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Central Tendency
Statistics 2126
Introduction
• As useful as like histograms and such
are, it would be nice to describe data in
terms of Central Tendency
• A single number to describe a sample
• BTW, the Sample is a subset of the
population
• We are almost always dealing with
samples
Back when I was in first
year…
• 77 80 83 70 90
• Would be nice to describe how I did in
first year with a number
• Well the one we are all pretty used to is
the mean or arithmetic average
• The sum of all of the data points,
divided by the number of data points
The formula
n
x
x
i1
n
n
77  80  83  70  90

5
400

5
 80
The Mean
• Sort of a balancing point in the data
• Simply adding up the numbers and
dividing by the number of observations
(n)
• X bar is for the sample
• We might want to consider my first year
marks as a population
For a population
• The formula does not change, but the
symbol does
• We use statistics for samples
• We use parameters for populations
• 
• The formula is the same really
The mean is not mean
• In the population, the mean does not
change
• The sample, yeah it changes, sample to
sample
• Parameters do NOT CHANGE
However, the lecture is getting
meaner
• If you sample from a population you will
get different values for x bar each time
• We don’t care about samples in the long
run, we care about populations
• Calculating  is pretty hard, umm it
takes forever
• Used sometimes, elections, the census
Samples vs. populations
• A good sample will give you a killer
estimate of the population
• The census could be done via sampling
actually
• This is because x bar is an unbiased
estimator of 
• It overestimates as often as it
underestimates
• Weighted averages sometimes
• Some assignments worth more than
others for example
• There are other measures of central
tendency though
The median
•
•
•
•
No need for a formula here
50th percentile
Midpoint
Half below, half above
The mode
•
•
•
•
•
The most common observation
Virtually useless
Example 25 25 37 42 25
The mode is 25
Tough eh…
If….
• If the median =
mean = mode we
have a unimodal,
symmetrical
distribution
• Say IQ in the
population, all
measures of central
tendency = 100
Normal distribution
• You don’t have to get a normal
distribution when you have a unimodal,
symmetrical distribution
• It is probably the most common one
though
Why?
• Why do we need all of these measures
of central tendency?
• They all have different properties
• The mode is useless…
• So let’s move on
Median vs. the Mean
•
•
•
•
Say you have five numbers
12345
The mean is 3, as is the median
(BTW, the mode is umm well there are 5
of them)
• Add another value
• 750
Mean vs median in a final all
out battle to the death
• Now the mean is 127.5
• So adding an extreme value really
affects the mean
• Median is now umm let’s see
• 1 2 3 4 5 750
• 3.5
• cool
Median for the win
• So sometimes it is good
• Think about say union negotiations
• Both sides can talk about average
salary
• Both are right!
• In this case the median is more useful
So the median is useful
• Especially when there are outliers
• However you want to leave them in
• When you want to take all of the scores
into account though you have to use the
mean really
• All of our techniques are about means
• The median is, pretty much, a dead end
statistically
Running out of pithy titles
• The mean is most useful for
symmetrical distributions
• Most distributions we deal with will be
like this
• Most are pretty much symmetrical, more
or less