Download STA 291 Summer 2010

Lecture 4
Dustin Lueker
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The population distribution for a continuous
variable is usually represented by a smooth curve
◦ Like a histogram that gets finer and finer
 Similar to the idea of using smaller and smaller rectangles to
calculate the area under a curve when learning how to
integrate
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Symmetric distributions
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Not symmetric distributions:
◦ Bell-shaped
◦ U-shaped
◦ Uniform
◦ Left-skewed
◦ Right-skewed
◦ Skewed
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Center of the data
◦ Mean
◦ Median
◦ Mode
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Dispersion of the data
 Sometimes referred to as spread
◦ Variance, Standard deviation
◦ Interquartile range
◦ Range
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Mean
◦ Arithmetic average
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Median
◦ Midpoint of the observations when they are
arranged in order
 Smallest to largest
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Mode
◦ Most frequently occurring value
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Sample size n
Observations x1, x2, …, xn
Sample Mean “x-bar”
x  ( x1  x2  ...  xn ) / n
n
1
  xi
n i 1
  SUM
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Population size N
Observations x1 , x2 ,…, xN
Population Mean “mu”
  ( x1  x2  ...  xN ) / N
1

N
N
 xi
  SUM
i 1
 Note: This is for a finite population of size N
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Requires numerical values
◦ Only appropriate for quantitative data
◦ Does not make sense to compute the mean for
nominal variables
◦ Can be calculated for ordinal variables, but this does not
always make sense
 Should be careful when using the mean on ordinal variables
 Example “Weather” (on an ordinal scale)
Sun=1, Partly Cloudy=2, Cloudy=3,
Rain=4, Thunderstorm=5
Mean (average) weather=2.8
 Another example is “GPA = 3.8” is also a mean of observations
measured on an ordinal scale
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Center of gravity for the data set
Sum of the differences from values above the
mean is equal to the sum of the differences
from values below the mean
◦ 3+2+2 = 3 + 4
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Mean
◦ Sum of observations divided by the number of
observations
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Example
◦ {7, 12, 11, 18}
◦ Mean =
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Highly influenced by outliers
◦ Data points that are far from the rest of the data
◦ Example
 Monthly income for five people
1,000 2,000 3,000 4,000 100,000
 Average monthly income =
 What is the problem with using the average to describe
this data set?
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
Measurement that falls in the middle of the
ordered sample
When the sample size n is odd, there is a
middle value
◦ It has the ordered index (n+1)/2
 Ordered index is where that value falls when the
sample is listed from smallest to largest
 An index of 2 means the second smallest value
◦ Example
 1.7, 4.6, 5.7, 6.1, 8.3
n=5, (n+1)/2=6/2=3, index = 3
Median = 3rd smallest observation = 5.7
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When the sample size n is even, average the
two middle values
◦ Example
 3, 5, 6, 9, n=4
(n+1)/2=5/2=2.5, Index = 2.5
Median = midpoint between 2nd and 3rd smallest
observations = (5+6)/2 = 5.5
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For skewed distributions, the median is often
a more appropriate measure of central
tendency than the mean
The median usually better describes a “typical
value” when the sample distribution is highly
skewed
Example
◦ Monthly income for five people
1,000 2,000 3,000 4,000 100,000
◦ Median monthly income:
 Why is the median better to use with this data than the
mean?
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Mean - Arithmetic Average
 Mean of a Sample - x

Mean of a Population - μ
Median - Midpoint of
the observations when
they are arranged in
increasing order
Notation: Subscripted variables
n = # of units in the sample
N = # of units in the population
x = Variable to be measured
xi = Measurement of the ith unit
Mode - Most frequent value.
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Example: Highest Degree Completed
Highest Degree
Frequency
Percentage
Not a high school
graduate
38,012
21.4
High school only
65,291
36.8
Some college, no
degree
33,191
18.7
Associate, Bachelor,
Master, Doctorate,
Professional
41,124
23.2
Total
177,618
100
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n = 177,618
(n+1)/2 = 88,809.5
Median = midpoint between the 88809th
smallest and 88810th smallest observations
◦ Both are in the category “High school only”
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Mean wouldn’t make sense here since the
variable is ordinal
Median
◦ Can be used for interval data and for ordinal data
◦ Can not be used for nominal data because the
observations can not be ordered on a scale
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Mean
◦ Interval data with an approximately symmetric
distribution
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Median
◦ Interval data
◦ Ordinal data
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Mean is sensitive to outliers, median is not
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Symmetric distribution
◦ Mean = Median
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Skewed distribution
◦ Mean lies more toward the direction which the
distribution is skewed
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While the median is better than the mean for
skewed distributions there is one large
disadvantage to using the median
◦ Insensitive to changes within the lower or upper
half of the data
◦ Example
 1, 2, 3, 4, 5
 1, 2, 3, 100, 100
◦ Sometimes, the mean is more informative even
when the distribution is skewed
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Keeneland Sales
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The deviation of the ith observation xi from
the sample mean x is the difference between
them, ( xi  x )
◦ Sum of all deviations is zero
◦ Therefore, we use either the sum of the absolute
deviations or the sum of the squared deviations as
a measure of variation
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Variance of n observations is the sum of the
squared deviations, divided by n-1
s
2
(x


i
x)
2
n 1
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Observation
Mean
Deviation
Squared
Deviation
1
3
4
7
10
Sum of the Squared Deviations
n-1
Sum of the Squared Deviations / (n-1)
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About the average of the squared deviations
◦ “average squared distance from the mean”
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Unit
◦ Square of the unit for the original data
 Difficult to interpret
◦ Solution
 Take the square root of the variance, and the unit is
the same as for the original data
 Standard Deviation
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s≥0
◦ s = 0 only when all observations are the same
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If data is collected for the whole population
instead of a sample, then n-1 is replaced by
N
s is sensitive to outliers
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Sample
◦ Variance
s2 
2
(
x
i

x
)

n 1
◦ Standard Deviation
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Population
◦ Variance
2 
s
2
(
x
i

x
)

n 1
2
(
x
i


)

◦ Standard Deviation
N

2
(
x
i


)

N
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Population mean and population standard deviation
are denoted by the Greek letters μ (mu) and σ
(sigma)
◦ They are unknown constants that we would like to estimate
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Sample mean and sample standard deviation are
denoted by x and s
◦ They are random variables, because their values vary
according to the random sample that has been selected
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If the data is approximately symmetric and
bell-shaped then
◦ About 68% of the observations are within one
standard deviation from the mean
◦ About 95% of the observations are within two
standard deviations from the mean
◦ About 99.7% of the observations are within three
standard deviations from the mean
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Scores on a standardized test are scaled so
they have a bell-shaped distribution with a
mean of 1000 and standard deviation of 150
◦ About 68% of the scores are between
◦ About 95% of the scores are between
◦ If you have a score above 1300, you are in the
top
%
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