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Lecture 5
Dustin Lueker
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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 (mu)
 (sigma)

population mean
population standard deviation
2
(sigma-squared)
population variance
x or xi (x-i) observation
x (x-bar)
s
2
s

sample mean
sample standard deviation
sample variance
summation symbol
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
Sample
◦ Variance
s2 
2
(
x
i

x
)

n 1
◦ Standard Deviation

Population
◦ Variance
2 
2
(
x
i

x
)

s
n 1
2
(
x
i


)

◦ Standard Deviation
N

2
(
x
i


)

N
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1.
2.
3.
4.
5.
Calculate the mean
For each observation, calculate the deviation
For each observation, calculate the squared
deviation
Add up all the squared deviations
Divide the result by (n-1)
Or N if you are finding the population variance
(To get the standard deviation, take the square root of the result)
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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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
The pth percentile (Xp) is a number such that
p% of the observations take values below it,
and (100-p)% take values above it
◦ 50th percentile = median
◦ 25th percentile = lower quartile
◦ 75th percentile = upper quartile

The index of Lp
◦ (n+1)p/100
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
25th percentile
◦ lower quartile
◦ Q1
◦ (approximately) median of the observations
below the median

75th percentile
◦ upper quartile
◦ Q3
◦ (approximately) median of the observations
above the median
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
Find the 25th percentile of this data set
◦ {3, 7, 12, 13, 15, 19, 24}
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


Use when the index is not a whole number
Want to start with the closest index lower
than the number found then go the distance
of the decimal towards the next number
If the index is found to be 5.4 you want to go
to the 5th value then add .4 of the value
between the 5th value and 6th value
◦ In essence we are going to the 5.4th value
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
Find the 40th percentile of the same data set
◦ {3, 7, 12, 13, 15, 19, 24}
 Must use interpolation
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

Five Number Summary
◦
◦
◦
◦
◦
Minimum
Lower Quartile
Median
Upper Quartile
Maximum
◦
◦
◦
◦
◦
minimum=4
Q1=256
median=530
Q3=1105
maximum=320,000.
Example
 What does this suggest about the shape of the distribution?
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
The Interquartile Range (IQR) is the difference
between upper and lower quartile
◦ IQR = Q3 – Q1
◦ IQR = Range of values that contains the middle 50%
of the data
◦ IQR increases as variability increases

Murder Rate Data
◦ Q1= 3.9
◦ Q3 = 10.3
◦ IQR =
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



Displays the five number summary (and
more) graphical
Consists of a box that contains the central
50% of the distribution (from lower quartile to
upper quartile)
A line within the box that marks the median,
And whiskers that extend to the maximum
and minimum values
 This is assuming there are no outliers in the data set
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
An observation is an outlier if it falls
◦ more than 1.5 IQR above the upper quartile
or
◦ more than 1.5 IQR below the lower quartile
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

Whiskers only extend to the most extreme
observations within 1.5 IQR beyond the
quartiles
If an observation is an outlier, it is marked by
an x, +, or some other identifier
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
Values






Min = 148
Q1 = 158
Median = Q2 = 162
Q3 = 182
Max = 204
Create a box plot
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


On right-skewed distributions, minimum, Q1,
and median will be “bunched up”, while Q3
and the maximum will be farther away.
For left-skewed distributions, the “mirror” is
true: the maximum, Q3, and the median will
be relatively close compared to the
corresponding distances to Q1 and the
minimum.
Symmetric distributions?
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
Value that occurs most frequently
◦ Does not need to be near the center of the distribution
 Not really a measure of central tendency
◦ Can be used for all types of data (nominal, ordinal, interval)

Special Cases
◦ Data Set
 {2, 2, 4, 5, 5, 6, 10, 11}
 Mode =
◦ Data Set
 {2, 6, 7, 10, 13}
 Mode =
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
Mean
◦ Interval data with an approximately symmetric
distribution

Median
◦ Interval or ordinal data

Mode
◦ All types of data
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
Mean is sensitive to outliers
◦ Median and mode are not
 Why?

In general, the median is more appropriate
for skewed data than the mean
◦ Why?


In some situations, the median may be too
insensitive to changes in the data
The mode may not be unique
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
“How often do you read the newspaper?”
Response
Frequency
every day
969
a few times a
week
452
once a week
261
less than once a
week
196
Never
76
TOTAL
1954
• Identify the
mode
• Identify the
median
response
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
Statistics that describe variability
◦ Two distributions may have the same mean
and/or median but different variability
 Mean and Median only describe a typical value, but
not the spread of the data
◦
◦
◦
◦
Range
Variance
Standard Deviation
Interquartile Range
 All of these can be computed for the sample or
population
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
Difference between the largest and smallest
observation
◦ Very much affected by outliers
 A misrecorded observation may lead to an outlier, and
affect the range

The range does not always reveal different
variation about the mean
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
Sample 1
◦ Smallest Observation: 112
◦ Largest Observation: 797
◦ Range =

Sample 2
◦ Smallest Observation: 15033
◦ Largest Observation: 16125
◦ Range =
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