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Measures of Position
MATH 130, Elements of Statistics I
J. Robert Buchanan
Department of Mathematics
Spring 2014
J. Robert Buchanan
Measures of Position
Introduction
A measure of position gives you some idea of
1
where particular data values would rank in an ordering of a
data set, or
2
where a data value falls with respect to the mean of the
sample or population.
J. Robert Buchanan
Measures of Position
z-Scores
Definition
The z-score represents the distance that a data value is from
the mean in terms of standard deviations.
Population z-score: z =
Sample z-score: z =
x −µ
σ
x −x
s
The z-score has no units. It has a mean of 0 and a standard
deviation of 1.
J. Robert Buchanan
Measures of Position
Example
Consider the data
16
21
22
23
24
25
25
27
28
29
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
26
27
28
30
18
21
22
23
24
25
26
27
28
30
19
21
22
23
24
25
26
27
29
30
19
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
27
29
31
We can calculate µ = 24.7 and σ = 3.4.
J. Robert Buchanan
Measures of Position
20
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
33
Example
Consider the data
16
21
22
23
24
25
25
27
28
29
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
26
27
28
30
18
21
22
23
24
25
26
27
28
30
19
21
22
23
24
25
26
27
29
30
19
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
We can calculate µ = 24.7 and σ = 3.4. Thus the z-score
corresponding to a data value of 27 is
z=
27 − 24.7
= 0.68.
3.4
J. Robert Buchanan
Measures of Position
21
22
22
23
24
25
26
28
29
33
Example
If µ = 24.7 and σ = 3.4, find the z-scores corresponding to
1
x = 17
2
x = 22
3
x = 30
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the z-scores corresponding to
1
x = 17
z=
2
x = 22
3
x = 30
17 − 24.7
−7.7
=
= −2.26
3.4
3.4
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the z-scores corresponding to
1
2
3
x = 17
z=
17 − 24.7
−7.7
=
= −2.26
3.4
3.4
z=
22 − 24.7
−2.7
=
= −0.79
3.4
3.4
x = 22
x = 30
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the z-scores corresponding to
1
2
3
x = 17
z=
17 − 24.7
−7.7
=
= −2.26
3.4
3.4
z=
22 − 24.7
−2.7
=
= −0.79
3.4
3.4
x = 22
x = 30
z=
30 − 24.7
5.3
=
= 1.56
3.4
3.4
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the data values corresponding to
z-scores of
1
z = −2.0
2
z = 1.51
3
z = 2.575
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the data values corresponding to
z-scores of
1
z = −2.0
x − 24.7
3.4
(−2.0)(3.4) + 24.7 = x
−2.0 =
x
2
z = 1.51
3
z = 2.575
J. Robert Buchanan
= 17.9
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the data values corresponding to
z-scores of
1
z = −2.0
x − 24.7
3.4
(−2.0)(3.4) + 24.7 = x
−2.0 =
x
2
= 17.9
z = 1.51
x − 24.7
3.4
= 29.8
1.51 =
x
3
z = 2.575
J. Robert Buchanan
Measures of Position
Example
If µ = 24.7 and σ = 3.4, find the data values corresponding to
z-scores of
1
z = −2.0
x − 24.7
3.4
(−2.0)(3.4) + 24.7 = x
−2.0 =
x
2
= 17.9
z = 1.51
x − 24.7
3.4
= 29.8
1.51 =
x
3
z = 2.575
J. Robert Buchanan
Measures of Position
Percentiles
Definition
The k th percentile, denoted Pk , of a set of data divides the
lower k % of a data set from the upper (100 − k )%. There are
99 percentiles.
J. Robert Buchanan
Measures of Position
Percentiles
Definition
The k th percentile, denoted Pk , of a set of data divides the
lower k % of a data set from the upper (100 − k )%. There are
99 percentiles.
Interpretation: The data value at the 40th percentile separates
the lower 40% of the data from the upper 60% of the data.
J. Robert Buchanan
Measures of Position
Determining the k th Percentile
1
Arrange the data in ascending order.
2
Compute an index i using the formula:
k
i=
(n + 1)
100
where k is the percentile and n is the number of
observations in the data set.
3
If i is a whole number, the k th percentile is the ith data
value. If i is not a whole number, the k th percentile is the
mean of the observations on either side of i.
