1. Chapter 4 Measures of Central Tendency

2. 2 Central Tendency Major Points Measures of central tendency summarize the average level or magnitude of a set of scores. Measures of central tendency summarize the average level or magnitude of a set of scores. Mode Mode Median Median Mean Mean Relative advantages and disadvantages Relative advantages and disadvantages Review questions Review questions

3. 3 Central TendencyMode The most commonly occurring value The most commonly occurring value There may be several There may be several Bimodal distribution has two distinct modes (peaks). Bimodal distribution has two distinct modes (peaks).

4. 4 Central TendencyMedian Center value in an ordered series, The score in the exact center of the distribution, at 50 th percentile Center value in an ordered series, The score in the exact center of the distribution, at 50 th percentile Average of two center values for an even number of points Average of two center values for an even number of points Median location Median location Even : take the average of the two center values Even : take the average of the two center values Ex 16 scores, Median = (8 th + 9 th )/2 Ex 16 scores, Median = (8 th + 9 th )/2 Odd number of scores: defined as (N + 1)/2 Odd number of scores: defined as (N + 1)/2 Ex 15 scores (15 + 1)/2 = 8 th score Ex 15 scores (15 + 1)/2 = 8 th score

5. 5 Central TendencyMean What we normally call the “average” What we normally call the “average” Denoted as “xbar” Denoted as “xbar” Calculated as Calculated as This will be our most commonly used statistic This will be our most commonly used statistic

6. 6 Central Tendency 14 Measures of Central Tendency Each measure of central tendency provides different informationEach measure of central tendency provides different information X Mean: Interval & Ratio Scale X Median: Interval Scale X Mode: Nominal Scale, lowest Each measure of CT is affected by the shape of the distribution differentlyEach measure of CT is affected by the shape of the distribution differently X If symmetrical & single peak (normal distribution), Mean is least biased estimator of the population. X If skewed (more at one end of scale) median or mode preferred. Con’t Can always use lower level, but never higher Can always use lower level, but never higher

7. 7 Central Tendency Relative Advantages Mode Mode The most frequently obtained score The most frequently obtained score Not influenced by extreme scores Not influenced by extreme scores Median Median Little influenced by extreme scores Little influenced by extreme scores Reasonable estimate of what most people mean by the center of a distribution Reasonable estimate of what most people mean by the center of a distribution Cont.

8. 8 Central Tendency Advantages --cont. Mean Mean Most commonly used statistic Most commonly used statistic Easily manipulated algebraically Easily manipulated algebraically Good statistical properties Good statistical properties

9. 9 Central TendencyDisadvantages Mode Mode May not represent a large proportion of the scores May not represent a large proportion of the scores Ignores extreme values completely Ignores extreme values completely Median Median Slightly less desirable statistical properties than mean Slightly less desirable statistical properties than mean May not be good to ignore extreme values May not be good to ignore extreme values Cont.

10. 10 Central Tendency Disadvantages --cont. Mean Mean Too easily influenced by extreme scores Too easily influenced by extreme scores Outliers: extreme values that stand out from the rest of the distribution, pull the mean Outliers: extreme values that stand out from the rest of the distribution, pull the mean Requires some faith in interval properties of measurement. Requires some faith in interval properties of measurement.

11. 11 Chapter 4 Central Tendency Review Questions List the advantages and disadvantages of each measure of central tendency. List the advantages and disadvantages of each measure of central tendency. Which measure would you use if you wanted to discuss performance on the next class test? Which measure would you use if you wanted to discuss performance on the next class test? If your class contains many older students, what measure of average income is appropriate? If your class contains many older students, what measure of average income is appropriate?