032 Measures of Position

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

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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

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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)


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

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First Quartile QFirst Quartile Q11

Q1 =L + (N s ) x c4 f

L = LCL of Q1 class (= 64.5)


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


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

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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

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Class

Interval


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


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


s = total frequency before class P75

f = frequency of class P75

c = class size

P75 = 74.5 + (3x50 33) x (74.5 69.5) =

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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 ==

+=

+=

QQ

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

032 Measures of Position

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