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.

• 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

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

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

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).

Example

Pred. Actual Student Score Score X Y

1 82 84 2 66 51 3 70 72 4 81 56 5 61 73

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 = 26262One can also see it as (X2)

(X)2 = 3602 = 129600

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 Location = N + 12

Median

Median Location = 9 + 12

= 5

• 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.

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

303.33

9

XX

N

• 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?

Mode = 2 slices per week

Median = 4 slices per week

Mean = 5.7 slices per week

Example: Slices of Pizza Eaten Last Week

Value Freq Value Freq

0 4 8 51 2 10 22 8 15 13 6 16 14 6 20 15 6 40 16 5

• This raises the issue of which measure is best

Other Means

• Geometric mean

• Harmonic mean

• Compare both to the Arithmetic mean of 3.8

nnxxxxxGM ...4321

448.348054622

5,4,6,2,255

GM

X i

nxxxxx

nHM

1...

4

1

3

1

2

1

1

1

093.35

5,4,6,2,2

5

1

4

1

6

1

2

1

2

1

HM

X i

Other Means

• Weighted mean• Multiply each score

by the weight, sum those then divide by the sum of the weights.

i

ii

w

xwWM

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• 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?

122233333344444556810

33333333334444455555

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

1.28/ .675

X M

MAD

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-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

1.28( )( )MADN U L BMest

n L U

M-estimators

• Compare with the mean of 181

226.14106

55)01)(6.12(28.1

estM