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Lesson 3 - 4
Measures of Position
Objectives
• Determine and interpret z-scores
• Determine and interpret percentiles
• Determine and interpret quartiles
• Check a set of data for outliers
Vocabulary
• Z-Score – the distance that a data value is from the
mean in terms of the number of standard deviations
• K Percentile – (Pk) divides the lower kth percentile of
a set of data from the rest
• Quartiles – (Qi) divides the whole data into four
(25%) sets of data
• Outliers – extreme observations
• IQR (Interquartile range) – difference between third
and first quartiles (IQR = Q3 – Q1)
• Lower fence – Q1 – 1.5(IQR)
• Upper fence – Q3 – 1.5(IQR)
Z-Scores
Population z-Score
x–μ
z = -----------σ
Sample z-Score
x–x
z = -----------s
Mean of z is 0 and the standard deviation of z is 1.
Allows comparisons of different distributions.
Quartiles
Smallest
Data Value
Median
Q1
25% of
the data
Q2
25% of
the data
Q3
25% of
the data
Largest
Data Value
25% of
the data
The index, i (position in sorted list), for the y%-tile will be
i = (y/100)(n + 1)
Where y is the percent and n is the number in the data set
Interquartile Range (IQR)
• IQR = Q3 – Q1
It is a measure of the spread of the data.
It is used to help determine outlying data
(data beyond the upper or lower fences).
• Upper Fence = Q3 + 1.5 • IQR
• Lower Fence = Q1 – 1.5 • IQR
Example 1
Which player had a better year in 1967?
Carl Yastrzemski
Roberto Clemente
AL average 0.236
AL stdev 0.01072
AL Batting Champ 0.326
NL Batting Champ 0.357
NL average 0.249
NL stdev 0.01257
Roberto did, barely.
His z-score was 8.60 and Yaz’s was 8.14
Example 2
Given the following set of data:
70,
28,
56,
63,
56,
35,
51,
50,
48,
58,
46,
46,
48,
62,
39,
69,
53,
45,
56,
53,
52,
60,
32,
70,
66,
38,
44,
33,
48,
73,
60,
54,
36,
45,
51,
55,
What is the median?
51
What is the Q1?
45
What is the Q3?
57
What is the IQR?
57- 45 = 12
49,
51,
44,
52
Example 2 continued
What is the upper fence?
Q3 + 1.5(IQR) = 57 + 1.5(12) = 75
What is the lower fence?
Q1 + 1.5(IQR) = 45 - 1.5(12) = 27
Are there any outliers?
No! UF > max and LF > min
Summary and Homework
• Summary
– Data sets should be checked for outliers as the
mean and standard deviation are not resistant
statistics and any conclusions drawn from a set
of data that contains outliers can be flawed
– Fences serve as cutoff points for determining
outliers (data values less than lower or greater
than upper fence are considered outliers)
• Homework: pg 172 - 174: 9-12, 14, 19