SOCY201 Lecture Notes 2 9 2012 Measures of Central Tendency Measures of Central Tendency We re familiar with average value concepts from many situations College GPA Goals Against Average GAA o GAA goals mins 60 Baseball batting averages Dow Jones stock market index Unemployment rates A measure of central tendency is a single number used to represent the center of a group of Different variables may possess different numerical characteristics so different measures of central tendency should be considered The basic measures are the Less information data o Mode o Median o Mean This class of measures can be calculated on grouped or ungrouped data The difference is in how the data values are weighted More information The Mode The mode is the group that contains the most cases in grouped data Or the most frequently occurring value in a group of raw scores Ex In the majority 75 of dual earner married couples the husband earns more money than his wife U S Department of Labor Bureau of Labor Statistics Mode for Grouped Data A grouped frequency distribution displays the numbers of cases with the same value or score The grouped data mode is either the category or score with the largest frequency or percentage Mode for Ungrouped Data Mode the single category among the k categories in a distribution with the largest number or highest percentage of observations Median Median md an outcome dividing an ordered frequency distribution exactly into halves o Caution Don t try to compute the median of an unordered variable because it doesn t make sense SOCY201 Lecture Notes 2 9 2012 Measures of Central Tendency We re familiar with average value concepts from many situations College GPA Goals Against Average GAA o GAA goals mins 60 Baseball batting averages Dow Jones stock market index Unemployment rates A measure of central tendency is a single number used to represent the center of a group of Different variables may possess different numerical characteristics so different measures of central tendency should be considered The basic measures are the Less information data o Mode o Median o Mean This class of measures can be calculated on grouped or ungrouped data The difference is in how the data values are weighted More information The Mode The mode is the group that contains the most cases in grouped data Or the most frequently occurring value in a group of raw scores Ex In the majority 75 of dual earner married couples the husband earns more money than his wife U S Department of Labor Bureau of Labor Statistics Mode for Grouped Data A grouped frequency distribution displays the numbers of cases with the same value or score The grouped data mode is either the category or score with the largest frequency or percentage Mode for Ungrouped Data Mode the single category among the k categories in a distribution with the largest number or highest percentage of observations Median Median md an outcome dividing an ordered frequency distribution exactly into halves o Caution Don t try to compute the median of an unordered variable because it doesn t make sense Median for Grouped Data Find the intervals Divide intervals by cases Count the sub intervals from the lower true limit until you come to the median Pi Lp pi n cp fp Wi o Pi the score of the ith percentile o Lp true lower limit of the interval containing the ith percentile o pi ith percentile written as a proportion o n total number of observations o cp cumulative frequency up to but not including the interval containing pi o fp frequency in the intercal containing the ith percentile o Wi width of the interval containing pnW UpLp Median for Ungrouped Data The median is defined as the middle value case of n values of Y objects arranged in order of size For an odd number of cases the middle case will be equal to the n 1 2 case For an even number of cases the middle case will be halfway between the n 2 and the n 2 1 case Mean For populations Y Y1 Y2 YN N Variable Y has n total observations cases The ith case is denoted by subscript i The ith case s score is Yi Summation add all values from i 1st to nth case Divide this sum of all scores by n observations The Mean as the Center of Gravity The mean has the algebraic property that the sum of the deviations of each score from the mean will always be zero The sum of the squared deviations of each score from the mean is less than the sum of the squared deviations from any other constant number