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7/27/2019 032 Measures of Position
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Measures of PositionMeasures of PositionMeasures of PositionMeasures of PositionMeasures of PositionMeasures of PositionMeasures of PositionMeasures of Position
byHj Ahmad Zawawi bin Abdullah
Measures of position are used to describethe relative location of an observation
Quartiles and percentiles are two of themost popular measures of position
An additional measure of central tendency,the midquartile, is defined using quartiles
Quartiles are part of the 5-numbersummary
Measures of PositionMeasures of Position
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25% 25% 25% 25%
L Q1
Q2
Q3
H
Ranked data, increasing order
1. The first quartile, Q1, is a number such that atmost 25% of the data are smaller in value than Q1and at most 75% are larger
2. The second quartile, Q2, is the median
3. The third quartile, Q3, is a number such that atmost 75% of the data are smaller in value than Q3and at most 25% are larger
Quartiles: Values of the variable that divide the ranked
data into quarters; each set of data has three quartiles
QuartilesQuartiles
Median of Grouped DataMedian of Grouped Data
Median =L + (N s ) x c2 f
L = LCL of median class (= 69.5)
N = f = total frequency (= 20)
s = total frequency before median class (= 9)
f = frequency of median class (= 5)
c = class size = (74.5 69.5 = 5)
Median = 69.5 + (20 9) x (74.5 69.5) = 70.5
2 5
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Class
Interval
Class Limit Class Mid-
point (m)
Frequency (less
than UCL) (f) cf
50 54 49.5 54.5 52 1 1
55 59 54.5 59.5 57 1 2
60 64 59.5 64.5 62 2 4
65 69 64.5 69.5 67 5 9
70 74 69.5 74.5 72 5 14
75 79 74.5 79.5 77 2 16
80 85 79.5 84.5 82 2 18
85 89 84.5 89.5 87 2 20
Median = 69.5 + (20 9) x (74.5 69.5) = 70.5
2 5
Median
Class
Second Quartile QSecond Quartile Q22
= Median= Median
Q2 = Median =L + (N s ) x c2 f
L = LCL of median class Q2 (= 69.5)
N = f = total frequency (= 20)
s = total frequency before median classQ2 (= 9)
f = frequency of median class Q2 (= 5)
c = class size (=74.5 69.5 = 5)
Q2 = 69.5 + (20 9) x (74.5 69.5) = 70.5
2 5
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First Quartile QFirst Quartile Q11
Q1 =L + (N s ) x c4 f
L = LCL of Q1 class (= 64.5)
N = f = total frequency (= 20)
s = total frequency before Q1 class (= 4)
f = frequency of Q1 class (= 5)
c = class size (= 69.5 64.5 = 5)
Q1 = 64.5 + (20 4) x (69.5 64.5) = 65.54 5
Class
Interval
Class Limit Class Mid-
point (m)
Frequency (less
than UCL) (f) cf
50 54 49.5 54.5 52 1 1
55 59 54.5 59.5 57 1 2
60 64 59.5 64.5 62 2 4
65 69 64.5 69.5 67 5 9
70 74 69.5 74.5 72 5 14
75 79 74.5 79.5 77 2 16
80 85 79.5 84.5 82 2 18
85 89 84.5 89.5 87 2 20
Q1 = 64.5 + (20 4) x (69.5 64.5) = 65.5
4 5
Q1Class
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Third Quartile QThird Quartile Q33
Q3 =L + (3N s ) x c4 f
L = LCL of Q3 class (=74.5)
N = f = total frequency (=20)
s = total frequency before Q3 class (=14)
f = frequency of Q3 class (= 2)
c = class size (=79.5 74.5 = 5)
Q3 = 74.5 + (3x20 14) x (79.5 74.5) = 76.0
4 2
Class
Interval
Class Limit Class Mid-
point (m)
Frequency (less
than UCL) (f) cf
50 54 49.5 54.5 52 1 1
55 59 54.5 59.5 57 1 2
60 64 59.5 64.5 62 2 4
65 69 64.5 69.5 67 5 9
70 74 69.5 74.5 72 5 14
75 79 74.5 79.5 77 2 16
80 85 79.5 84.5 82 2 18
85 89 84.5 89.5 87 2 20
Q3 = 74.5 + (3x20 14) x (79.5 74.5) = 76.04 2
Q3Class
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Percentiles: Values of the variable that divide a set of
ranked data into 100 equal subsets; each set of data
has 99 percentiles. The kth percentile, Pk, is a value
such that at most k% of the data is smaller in value
than Pk
and at most (100 k)% of the data is larger.
at most k% at most (100 - k)%
Pk
L H
~x Q P= =2 50
Notes:
The 1st quartile and the 25th percentile are the same: Q1
= P25
The median, the 2nd quartile, and the 50th percentile are
all the same:
PercentilesPercentiles
1% 1% 1% 1% 1% 1%
P1
P2
P3
P97
P98
P99
Pk= the th value
100
kN
PercentilesPercentiles
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Procedure for finding Pk:
1. Rank the nobservations, lowest to highest
2. Compute A = (nk)/100
3. If A is an integer:
d(Pk) = A.5 (depth)
Pk is halfway between the value of the data in the Athposition and the value of the next data
If A is a fraction: d(Pk) = B, the next larger integer
Pk is the value of the data in the Bth position
PercentilePercentile Pk of Ungrouped Dataof Ungrouped Data
1) k= 25: (20) (25) / 100 = 5, depth = 5.5, Q1 = 6
5.6 5.6 5.8 5.9 6.0
6.0 6.1 6.2 6.3 6.46.7 6.8 6.8 6.8 6.9
7.0 7.3 7.4 7.4 7.5
Example:The following data represents the pH levels of arandom sample of swimming pools in a town.
