1. Measures of Central Tendency

2. Central Tendency = values that summarize/ represent the majority of scores in a distribution 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) MedianMode

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

4. Measures of Central Tendency Mode = (3+4)/2 = 3.5 Data Point Frequency 02 15 27 314 415 58 65

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

6. Measures of Central Tendency Median Median Location = (N +1)/2 = (56 + 1)/2 = 28.5 Median Location = (N +1)/2 = (56 + 1)/2 = 28.5 Median = (3+4)/2 = 3.5 Median = (3+4)/2 = 3.5 Data Point Frequency 02 15 27 314 415 58 65

7. Measures of Central Tendency Mean = Average =  X/N  X = 191Mean = 191/56 = 3.41  X = 191Mean = 191/56 = 3.41 Data Point FrequencyX 020 155 2714 31442 41560 5840 6530

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. curving a test i.e. curving a test What do you think would happen to the average score if 4 points were added to each score? 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? 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 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 = 751Mean = 751/56 = 13.41 Data Point + 10 FrequencyX 010220 111555 212784 31314182 41415210 5158120 616580

10. Measures of Central Tendency Characteristics of the Mean Multiplying or dividing a constant from each score has similar effects upon 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 Data Point x10 x10FrequencyX 0020 110550 2207140 33014420 44015600 5508400 6605300

11. Measures of Central Tendency Advantages and Disadvantages of the Measures: Mode Mode 1.Typically a number that actually occurs in dataset 2.Has highest probability of occurrence 3.Applicable to Nominal, as well as Ordinal, Interval and Ratio Scales 4.Unaffected by extreme scores 5.But not representative if multimodal with peaks far apart (see next slide)

12. Measures of Central Tendency Mode

13. Advantages and Disadvantages of the Measures: Median Median 1.Also unaffected by extreme scores Data: 5 8 11 Median = 8 Data: 5 8 5 million Median = 8 2.Usually its value actually occurs in the data 3.But cannot be entered into equations, because there is no equation that defines it 4.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 Mean 1.Defined algebraically 2.Stable from sample to sample 3.But usually does not actually occur in the data 4.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 Mean Sums/totals vs. average or mean values i.e. Basketball player has 134 total points this season, while average of other players is 200 points i.e. Basketball player has 134 total points this season, while average of other players is 200 points What would most people reasonably conclude? What would most people reasonably conclude?

16. Measures of Central Tendency What if he played fewer games than other players (due to injury)? What if he played fewer games than other players (due to injury)? Looking at averages, the player actually averaged ~50 pts. per game, but has only played three games, whereas other players average 20 or less pts. over more games Looking at averages, the player actually averaged ~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 Using this much richer information, our conclusions would be completely different – AVERAGES ARE ALWAYS MORE INFORMATIVE THAN SIMPLE SUMS