aim: measures of dispersion course: alg. 2 & trig. do now: aim: how do we use measures of...
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Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Do Now:
Aim: How do we use measures of dispersion: range, variance, and standard deviation?
Find the average of each set
1 2 3 4 5 1 2 3 4 5
1 2 3 4 5 1 2 3 4 5
3x 3x
3x 3x
average does not
give sufficient info about
data
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
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.
STAT EDIT 1
STAT CALC 1 ENTER
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
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 ameasure of dispersion
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Quartiles
Find the lower and upper quartile of
Range - lowest to highest amount
Q3Q1 Median25% 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.
Paper Grade Problem n = 32
Lower quartile is the median of the first 16 numbers
Upper quartile is the median of the last 16 numbers
(86 + 89)/2 = 87.5
Average of the 8th & 9th numbers
Average of the 24th & 25th numbers
(70 + 72)/2 = 71
78 10045 71 87.5
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Percentiles
Q3Q1 MedianLower
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
Paper Grade Problem
• Lower quartile - 71 - 8.5th # • Lower extreme - 45 - 1st #
• Upper extreme - 100 - 32th #
• Median - 78 - 18.5th # • Upper quartile - 87.5 - 24.5th #
What percentile is the score of 70?
70 is the 7th lowest of the 32 scores7/32 = .21875 =
21.875% 22%
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Interquartile Range
Max
Mi n 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
Q3Q1
Box-Whisker Plot
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Box-Whisker Plot
70 80 90 10060 45
Interquartile Range
MaxMi n Q3Q2 Median
78 10087.571 45
Paper Grade Problem
Make a Box-and-Whisker Plot of
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Mean Absolute Deviation
1Mean Absolute Deviation =
n
ii
x x
n
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
1x xx 1x x
516ix
0
ix x 30
x x
the sum of the differences between each entry in a sample and the mean of that sample is
always equal to 0
6
1 30M A D = 5
6 6
ii
x x
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
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.
2
1
n
ii
x xv
n
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Variance – Model Problem
2
1
n
ii
x xv
n
Example: on 5 test scores,
Fred earned gradesof 92, 86, 95, 84, and 78. Find the variance. 1. Write # in order
2. Find mean3. Find differences4. Square differences5. Apply formula
Example: on 5 test scores, Fred earned gradesof 78, 84, 86, 92, and 95. Find the variance.
xi
78 87
84 87
86 87
92 87
95 87
x ix x 2
ix x
5
1
435
ii
x
5
1 435
5 587
ii
xx
-9
-3-1
5
8
81
91
25
64
25
1
180
ii
x x
18036
5v
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Standard Deviation
Definition: the standard deviation, , of a set of data is equal to the square root of the variance.
2
1
n
ii
x xv
n
Result is in terms of original data, not the square of the values.
Most important and widely used measure of dispersion in the world.
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Model Problem
Example: on 5 test scores, Fred earned grades of 78, 84, 86, 92, and 95. Find standard deviation.
5 2
1 18036
5 5
ii
x xv
2
1 6
n
ii
x xv
n
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
Calculator and Model Problem
STAT EDIT 1 STAT CALC 1 ENTER
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
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.
value mean-score =
standard deviation76 85
substitute6
9= simplify
6= 1.5
z
Aim: Measures of Dispersion Course: Alg. 2 & Trig.
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?
value = mean + (z-score)(standard deviation)
value mean-score =
standard deviationz
value = 78.125 + (1.25)(13.7426)
value = 95.30325 95
Aim: Measures of Dispersion Course: Alg. 2 & Trig.