Find: 1) the first quartile, 2) the third quartile,and 3) the 37th percentile:
2) k= 75: (20) (75) / 100 = 15, depth = 15.5, Q3 = 6.95
3) k= 37: (20) (37) / 100 = 7.4, depth = 8, P37 = 6.2
Solutions:
ExampleExample
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Percentile of Ungrouped DataPercentile of Ungrouped Data
Example
kth percentile, Pk= value at position
100
kN
53 58 68 73 75 76 79 80 85 88 91 99
Depth of 62th percentile = = at position 7.44
100
kN ( )
100
1262
P62 = 62 th percentile = = 79.52
8079 +
62% lies below 79.5 or 38% is above 79.5.
Percentile PPercentile Pkk
of Grouped Dataof Grouped Data
Pk = L + (kN s ) x c100 f
L = LCL of Pk class
N = f = total frequency
s = total frequency before Pkclass
f = frequency of Pkclass
c = class size
Pk= L + (kxN s) x c =
100 f
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Percentile PPercentile Pkk
X f cf cf %
38 1 125 100
37 1 124 99
36 3 123 98
35 5 120 96
34 9 115 92
33 8 106 85
32 17 98 78
31 23 81 65
30
29
24 58 46
18 34 27
28 10 16 13
27 3 6 526 1 3 3
25 0 2 2
24 2 2 2
N= 125
kN- cff
P25
= 29.35
P25
= L +
= 28.5 + 0.85
k= 25/100
Percentile PPercentile P2525
= Quartile Q= Quartile Q11
P25 = Q1 =L + (N s ) x c4 f
L = LCL of class P25
N = f = total frequency
s = total frequency before class P25
f = frequency of class P25
c = class size
P25 = 64.5 + (50 3) x (64.5 59.5) =4 10
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Percentile PPercentile P5050 = Quartile Q= Quartile Q22
= Median= MedianP50 = Q2 = Median =L + (N s ) x c
2 f
L = LCL of median class P50
N = f= total frequency
s = total frequency before median class P50
f = frequency of median class P50
c = class size
P50 = 69.5 + (50 13) x (74.5 69.5) = 72.5
2 20
Percentile PPercentile P7575
= Quartile Q= Quartile Q33
P75 = Q3 = L + (3N s ) x c4 f
L = LCL of class P75
N = f = total frequency
s = total frequency before class P75
f = frequency of class P75
c = class size
P75 = 74.5 + (3x50 33) x (74.5 69.5) =
4 15
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Midquartile: The numerical value midway between the firstand third quartile:
midquartile =+Q Q
1 3
2
475.62
95.12
2
95.66
2emidquartil 31 ==
+=
+=
Example: Find the midquartile for the 20 pH values inthe previous example:
Note:The mean, median, midrange, and midquartile are all
measures of central tendency. They are notnecessarily equal. Can you think of an example whenthey would be the same value?
MidquartileMidquartile
5-Number Summary: The 5-number summary is composedof: 1. L, the smallest value in the data set
2. Q1, the first quartile (also P25)
3. , the median (also P50 and 2nd quartile)
4. Q3, the third quartile (also P75)
5. H, the largest value in the data set
~x
Notes:
The 5-number summary indicates how much the data is
spread out in each quarter The interquartile range is the difference between the
first and third quartiles. It is the range of the middle 50%of the data
55--Number SummaryNumber Summary
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Box-and-Whisker Display: A graphic representation of the5-number summary:
The five numerical values (smallest, first quartile, median, thirdquartile, and largest) are located on a scale, either vertical orhorizontal
The box is used to depict the middle half of the data that liesbetween the two quartiles
The whiskers are line segments used to depict the other half ofthe data
One line segment represents the quarter of the data that is
smaller in value than the first quartile The second line segment represents the quarter of the data
that is larger in value that the third quartile
BoxBox--andand--Whisker DisplayWhisker Display
63 64 76 76 81 83 85 86 8889 90 91 92 93 93 93 94 9799 99 99 101 108 109 112
Example: A random sample of students in a sixth gradeclass was selected. Their weights are given in the tablebelow. Find the 5-number summary for this data andconstruct a boxplot:
63 85 92 99 112
L HQ1 Q3~x
Solution:
ExampleExample
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Weights from Sixth Grade Class
11010090807060
L Q1
~x Q3
H
Weight
Boxplot for Weight DataBoxplot for Weight Data