Measures of Central Tendency Measures of Central Tendency

Measures of Central Tendency

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

Measures of Central Tendency Mode = most frequently occurring data point

Measures of Central Tendency Mode = (3+4)/2 = 3. 5 Data Point Frequency 0 2 1 5 2 7 3 14 4 15 5 8 6 5

Measures of Central Tendency Median = the middle number when data are arranged in numerical order n n n 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

Measures of Central Tendency Median n n Median Location = (N +1)/2 = (56 + 1)/2 = 28. 5 Median = (3+4)/2 = 3. 5 Data Point Frequency 0 2 1 5 2 7 3 14 4 15 5 8 6 5

Measures of Central Tendency Mean = Average = X/N n X = 191 Mean = 191/56 = 3. 41 Data Point Frequency X 0 2 0 1 5 5 2 7 14 3 14 42 4 15 60 5 8 40 6 5 30

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 n n n 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?

Measures of Central Tendency Characteristics of the Mean n 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 Data Point + 10 Frequency X 0 10 2 20 1 11 5 55 2 12 7 84 3 13 14 182 4 14 15 210 5 15 8 120 6 16 5 80

Measures of Central Tendency Characteristics of the Mean n 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 10 x X = 1910 Mean = 1910/56 = 34. 1 Data Point x 10 Frequency X 0 0 2 0 1 10 5 50 2 20 7 140 3 30 14 420 4 40 15 600 5 50 8 400 6 60 5 300

Measures of Central Tendency Advantages and Disadvantages of the Measures: n 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)

Measures of Central Tendency Mode

Measures of Central Tendency Advantages and Disadvantages of the Measures: n 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

Measures of Central Tendency Advantages and Disadvantages of the Measures: n Mean 1. 2. 3. 4. 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

Measures of Central Tendency Advantages and Disadvantages of the Measures: n 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 n 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