Transcript Lecture 4 Measurement

Measure of Location and
Variability
Histogram
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Multimodal
Mean, Median and Mode
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Mean: average value
Sample mean: denoted by x-bar (English)
Population mean: Mu (Greek)
Median: value in the middle when
the data are arranged in ascending
order (smallest to largest).
odd number of observations: the middle value
even number of observations: the average of the two
middle values
Mean, Median and Mode II
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Mode: the value that occurs with
greatest frequency (may be more
than one mode in a set of data).
Example:
Data: 1,1,2,2,2,2,3,3,4,4,5
Mode=2
Percentile
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the pth percentile is the value such that at
least p percent of the observations are
less than or equal to this value and at
least (100-p) percent of the
observations are greater than or
equal to this value.
For example: If you scored in the 90th
percentile on the verbal part of your
SAT’s, this would mean that you scored
above 90% of all verbal scores taken for
the SAT’s at that time.
Percentile II
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To calculate the pth percentile:
• arrange the data in ascending order
• compute an index i=(p/100)*n
• if i is not an integer, round up, the next
integer greater than i denotes the
position of the pth percentile
• if i is an integer the pth percentile is the
average of the values in positions i and
i+1.
Quartile
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used to divide the data into 4 parts.
Q1: first quartile, 25th percentile
Q2: second quartile, 50th
percentile, median
Q3: third quartile, 75th
percentile
5 Number Summary
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1.
2.
3.
4.
5.
Smallest value
first quartile (Q1)
Median (Q2)
third quartile (Q3)
Largest value
Variability
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We have talked about measure of location.
• Mean, median, mode
Why do we need to look at variability?
• Data with same mean or median may have different
variability
• Example: two sets of quiz grades
 11 12 14 15 17 18 20
mean=15.28571
 7 11 12 18 19 19 20
mean=15.14286
 The two means are similar but apparently the second
set is more spreadout  greater variability
How to quantify variability
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Range
Inter-Quantile Range (IQR)
Variance
• Population variance
• Sample variance
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Standard Deviation (S.D., std dev)
• Population S.D.
• Sample S.D.
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Coefficients of Variation
Range
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Range=max — min
In our example:
• Set 1: Range=20-11=9
• Set 2: Range=20-7=13
Inter-Quantile Range
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Set 1:
• 11 12 14 15 17 18 20
• Median=15, Q1=12, Q3=18
• IQR = 18-12=6
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Set 2:
• 7 11 12 18 19 19 20
• Median: 18, Q1=11, Q2=19
• IQR=19-11=8
Boxplot
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a graphical
summary of data
that is based on a
five-number
summary.
Example: