Using Measures of Position to Describe Spread? 1.
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Transcript of Using Measures of Position to Describe Spread? 1.
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Section 3.5/3.5
Using Measures of Position to Describe Spread?
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Percentile
The pth percentile is a value such that p percent of the observations fall below or at that value
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Finding Quartiles
Splits the data into four parts• Arrange the data in order• The median is the second quartile, Q2
• The first quartile, Q1, is the median of the lower half of the observations
• The third quartile, Q3, is the median of the upper half of the observations
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M = median = 3.4
Q1= first quartile = 2.2
Q3= third quartile = 4.35
1 0.62 1.23 1.64 1.95 1.56 2.17 2.38 2.39 2.5
10 2.811 2.912 3.313 3.414 3.615 3.716 3.817 3.918 4.119 4.220 4.521 4.722 4.923 5.324 5.625 6.1
Measure of spread: quartiles
The first quartile, Q1, is the value in the sample
that has 25% of the data at or below it and 75%
above
The second quartile is the same as the median of
a data set. 50% of the data are above the median
and 50% are below
The third quartile, Q3, is the value in the sample
that has 75% of the data at or below it and 25%
above
Quartiles divide a ranked data set into four equal parts.
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Quartile ExampleFind the first and third quartiles
Prices per share of 10 most actively traded stocks on NYSE (rounded to nearest $)
2 4 11 13 14 15 31 32 34 47
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Calculating Interquartile range The interquartile range is the distance
between the third quartile and first quartile:
IQR = Q3 Q1
IQR gives spread of middle 50% of the data
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Criteria for identifying an outlier
An observation is a potential outlier if it falls more than 1.5 x IQR below the first quartile or more than 1.5 x IQR above the third quartile
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5 Number Summary
The five-number summary of a dataset consists of the• “Minimum” value• First Quartile• Median• Third Quartile• “Maximum” value
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Boxplot
A box goes from the Q1 to Q3 A line is drawn inside the box at the median A line goes from the lower end of the box to the smallest
observation that is not a potential outlier and from the upper end of the box to the largest observation that is not a potential outlier
The potential outliers are shown separately
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Comparing Distributions
Box Plots do not display the shape of the distribution as clearly as
histograms, but are useful for making graphical comparisons of two
or more distributions
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Box Plots and Histograms
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Z-Score and outliers The z-score for an observation is the number of
standard deviations that it falls from the mean
An observation from a bell-shaped distribution is a potential outlier if its z-score < -3 or > +3
deviation standard
mean -n observatio z
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Homework
3.2; 1, 13, 19, 21, 37, 41
3.4; 3, 9, 11, 15, 19
3.5; 5, 7