Download STAT1010 – mean (or average)

STAT1010 – mean (or average)
Chapter 4: Describing data
!  4.1
Averages and measures of center
" Describing
!  4.2
" Describing
!  4.3
the center of a distribution
Shapes of distributions
the shape
Quantifying variation
" Describing
the spread of a distribution
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4.1 What is an average?
!  In
statistics, we generally use the term
mean instead of average, and the mean
has a specific formula…
mean =
sum of all values
total number of values
!  The
term average could be interpreted in
a variety of ways, thus, we’ll focus on the
mean of a distribution or set of numbers.
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Example:
Eight grocery stores sell the PR energy bar for the following
prices:
$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79
Find the mean of these prices.
Solution:
The mean price is $1.41:
mean =
$1.09 + $1.29 + $1.29 + $1.35 + $1.39 + $1.49 + $1.59 + $1.79
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= $1.41
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STAT1010 – mean (or average)
Example: Octane Rating
n = 40
87.4, 88.4, 88.7, 88.9, 89.3, 89.3, 89.6, 89.7
89.8, 89.8, 89.9, 90.0, 90.1, 90.3, 90.4, 90.4
90.4, 90.5, 90.6, 90.7, 91.0, 91.1, 91.1, 91.2
91.2, 91.6, 91.6, 91.8, 91.8, 92.2, 92.2, 92.2
92.3, 92.6, 92.7, 92.7, 93.0, 93.3, 93.7, 94.4
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Example: Octane Rating
Technical Note (short hand formula for the mean):
Let x1, x2, …, xn represent n values. Then,
" n %
$ ∑ xi '
Total sum x1 + x2 +... + xn # i=1 &
mean = x =
=
=
n
n
n
In the octane example,
mean = x =
x1= 87.4, x2 = 88.4, etc. and n = 40.
Total sum 3637.9
=
= 90.95
n
40
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Rounding Rule for
Statistical Calculations
Rounding Rule for Statistical Calculations
State your answers with one more decimal place of
precision than is found in the raw data.
Example: The mean of 2, 3, and 5 is 3.3333 . . . ,
which we round to 3.3. Because the raw data are
whole numbers, we round to the nearest tenth. As
always, round only the final answer and not any
intermediate values used in your calculations.
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STAT1010 – mean (or average)
Mean as a measure of center
!  A histogram
of octane rating would
balance at the position of its mean.
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Count
8
4
2
85
90
95
90.95
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Mean as a measure of center
!  A histogram,
in general, will balance at the
position of its mean.
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Effects of outliers on mean
!  Consider
the following ordered data set:
110, 111, 116, 117, 118, 122, 123, 125, 126, 175
x=
1243
= 124.3
10
!  It
turns out that the 175 was incorrectly written
down, and was actually 135, then
1203
xcorrect =
= 120.3
10
!  Did
the mistake change the mean very much?
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STAT1010 – mean (or average)
Effects of outliers on mean
!  There’s
some opinion on what constitutes
a large change, but in general, outliers
can greatly affect the mean.
Definition
An outlier in a data set is a value that is much higher or
much lower than almost all others.
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Effects of outliers on mean
!  Here
is a visual along the real number line
that shows the obvious outlier:
110
120
130
140
150
160
170
value
!  More
on this topic in part 2…
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Weighted Mean
!  Sometimes
we weigh certain data points
heavier than others in computing a mean.
!  For instance, final grades are often
computed using a weighted mean.
Definition
A weighted mean accounts for variations in the relative
importance of data values. Each data value is assigned
a weight and the weighted mean is
weighted mean =
sum of (each data value x its weight)
sum of all weights
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STAT1010 – mean (or average)
Example: Weighted average,
Final grades
Category
HW
SchQuiz
UnschQuiz
Discussion
Exams
Weight
0.10
0.10
0.05
0.05
0.70
DataValue
95
80
65
70
78
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Example: Weighted average,
Final grades
Data DataValuex
Weight
Category Weight Value
HW
0.10
95
9.50
SchQuiz
0.10
80
8.00
UnschQuiz 0.05
65
3.25
Discussion 0.05
70
3.50
Exams
0.70
78
54.60
78.85
78.85
1.00 =78.85
sum of (each data
value x its weight)
sum of (each data value x its weight)
sum of all weights
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What about other measures of center?
!  Median
!  Mode
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STAT1010 – mean (or average)
Median
!  A value
that divides the data into a lower
half and an upper half.
!  About half the data values are greater than
the median about half are less than the
median.
!  Perhaps a better measure of center than
the mean for skewed distributions.
More on this later.
Example:
Eight grocery stores sell the PR energy bar for the following
prices:
$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79
Find the median of these prices.
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Solution:
To find the median, we first sort the data in ascending order:
3 values below
2 middle values
3 values above
Because there are eight prices (an even number), there are two
values in the middle of the list: $1.35 and $1.39. Therefore the
median lies halfway between these two values, which we
calculate by adding them and dividing by 2:
median =
$1.35 + $1.39 = $1.37
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Copyright © 2009 Pearson Education, Inc.
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STAT1010 – mean (or average)
Solution:
To find the median, we first sort the data in ascending order:
3 values below
2 middle values
3 values above
Because there are eight prices (an even number), there are two
values in the middle of the list: $1.35 and $1.39. Therefore the
median lies halfway between these two values, which we
calculate by adding them and dividing by 2:
median =
$1.35 + $1.39 = $1.37
2
Using the rounding rule, we could express the median as $1.370
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Copyright © 2009 Pearson Education, Inc.
n
is the
number of
observations
in the data set.
Median
!  When
" The
!  When
n is odd:
median IS one of the observations
25, 47, 55, 78, 110
n is even:
The median for the
5 data points
" The
median is BETWEEN the two
observations closest to the middle
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Median
(shown in a stem-n-leaf plot)
n = 12
36  |2
36*|5678899
37  |003
Median = (368+369)/2
37*|8
= 368.5 grams
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STAT1010 – mean (or average)
Recall the earlier example…
!  Consider
the following ordered data set:
110, 111, 116, 117, 118, 122, 123, 125, 126, 175
x=
1243
= 124.3
10
!  It
turns out that the 175 was incorrectly written
down, and was actually 135, then
1203
xcorrect =
= 120.3
10
!  Did
the mistake change the mean very much?
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Effects of outliers on median
!  How
does it change the median?
110, 111, 116, 117, 118, 122, 123, 125, 126, 175
Median = (118+122)/2=120
110, 111, 116, 117, 118, 122, 123, 125, 126, 135
Median = (118+122)/2=120
!  It
800
The median divides
the distribution into a
lower and an upper
half.
Frequency
! 
600
The mean is the
balance point of the
distribution.
0
200
! 
1000
Mean or Median?
1200
Outliers can have a large affect on the mean, but the
median is not greatly affected by outliers.
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400
! 
didn’t change it all!!
Median = 2.88
0
5
10
15
bird count
Mean = 3.48
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STAT1010 – mean (or average)
Mode
!  The
mode is another way to find a
representative value in a data set or
distribution.
!  The
mode is the most common value in a
data set.
" If
two values are ‘most common’, then we say
the distribution is bimodal.
Example:
Eight grocery stores sell the PR energy bar for the following
prices:
$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79
Find the mode of these prices.
Solution:
The mode is $1.29 because this price occurs more times than
any other price.
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