Central tendency
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Central Tendency. Mechanics. 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.

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Central Tendency

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Central tendency

Central Tendency

Mechanics


Notation

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 X1

    • X = variable Xi = specific value


Example

Example

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

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


Summation

Summation

  • 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).


Example1

Example


Example2

Example

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


Calculations of measures of central tendency

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:


Median

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.


Median1

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.


The arithmetic mean

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?


Central tendency

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


Other means

Other Means

  • Geometric mean

  • Harmonic mean

  • Compare both to the Arithmetic mean of 3.8


Other means1

Other Means

  • Weighted mean

  • Multiply each score by the weight, sum those then divide by the sum of the weights.


Trimmed mean

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


Winsorized mean

Winsorized Mean

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?


M estimators

M-estimators

  • 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


M estimators1

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


M estimators2

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


M estimators3

M-estimators

  • Compare with the mean of 181


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