1. Central Tendency Mechanics

2. Notation When we describe a set of data corresponding to the values of some variable, we will refer to that set using an uppercase letter such as X or Y. When we want to talk about specific data points within that set, we specify those points by adding a subscript to the uppercase letter like X 1 –X = variable X i = specific value

3. Example 5,8, 12,3,6,8,7 X 1, X 2, X 3, X 4, X 5, X 6, X 7

4. Summation The Greek letter sigma, which looks like , means “add up” or “sum” whatever follows it. For example,  X i, means “add up all the X i s”. If we use the X i s from the previous example,  X i = 49 (or just  X).

5. Example

6.  X = 82 + 66 + 70 + 81 + 61 = 360  Y = 84 + 51 + 72 + 56 + 73 = 336  (X-Y) = (82-84) + (66-51) + (70-72) + (81-56) + (61-73) = -2 + 15 + (-2) + 25 + (-12) = 24  X 2 = 82 2 + 66 2 + 70 2 + 81 2 + 61 2 = 6724 + 4356 + 4900 + 6561 + 3721 = 26262 One can also see it as  (X 2 ) (  X) 2 = 360 2 = 129600

7. Calculations of Measures of Central Tendency Mode = Most commonly occurring value May have bimodal, trimodal etc. distributions. A uniform distribution is one in which every value has an equal chance of occurring Median The position of the median value can then be calculated using the following formula:

8. Median If there are an odd number of data points: (1, 2, 2, 3, 3, 4, 4, 5, 6) The median is the item in the fifth position of the ordered data set, therefore the median is 3.

9. Median If there are an even number of data points: (1, 2, 2, 3, 3, 4, 4, 5, 6, 793) The formula would tell us to look in the 5.5th place, which we can’t really do. However we can take the average of the 5th and 6th values to give us the median. In the above scenario 3 is in the fifth place and 4 is in the sixth place so we can use 3.5 as our median.

10. The Arithmetic Mean For example, given the data set that we used to calculate the median (odd number example), the corresponding mean would be: Note that they are not exactly the same. When would they be?

11. Example: Slices of Pizza Eaten Last Week ValueFreqValueFreq 0485 12102 28151 36161 46201 56401 65 This raises the issue of which measure is best

12. Other Means Geometric mean Harmonic mean Compare both to the Arithmetic mean of 3.8

13. Other Means Weighted mean Multiply each score by the weight, sum those then divide by the sum of the weights.

14. Trimmed mean You are very familiar with this in terms of the median, in which essentially all but the middle value is trimmed (i.e. a 50% trimmed mean) But now we want to retain as much of the data for best performance but enough to ensure resistance to outliers How much to trim? About 20%, and that means from both sides Example: 15 values..2 * 15 = 3, remove 3 largest and 3 smallest

15. Winsorized Mean Make some percentage of the most extreme values the same as the previous, non- extreme value Think of the 20% Winsorized mean as affecting the same number of values as the trimming Median = 3.5 Huber’s M 1 = 3.56 M.20 = 3.533 WM.20 = 3.75 Mean = 3.95 Which of these best represents the sample’s central tendency? 1 2 3 4 5 6 8 10  3333333333444445555533333333334444455555

16. M-estimators Wilcox’s text example with more detail, to show the ‘gist’ of the calculation 1 Data = 3,4,8,16,24,53 We will start by using a measure of outlierness as follows What it means: –M = median –MAD = median absolute deviation Order deviations from the median, pick the median of those outliers –.6745 = dividing by this allows this measure of variance to equal the population standard deviation When we do will call it MADN in the upcoming formula –So basically it’s the old ‘Z score > x’ approach just made resistant to outliers

17. M-estimators Median = 12 Median absolute deviation –-9 -8 -4 4 12 41  4 4 8 9 12 41 –MAD is 8.5, 8.5/.6745 = 12.6 So if the absolute deviation from the median divided by 12.6 is greater than 1.28, we will call it an outlier In this case the value of 53 is an outlier –(53-12)/12.6 = 3.25 –If one used the poorer method of using a simple z-score > 2 (or whatever) based on means and standard deviations, it’s influence is such that the z-score of 1.85 would not signify it as an outlier

18. M-estimators L = number of outliers less than the median –For our data none qualify U = number of outliers greater than the median –For our data 1 value is an upper outlier B = sum of values that are not outliers Notice that if there are no outliers, this would default to the mean

19. M-estimators Compare with the mean of 18 1