Download 3.4 - MAT143

Measures of Position
3.4
Z-SCORE
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The standard deviation is a measure of
dispersion that uses the same dimensions as
the data (remember the empirical rule)
The distance of a data value from the mean,
calculated as the number of standard
deviations, would be a useful measurement
This distance is called the z-score
Z-SCORE
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If the mean was 20 and the standard deviation
was 6
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The value 26 would have a z-score of 1.0 (1.0
standard deviation higher than the mean)
The value 14 would have a z-score of –1.0 (1.0
standard deviation lower than the mean)
The value 17 would have a z-score of –0.5 (0.5
standard deviations lower than the mean)
The value 20 would have a z-score of 0.0
Z-SCORE
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The population z-score is calculated using the
population mean and population standard
deviation
z
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x
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The sample z-score is calculated using the
sample mean and sample standard deviation
xx
z
s
Z-SCORE
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z-scores can be used to compare the relative positions
of data values in different samples
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Pat received a grade of 82 on her statistics exam where the
mean grade was 74 and the standard deviation was 12
Pat received a grade of 72 on her biology exam where the
mean grade was 65 and the standard deviation was 10
Pat received a grade of 91 on her kayaking exam where the
mean grade was 88 and the standard deviation was 6
Calculate each z-score and see what class has
the highest RELATIVE grade.
Z-SCORE
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Statistics
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Biology
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Grade of 72
z-score of (72 – 65) / 10 = .70
Kayaking
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Grade of 82
z-score of (82 – 74) / 12 = .67
Grade of 81
z-score of (91 – 88) / 6 = .50
Biology was the highest relative grade
PERCENTILE
The median divides the lower 50% of the data
from the upper 50%
 The median is the 50th percentile
 If a number divides the lower 34% of the data
from the upper 66%, that number is the 34th
percentile
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QUARTILES
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The quartiles are the 25th, 50th, and 75th
percentiles
 Q1
= 25th percentile
 Q2 = 50th percentile = median
 Q3 = 75th percentile
Quartiles are the most commonly used
percentiles
 The 50th percentile and the second quartile Q2
are both other ways of defining the median
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QUARTILES
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Quartiles divide the data set into four equal parts
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The top quarter are the values between Q3 and the
maximum
The bottom quarter are the values between the
minimum and Q1
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IQR
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Quartiles divide the data set into four equal parts
The interquartile range (IQR) is the difference
between the third and first quartiles
IQR = Q3 – Q1
 The IQR is a resistant measurement of dispersion
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CALCULATOR
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Can we find the Quartiles with a Calculator?
 Data
 1,2,3,4,5,6,8,10,15,20
OUTLIERS
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Extreme observations in the data are referred to as
outliers
Outliers should be investigated
Outliers could be
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Chance occurrences
Measurement errors
Data entry errors
Sampling errors
Outliers are not necessarily invalid data
FINDING OUTLIERS
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One way to check for outliers uses the quartiles
Outliers can be detected as values that are
significantly too high or too low, based on the known
spread
The fences used to identify outliers are
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Lower fence = LF = Q1 – 1.5  IQR
Upper fence = UF = Q3 + 1.5  IQR
Values less than the lower fence or more than the
upper fence could be considered outliers
FINDING OUTLIERS
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Are there any outliers?
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54
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Calculations (You can use your Calculator to find
these!)
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Q1 = 7
Q3 = 27
IQR = 20
Lower Fence = Q1 – 1.5  IQR
 Upper Fence = Q3 + 1.5  IQR
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RECAP!
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z-scores
Measures the distance from the mean in units of
standard deviations
 Can compare relative positions in different samples
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Percentiles and quartiles
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Divides the data so that a certain percent is lower and
a certain percent is higher
Outliers
Extreme values of the variable
 Can be identified using the upper and lower fences
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