Aim: How do we use measures of dispersion: range, variance, and standard deviation?

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Aim How do we use measures of dispersion
range, variance, and standard deviation?
Do Now
Find the average of each set
average does not give sufficient info about data
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Recall Model Problem
A teacher marked a set of 32 papers. The grades
were as follows 90, 85, 74, 86, 65, 62, 100,
95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73,
100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72,
75.
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Range
Range the difference between the highest value
and the lowest value in a set of data.
A teacher marked a set of 32 papers. The grades
were as follows 90, 85, 74, 86, 65, 62, 100,
95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73,
100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72,
75.
Range 100 45 55
Often unreliable as a measure of dispersion
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Quartiles
Range - lowest to highest amount
71
87.5
Q3
Q1
Median
25
50
75
Quartiles break a data group into 4 equal parts.
The lower quartile is the median of the lower
half. The upper quartile is median of the upper
half.
Lower quartile is the median of the first 16
numbers
Average of the 8th 9th numbers
(70 72)/2 71
Upper quartile is the median of the last 16
numbers
(86 89)/2 87.5
Average of the 24th 25th numbers
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Percentiles
Q1
Q3
Median
Lower quartile 25
Second quartile 50
Upper quartile 75
Percentile is a number that tells us what percent
of the total number of data values lie at or
below a given measure. Ranking
What percentile is the score of 70?
70 is the 7th lowest of the 32 scores
7/32 .21875 21.875 ? 22
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Box-Whisker Plot
Median
Box-and-Whisker Plots
show 5 important values from the data set.

• Lower extreme - lowest value

• Upper extreme - highest value

• Median - middle value

• Lower quartile - 25th percentile value

• Upper quartile - 75th percentile value

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Box-Whisker Plot
Interquartile Range
Max
Mi n
Q3
Q2
Median
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Mean Absolute Deviation
Set of data 72, 85, 87, 89, 90, 93
xi
93 86 7 7
90 86 4 4
89 86 3 3
87 86 1 1
85 86 -1 1
72 86 -14 14

the sum of the differences between each entry in
a sample and the mean of that sample is always
equal to 0
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Variance
Variance A measure of dispersion that uses the
squares of the deviations from the mean and gives
greatest weight to scores farthest from the mean.
Definition The variance, v, of a set of data is
the average of the squares of the deviation from
the Mean.
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Variance Model Problem
Example on 5 test scores, Fred earned grades of
92, 86, 95, 84, and 78. Find the variance.
Example on 5 test scores, Fred earned grades of
78, 84, 86, 92, and 95. Find the variance.

1. Write in order
2. Find mean
3. Find differences
4. Square differences
5. Apply formula

xi
78 87
84 87
86 87
92 87
95 87

-9
81
9
-3
-1
1
5
25
64
8
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Standard Deviation
Definition the standard deviation, ?, of a set
of data is equal to the square root of the
variance.
Result is in terms of original data, not the
square of the values.
Most important and widely used measure of
dispersion in the world.
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Model Problem
Example on 5 test scores, Fred earned grades of
78, 84, 86, 92, and 95. Find standard deviation.

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Calculator and Model Problem
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z-score
Definition the z-score is the number of
standard deviations that a value is from the mean
Example A set of values has a mean of 85 and a
standard deviation of 6. Find the z-score of the
value 76.
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Our Favorite Model Problem
A teacher marked a set of 32 papers. The grades
were as follows 90, 85, 74, 86, 65, 62, 100,
95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73,
100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72,
75.
What value has a z-score of approximately 1.25?
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