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Central Tendency

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Central Tendency

Mechanics

- 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 X1
- X = variable Xi = specific value

5,8, 12,3,6,8,7

X1, X2, X3, X4, X5, X6, X7

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

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

X2= 822 + 662 + 702 + 812 + 612 = 6724 + 4356 + 4900 + 6561 + 3721 = 26262

One can also see it as (X2)

(X)2= 3602 = 129600

- 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:

- 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.

- 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.

- 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?

Example: Slices of Pizza Eaten Last Week

Value

Freq

Value

Freq

0

4

8

5

1

2

10

2

2

8

15

1

3

6

16

1

4

6

20

1

5

6

40

1

6

5

- This raises the issue of which measure is best

- Geometric mean
- Harmonic mean
- Compare both to the Arithmetic mean of 3.8

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

- 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

1

2

2

2

3

3

3

3

3

3

4

4

4

4

4

5

5

6

8

10

3

3

3

3

3

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

- 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 M1 = 3.56
- M.20 = 3.533
- WM.20 = 3.75
- Mean = 3.95
- Which of these best represents the sample’s central tendency?

- Wilcox’s text example with more detail, to show the ‘gist’ of the calculation1
- 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

- 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

- 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

- Compare with the mean of 181