Download Lecture 15 - Measuring Center

Measuring Center
Lecture 15
Sections 5.1 – 5.2
Tue, Feb 14, 2006
Measuring the Center
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Often, we would like to have one number that
that is “representative” of a population or
sample.
It seems reasonable to choose a number that is
near the “center” of the distribution rather than
in the left or right extremes.
But there is no single “correct” way to do this.
Measuring the Center
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Mean – the simple average of a set of numbers.
Median – the value that divides the set of
numbers into a lower half and an upper half.
Mode – the most frequently occurring value in
the set of numbers.
Measuring the Center
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In a unimodal, symmetric distribution, these
values will all be near the center.
In other-shaped distributions, they may be
spread out.
The Mean
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We use the letter x to denote a value from the
sample or population.
The symbol  means “add them all up.”
So,
x
means add up all the values in the population or
sample (depending on the context).
Then the sample mean is  x
n
The Mean
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We denote the mean of a sample by the
symbolx, pronounced “x bar”.
We denote the mean of a population by ,
pronounced “mu” (myoo).
Therefore,
x
x
n
x

N
TI-83 – The Mean
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Enter the data into a list, say L1.
Press STAT > CALC > 1-Var Stats.
Press ENTER. “1-Var-Stats” appears.
Type L1 and press ENTER.
A list of statistics appears. The first one is the
mean.
See p. 301 for more details.
Examples
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Use the TI-83 to find the mean of the data in
Example 5.1(a), p. 301.
Weighted Means
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Continuing the previous example, suppose we
surveyed another group of households and
found the following number of children:
3, 2, 5, 2, 6.
Find the average of this group by itself.
Combine the two averages into one average for
all 15 households.
The Median
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Median – The middle value, or the average of
the middle two values, of a sample or
population, when the values are arranged from
smallest to largest.
The median, by definition, is at the 50th
percentile.
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It separates the lower 50% of the sample from the
upper 50%.
The Median
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When n is odd, the median is the middle
number, which is in position (n + 1)/2.
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Find the median of 3, 2, 5, 2, 6.
When n is even, the median is the average of the
middle two numbers, which are in positions n/2
and n/2 + 1.
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Find the median of 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.
The Median
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Alternately, we could calculate (n + 1)/2 in all
cases.
If it is a whole number, then use that position.
 If it is halfway between two whole numbers, take use
both positions and take the average.
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TI-83 – The Median
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Follow the same procedure that was used to find
the mean.
When the list of statistics appears, scroll down
to the one labeled “Med.” It is the median.
Use the TI-83 to find the medians of the
samples
3, 2, 5, 2, 6
 2, 3, 0, 2, 1, 0, 3, 0, 1, 4
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The Median vs. The Mean
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In the last example, change 4 to 4000 and
recompute the mean and the median.
How did the change affect the median?
How did the change affect the mean?
Which is a better measure of the “center” of
this sample?
The Mode
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Mode – The value in the sample or population
that occurs most frequently.
The mode is a good indicator of the
distribution’s central peak, if it has one.
Mode
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The problem is that many distributions do not
have a peak or have several peaks.
In other words, the mode does not necessarily
exist or there may be several modes.
Mean, Median, and Mode
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If a distribution is symmetric, then the mean,
median, and mode are all the same and are all at
the center of the distribution.
Mean, Median, and Mode
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However, if the distribution is skewed, then the
mean, median, and mode are all different.
Mean, Median, and Mode
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However, if the distribution is skewed, then the
mean, median, and mode are all different.
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The mode is at the peak.
Mode
Mean, Median, and Mode
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However, if the distribution is skewed, then the
mean, median, and mode are all different.
The mode is at the peak.
 The mean is shifted in the direction of skewing.
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Mode
Mean
Mean, Median, and Mode
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However, if the distribution is skewed, then the
mean, median, and mode are all different.
The mode is at the peak.
 The mean is shifted in the direction of skewing.
 The median is (typically) between the mode and the
mean.
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Mode Median Mean
Let’s Do It!
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Let’s Do It! 5.6, p. 309 – A Different
Distribution.
Do the same for the distribution
1/2
x
0
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