Download Lecture 5 Box plots and percentiles

Review
• Measures of central tendency
• Mean
• Median
• Mode
• Measures of dispersion or variation
• Range
• Variance
• Standard Deviation
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Interpreting the Standard
Deviation
Chebyshev’s Theorem
The proportion (or fraction) of any data set lying
within K standard deviations of the mean is always
at least 1-1/K2, where K is any positive number
greater than 1.
For K=2 we obtain, at least 3/4 (75 %) of all scores
will fall within 2 standard deviations of the mean,
i.e. 75% of the data will fall between
x  2s and x  2s
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Interpreting the Standard
Deviation
Chebyshev’s Theorem
The proportion (or fraction) of any data set lying
within K standard deviations of the mean is always
at least 1-1/K2, where K is any positive number
greater than 1.
For K=3 we obtain, at least 8/9 (89 %) of all scores
will fall within 3 standard deviations of the mean,
i.e. 89% of the data will fall between
x  3s and x  3s
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This Data is Symmetric, Bell
Shaped (or Normal Data)
x M
Relative
Frequency
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
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The Empirical Rule
The Empirical Rule states that for bell shaped
(normal) data:
68% of all data points are within 1 standard deviations of the mean
95% of all data points are within 2 standard deviations of the mean
99.7% of all data points are within 3 standard deviations of the mean
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The Empirical Rule
The Empirical Rule states that for bell shaped
(normal) data, approximately:
68% of all data points are within 1 standard deviations of the mean
95% of all data points are within 2 standard deviations of the mean
99.7% of all data points are within 3 standard deviations of the mean
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Z-Score
To calculate the number of standard
deviations a particular point is away from the
standard deviation we use the following
formula.
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Z-Score
To calculate the number of standard
deviations a particular point is away from the
standard deviation we use the following
formula.
z
x

or
xx
z
s
The number we calculate is called the z-score
of the measurement x.
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Example – Z-score
Here are eight test scores from a previous
Stats 201 class:
35, 59, 70, 73, 75, 81, 84, 86.
The mean and standard deviation are 70.4 and
16.7, respectively.
a) Find the z-score of the data point 35.
b) Find the z-score of the data point 73.
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Example – Z-score
Here are eight test scores from a previous
Stats 201 class:
35, 59, 70, 73, 75, 81, 84, 86.
The mean and standard deviation are 70.4 and
16.7, respectively.
a) Find the z-score of the data point 35.
z = -2.11
b) Find the z-score of the data point 73.
z = 0.16
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Interpreting Z-scores
The further away the z-score is from zero the
more exceptional the original score.
Values of z less than -2 or greater than +2 can
be considered exceptional or unusual (“a
suspected outlier”).
Values of z less than -3 or greater than +3 are
often exceptional or unusual (“a highly
suspected outlier”).
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Percentiles
Another method for detecting outliers
involves percentiles.
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Percentiles
Another method for detecting outliers involves
percentiles.
The pth percentile ranking is a number so that
p% of the measurements fall below the pth
percentile and 100 – p% fall above it.
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Example
If your score on a class quiz of 200
students places you in the 80th
percentile, then only 40 students
received a higher mark then you
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Important Percentiles
Memorize:
The 25th percentile is called the lower
quartile (QL)
The 75th percentile is called the upper
quartile (QU)
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Important Percentiles
Memorize:
The 25th percentile is called the lower
quartile (QL)
The 75th percentile is called the upper
quartile (QU)
The 50th percentile is called the
16
Important Percentiles
Memorize:
The 25th percentile is called the lower
quartile (QL)
The 75th percentile is called the upper
quartile (QU)
The 50th percentile is called the median (M)
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Quick way to find quartiles
1. Arrange the data in increasing order
2. The middle number (or average of the
two middle numbers) is the 50th
percentile.
3. Find the middle number in the set of
numbers greater than the median. This is
the upper quartile.
