CHAPTER 3: MEASURES OF CENTRAL TENDENCY Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chapter 3: Measures of Central Tendency  The Mode  The Median  The Mean  Finding the Mean in a Frequency Distribution  The Shape of the Distribution  Considerations for Choosing a Measure of Central Tendency  Statistics in Practice: Representing Income

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications What is a measure of Central Tendency? Numbers that describe what is average or typical of the distribution You can think of this value as where the middle of a distribution lies.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications The Mode The category or score with the largest frequency (or percentage) in the distribution. The mode can be calculated for variables with levels of measurement that are: nominal, ordinal, or interval-ratio.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications The Mode: An Example  Example: Number of Votes for Candidates for Mayor of Camarillo, California. The mode, in this case, gives you the “central” response of the voters: the most popular candidate. Sheriff Tupper – 11,769 votes The Mode: Jessica Fletcher – 39,443 votes“Dr. Seth Hazlett” Dr. Seth Hazlett – 78,331 votes

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications The Median  The score that divides the distribution into two equal parts, so that half the cases are above it and half below it.  The median is the middle score, or average of middle scores in a distribution.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Median Exercise #1 (N is odd) Calculate the median for this hypothetical distribution: Job Satisfaction Frequency Very High2 High3 Moderate5 Low7 Very Low4 TOTAL21

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Median Exercise #2 (N is even) Calculate the median for this hypothetical distribution: Satisfaction with Health Frequency Very High5 High7 Moderate6 Low7 Very Low3 TOTAL28

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Percentiles  A score below which a specific percentage of the distribution falls.  For example, the 75 th percentile is a score that divides the distribution so that 75% of the cases are below it.  For example, the 25 th percentile is a score that divides the distribution so that 25% of the cases are below it.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications The Mean  The arithmetic average obtained by adding up all the scores and dividing by the total number of scores.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Formula for the Mean Where Σ Y = sum of all scores N = the number of scores.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Calculating the mean with grouped scores where: f Y = a score multiplied by its frequency

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Example: Mean of Grouped Scores

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Mean: Grouped Scores

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Grouped Data: the Mean & Median Number of People Age 18 or older living in a U.S. Household in 1996 (GSS 1996) Number of People Frequency TOTAL581 Calculate the median and mean for the grouped frequency below.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Shape of the Distribution  Symmetrical (mean is about equal to median)  Skewed  Negatively (example: years of education) mean < median  Positively (example: income) mean > median  Bimodal (two distinct modes)  Multi-modal (more than 2 distinct modes)

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Distribution Shape

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Considerations for Choosing a Measure of Central Tendency  For a nominal variable, the mode is the only measure that can be used.  For ordinal variables: -Use the mode to show what is the most common value in the distribution. -Use the median to show which value is located exactly in the middle of the distribution.  For interval-ratio variables, the mode, median, and mean may all be calculated. The mean provides the most information about the distribution, but the median is preferred if the distribution is skewed.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Central Tendency