J. Robert Buchanan
Measures of Position
Example
Consider the data:
16
21
22
23
24
25
25
27
28
29
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
26
27
28
30
18
21
22
23
24
25
26
27
28
30
19
21
22
23
24
25
26
27
29
30
19
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
27
29
31
Find P23 , P45 , P75 , and P90 .
J. Robert Buchanan
Measures of Position
20
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
33
Solution
To find P23 :
23
Index, i =
(150 + 1) = 34.73 (not a whole number)
100
The 34th and 35th values of the variable are respectively,
22 and 22.
The mean of 22 and 22 is 22, thus P23 = 22.
J. Robert Buchanan
Measures of Position
Solution
To find P23 :
23
Index, i =
(150 + 1) = 34.73 (not a whole number)
100
The 34th and 35th values of the variable are respectively,
22 and 22.
The mean of 22 and 22 is 22, thus P23 = 22.
To find P75 :
Index, i =
75
100
(150 + 1) = 113.25 (not a whole
number)
The 113th and 114th values of the variable are
respectively, 27 and 27.
The mean of 27 and 27 is 27, thus P75 = 27.
J. Robert Buchanan
Measures of Position
Finding the Percentile of a Given Data Value
1
Arrange the data in ascending order.
2
Use the following formula:
Percentile of x =
number of data values less than x
× 100
n
Round to the nearest whole number.
J. Robert Buchanan
Measures of Position
Example
Consider the data:
16
21
22
23
24
25
25
27
28
29
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
26
27
28
30
18
21
22
23
24
25
26
27
28
30
19
21
22
23
24
25
26
27
29
30
19
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
28
29
32
Find the percentile scores of 24, 25, 29, and 30.
J. Robert Buchanan
Measures of Position
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
33
Solution
Percentile of 24:
In other words, P40
60
× 100 = 40
150
= 24.
J. Robert Buchanan
Measures of Position
Solution
Percentile of 24:
In other words, P40
60
× 100 = 40
150
= 24.
Percentile of 30:
136
× 100 = 90.7 ≈ 91
150
In other words, P91 = 30.
J. Robert Buchanan
Measures of Position
Quartiles
Quartiles divide data sets into fourths.
Q1 = P25
Q2 = P50 = M = x̃
Q3 = P75
J. Robert Buchanan
Measures of Position
Example
Consider the data:
16
21
22
23
24
25
25
27
28
29
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
25
27
28
30
17
21
22
23
24
25
26
27
28
30
18
21
22
23
24
25
26
27
28
30
19
21
22
23
24
25
26
27
29
30
19
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
30
20
21
22
23
24
25
26
27
29
31
20
22
22
23
24
25
26
27
29
31
Find Q1 , Q2 , and Q3 .
J. Robert Buchanan
Measures of Position
20
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
32
21
22
22
23
24
25
26
28
29
33
Solution
i =
25
100
(150 + 1) = 37.75
22 + 22
Q1 = P25 =
= 22
2
50
(150 + 1) = 75.5
i =
100
24 + 25
Q2 = P50 =
= 24.5
2
75
i =
(150 + 1) = 113.25
100
27 + 27
Q3 = P75 =
= 27
2
J. Robert Buchanan
Measures of Position
Outliers
Extreme data values are called outliers. We may check for
them using the following procedure.
1
Determine Q1 and Q3 .
2
Compute the interquartile range:
IQR = Q3 − Q1
3
Determine the fences:
lower fence = Q1 − 1.5(IQR)
upper fence = Q3 + 1.5(IQR)
4
An outlier is a data value smaller than the lower fence or
larger than the upper fence.
J. Robert Buchanan
Measures of Position
Example
Determine if there are any outliers in the following data set.
3.02
4.90
7.90
9.11
11.33
3.79
5.50
7.92
9.29
11.56
3.91
7.00
8.05
9.60
11.72
3.99
7.11
8.37
9.81
11.72
4.60
7.31
8.50
10.30
11.80
4.71
7.45
8.50
10.72
11.97
Hint: Q1 = 7.16 and Q3 = 11.72.
J. Robert Buchanan
Measures of Position
4.80
7.66
8.79
10.74
12.57
4.85
7.85
9.10
10.89
12.89
Solution
1
Q1 = 7.16 and Q3 = 11.72.
2
Interquartile range: IQR = 11.72 − 7.16 = 4.56.
3
Fences:
lower fence = 7.16 − (1.5)(4.56) = 0.32
upper fence = 11.32 + (1.5)(4.56) = 18.16
4
There are no outliers in the data set.
J. Robert Buchanan
Measures of Position