4. Similarly, find the lower quartile
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Important Percentiles
The interquartile range (IQR) is defined to
be:
IQR = QU -QL
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Example - Fax
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Example - Fax
Here are the number of pages faxed by each
fax sent from our Math and Stats department
since April 24th, in the order that they
occurred.
5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13,
2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25
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Example - Fax
Here are the number of pages faxed by each
fax sent from our Math and Stats department
since April 24th, in the order that they
occurred.
5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13,
2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25
Find QU , QL , M and IQR.
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Example - Fax
1) Rank the n points of data from lowest to
highest
5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13,
2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25
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Example - Fax
1) Rank the n points of data from lowest to
highest
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5,
5, 5, 6, 6, 6, 8, 10, 13, 25, 37
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5,
5, 5, 6, 6, 6, 8, 10, 13, 25, 37
To compute QU and QL , M.
Find the Median, divide the data into two
equal parts and then the Medians of these.
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5,
5, 5, 6, 6, 6, 8, 10, 13, 25, 37
N = 28
Therefore, median is half way between the
14th and 15th number.
Median = 50th percentile = 3
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3 | 3, 4, 5, 5
5, 5, 6, 6, 6, 8, 10, 13, 25, 37
M =3
QU = 6
QL = 2
IQR=6-2=4.
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Percentiles
Sometimes the IQR, is a better measure of
variance then the standard deviation since it
only depends on the center 50% of the data.
That is, it is not effected at all by outliers.
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Percentiles
Sometimes the IQR, is a better measure of
variance then the standard deviation since it
only depends on the center 50% of the data.
That is, it is not effected at all by outliers.
To use the IQR as a measure of variance we
need to find the Five Number Summary of
the data and then construct a Box Plot.
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Five Number Summary and
Outliers
The Five Number Summary of a data set
consists of five numbers,
– MIN, QL , M, QU , Max
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Five Number Summary and
Outliers
The Five Number Summary of a data set
consists of five numbers,
– MIN, QL , M, QU , Max
Suspected Outliers lie
– Above 1.5 IQRs but below 3 IQRs from the
Upper Quartile
– Below 1.5 IQRs but above 3 IQRs from the
Lower Quartile
Highly Suspected Outliers lie
– Above 3 IQRs from the Upper Quartile
– Below 3 IQRs from the Lower Quartile.
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Five Number Summary and
Outliers
The Inner Fences are:
– data between the Upper Quartile and 1.5 IQRs
above the Upper Quartile and
– data between the Lower Quartile and 1.5 IQRs
below the Lower Quartile
The Outer Fences are:
– data between 1.5 IQRs above the Upper
Quartile and 3 IQRs above the Upper Quartile
and
– data between 1.5 IQRs Lower Quartile and 3
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IQRs below the Lower Quartile
Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37
Min=1, QL = 2, M = 3, QU = 6, Max = 37.
IQR=6-2=4.
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37
Min=1, QL = 2, M = 3, QU = 6, Max = 37.
IQR=6-2=4.
Inner Fence extremes: -4, 12
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37
Min=1, QL = 2, M = 3, QU = 6, Max = 37.
IQR=6-2=4.
Inner Fence extremes: -4, 12
Outer Fence extremes: -10, 18
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Example - Fax
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37
Min=1, QL = 2, M = 3, QU = 6, Max = 37.
IQR=6-2=4.
Inner Fence extremes: -4, 12
Outer Fence extremes: -8, 18
Suspected Outliers: 13
Highly Suspected Outliers: 25, 37
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Definition: Boxplot
A boxplot is a graph of lines (from lowest point
inside the lower inner fence to highest point in
the upper inner fence) and boxes (from Lower
Quartile to Upper quartile) indicating the
position of the median.
*
Lowest data
Point more than
the lower inner
fence
Outliers
Median
Lower
Quartile
Highest data
Point less than
Upper
the upper inner
Quartile fence
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