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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 SX means to sum (add) the X scores • The symbol SX is pronounced “sum of X” Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 2 What Is Central Tendency? Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 3 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 4 The Mode Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 5 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 6 Unimodal Distributions When a polygon has one hump (such as on the normal curve) the distribution is called unimodal. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 7 Bimodal Distributions When a distribution has two scores that are tied for the most frequently occurring score, it is called bimodal. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 8 The Median Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 9 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 10 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 median is the score in the middle position. – If there are an even number of scores, the median is the average of the two scores in the middle. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 11 The Mean Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 12 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 13 Determining the Mean • The formula for the sample mean is SX X N Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 14 Central Tendency and Normal Distributions On a perfect normal distribution all three measures of central tendency are located at the same score. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 15 Central Tendency and Skewed Distributions Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 16 Deviations Around the Mean Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 17 Deviations • A score’s deviation is the distance separate the score from the mean • In symbols, this is S ( X X ) • The sum of the deviations around the mean always equals 0. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 18 More About Deviations • When using the mean to predict scores, a deviation ( X X ) indicates our error in prediction. • A deviation score indicates a raw score’s location and frequency relative to the rest of the distribution. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 19 Example • For the following data set, find the mode, the median, and the mean 14 14 13 15 11 15 13 10 12 13 14 13 14 15 17 14 14 15 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 20 Example Mode • The mode is the most frequently occurring score. • In this data set, the mode is 14 with a simple frequency of 6. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 21 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. 10 11 12 13 13 13 13 14 14 14 14 14 14 15 15 15 15 17 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 22 Example Median • 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. 14 14 14 20 • The median, then is 14. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 23 Example Mean • For the mean, we need SX and N. We know that N = 18. SX 246 SX 246 X 13.67 N 18 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 4 - 24