 - 23

  24. Bar Graphs Create a bar graph when the independent variable is a nominal or an ordinal variable. Chapter 4 - 24

  25. Inferring the Relationship in the Population • Compute each sample mean to summarize the scores and the relationship found in the experiment • Perform the appropriate inferential procedure • Determine the location of the population of score by estimating m for each condition Chapter 4 - 25

  26. Example For the following data set, find the mode, the median, and the mean Chapter 4 - 26

  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 Chapter 4 - 27

  28. 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. Chapter 4 - 28

  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 Chapter 4 - 29

  30. Example Mean For the mean, we need SX and N. We know that N = 18. Chapter 4 - 30

  31. Key Terms • median • mode • sum of the deviations around the mean • sum of X • unimodal distribution bar graph bimodal distribution deviation line graph mean measure of central tendency Chapter 4 - 31

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