Chapter 4 – 1 Chapter 4: Measures of Central Tendency What is a measure of central tendency? Measures of Central Tendency –Mode –Median –Mean Shape of the Distribution Considerations for Choosing an Appropriate Measure of Central Tendency

Chapter 4 – 2 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. What is a measure of Central Tendency?

Chapter 4 – 3 Central Tendency The mode The most frequently value. The median A value in the middle of distribution, 50% vs. 50%. The mean “the average”.

Chapter 4 – 4 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. The Mode

Chapter 4 – 5 The Mode: An Example Example: Number of Votes for Candidates for Mayor. The mode, in this case, gives you the “central” response of the voters: the most popular candidate. Candidate A – 11,769 votes The Mode: Candidate B – 39,443 votes “Candidate C” Candidate C – 78,331 votes

Chapter 4 – 6 Example Mode Mode = 7 Modal category = ’20-39’ X=678 f=243 age f=406055

Chapter 4 – 7 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.

Chapter 4 – 8 The Median Calculation 1 st, place the values in an array, from lowest to highest (sorting). 2 nd, find the median position by using the following formula Md. Pos = (n+1)/2 3 rd, find the value associated with the median position.

Chapter 4 – 9 Example X= Md. Pos = (9+1)/2 =5 Median = 7 X= Md. Pos = (6+1)/2 =3.5 Median=(7+7)/2=7 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p p1p1 p2p2 p3p3 p4p4 p5p5 p6p

Chapter 4 – 10 Median Exercise #1 (N is odd) Calculate the median for this hypothetical distribution: Job Satisfaction Frequency Very High2 High3 Moderate5 Low7 Very Low4 TOTAL21

Chapter 4 – 11 Median Exercise #2 (N is even) Calculate the median for this hypothetical distribution: Satisfaction with Health Frequency Very High 5 High 7 Moderate 6 Low 7 Very Low 3 TOTAL 28

Chapter 4 – 12 Finding the Median in Grouped Data L = lower limit of the interval containing the median W = width of the interval containing the median N = total number of respondents Cf = cumulative frequency corresponding to the lower limit f = number of cases in the interval containing the median

Chapter 4 – 13 Finding the Median in Grouped Data From this we can see that there are 50 results. To find the median add 1 to this number and halve the result. We are therefore looking for the 25.5th result. This lies in the fourth group, which starts at 24. We must work out how far though the group we must go. Our result is 1.5 into the group ( ) and the group has a frequency of 11. x(cm)(0,10](10,20](20,25](25,30](30,40](40,50] f cf

Chapter 4 – 14 Finding the Median in Grouped Data We therefore must go 1.5/11 of the way through the group The group covers values from 25cm to 30 cm, and therefore has a 5cm width. We need to find a fraction of 5cm. Therefore the median is 25+(1.5/11)=25.682

Chapter 4 – 15 Example x f cf

Chapter 4 – 16 Percentiles A score below which a specific percentage of the distribution falls. Finding percentiles in grouped data:

Chapter 4 – 17 The Mean The arithmetic average obtained by adding up all the scores and dividing by the total number of scores.

Chapter 4 – 18 The Mean Formula for the Mean the mean value of the variable x “X bar” equals the sum of all the scores, X, divided by the number of scores, N.

Chapter 4 – 19 The mean Formula for the mean w/ grouped scores where: f i X i = a score multiplied by its frequency

Chapter 4 – 20 Example X= Σx= =42 n=6 X= f= Σfx=2×8+2×7+6×2 =42 N=2+2+2=6

Chapter 4 – 21 Mean: Grouped Scores

Chapter 4 – 22 Mean: Grouped Scores

Chapter 4 – 23 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.

Chapter 4 – 24 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)

Chapter 4 – 25 Skewness When a curve is skewed, we indicate the direction of skewness with reference to the longer tail. Skewed to the right or positively skewed Skewed to the left or negatively skewed

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Chapter 4 – 31 Distribution Shape

Chapter 4 – 32 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, the mode and the median may be used. The median provides more information (taking into account the ranking of categories.) 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.

Chapter 4 – 33 Central Tendency