Measures of Central Tendency

1. Measures of Central Tendency

2. Measures of Central Tendency • Central Tendency = values that summarize/ represent the majority of scores in a distribution • Three main measures of central tendency: • Mean ( = Sample Mean; μ = Population Mean) • Median • Mode

3. Measures of Central Tendency • Mode = most frequently occurring data point

4. Measures of Central Tendency • Mode = (3+4)/2 = 3.5

5. Measures of Central Tendency • Median = the middle number when data are arranged in numerical order • Data: 3 5 1 • Step 1: Arrange in numerical order 1 3 5 • Step 2: Pick the middle number (3) • Data: 3 5 7 11 14 15 • Median = (7+11)/2 = 9

6. Measures of Central Tendency • Median • Median Location = (N +1)/2 = (56 + 1)/2 = 28.5 • Median = (3+4)/2 = 3.5

7. Measures of Central Tendency • Mean = Average = X/N • X = 191 Mean = 191/56 = 3.41

8. Measures of Central Tendency • Occasionally we may need to add or subtract, multiply or divide, a certain fixed number (constant) to all values in our dataset • i.e. this is essentially what is done when curving a test • What do you think would happen to the average score if 4 points were added to each score? • What would happen if each score was doubled?

9. Measures of Central Tendency • Characteristics of the Mean • Adding or subtracting a constant from each score also adds or subtracts the same number from the mean • i.e. adding 10 to all scores in a sample will increase the mean of these scores by 10 X = 751 Mean = 751/56 = 13.41

10. Measures of Central Tendency • Characteristics of the Mean • Multiplying or dividing a constant from each score has similar effects upon the mean • i.e. multiplying each score in a sample by 10 will increase the mean by 10x X = 1910 Mean = 1910/56 = 34.1

11. Measures of Central Tendency • Advantages and Disadvantages of the Measures: • Mode • Typically a number that actually occurs in dataset • Has highest probability of occurrence • Applicable to Nominal, as well as Ordinal, Interval and Ratio Scales • Unaffected by extreme scores • But not representative if multimodal with peaks far apart (see next slide)

12. Measures of Central Tendency • Mode

13. Measures of Central Tendency • Advantages and Disadvantages of the Measures: • Median • Also unaffected by extreme scores Data: 5 8 11 Median = 8 Data: 5 8 5 million Median = 8 • Usually its value actually occurs in the data • But cannot be entered into equations, because there is no equation that defines it • And not as stable from sample to sample, because dependent upon the number of scores in the sample

14. Measures of Central Tendency • Advantages and Disadvantages of the Measures: • Mean • Defined algebraically • Stable from sample to sample • But usually does not actually occur in the data • And heavily influenced by outliers Data: 5 8 11 Mean = 8 Data: 5 8 5 million Mean = 1,666,671

15. Measures of Central Tendency • Advantages and Disadvantages of the Measures: • Mean • Often you will see sums quoted instead of average or mean values, you should be wary of these statistics because they are easily skewed • i.e. Statistics for the performance of a basketball player are quoted in the newspaper, it says that he has 134 points over the course of the season, whereas other players average well over 200. • From this you would conclude that he is a mediocre player at best, however, it is possible that he has played fewer games than other players (due to injury) • Looking at averages, the player actually averages ~50 pts. per game, but has only played three games, whereas other players average 20 or less pts. over more games • Using this much richer information, our conclusions would be completely different – AVERAGES ARE ALWAYS MORE INFORMATIVE THAN SIMPLE SUMS