Something to Think About:

•“A pediatrician reports that a child is in the 90th

percentile for heights among children of that

age.”

Conclusion:

•This means 90% of all children of that age are shorter than the given child. The child is taller

from the average.

Chapter 5Measures of Position

Lesson 5.1Measures of Position for

Ungrouped Data

Measures of Position•These are used to describe

the location of a specific piece of data in relation to

the rest of the sample.

•It is also called as “Measures of Quantiles”

Measures of Position

•Quartiles

•Deciles

•Percentiles

1. Quartiles

•The values that divide a rank-ordered

data set into four equal parts.

Each set of data has 3 Quartiles:

•1st Quartile

•2nd Quartile

•3rd Quartile

1st Quartile •It is denoted by 𝑸𝟏.

•It is a number such that at most, ¼ of the data is smaller than 𝑸𝟏 and most ¾ of the data is more than 𝑸𝟏.

2nd Quartile•It is denoted by 𝑸𝟐.

•It is the media.

•It is a number such that half is less than 𝑸𝟐 and the other half is greater than 𝑸𝟐.

3rd Quartile•It is denoted by 𝑸𝟑.

•It is a number such that at most ¾ of the data is less than 𝑸𝟑 and at most ¼ of the data is greater than 𝑸𝟑.

2. Deciles

•The values of the variable that divide a

set of ranked data into ten equal parts.

Note:

•Each set of data has 9 deciles, such

that,𝑫𝟏𝒕𝒐𝑫𝟗.

Percentile

•These are number values of the variable that divide a set of ranked data into

100 equal parts.

Note:

•Each set of data has 99 deciles, such

that,𝑷𝟏𝒕𝒐𝑷𝟗𝟗.

Example 5.1:•Given the following data:

26 51 44 23 25

61 45 65 23 43

41 55 34 35

Find (a) 𝑫𝟒, (b) 𝑷𝟔𝟎, (c)𝑸𝟑, (d) 𝑫𝟕, (e) 𝑷𝟐𝟓 and (f) 𝑸𝟐.

Steps in Solving forMeasures of Position for

Ungrouped Data

Step 1:Arrange the data in

ascending or descending order

Example 5.1:

23 23 25 26 34

35 41 43 44 45

51 55 61 65

Step 2:Solve for the location of the given quartile using the formula for

the location.

Formula 5.1:

•𝑳𝒐𝒄𝒂𝒕𝒊𝒐𝒏 = 𝑷(𝒏+𝟏)

𝒒

Where:•𝒑 = 𝒅𝒆𝒔𝒊𝒓𝒆𝒅 𝒑𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆•𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒔𝒄𝒐𝒓𝒆𝒔•𝒒 = 𝒒𝒖𝒂𝒏𝒕𝒊𝒍𝒆 (𝟒, 𝟏𝟎, 𝟏𝟎𝟎)

Step 3:Locate the score

corresponding to the obtained location in

the distribution.

Step 4:If the obtained location is not

exact, interpolate.

Steps in

InterpolationMeasures of Position for

Ungrouped Data

Step 5:Locate the inexact

location, and get the difference between the upper and lower score.

Step 6:Multiply the difference by the decimal part of

the inexact location obtained.

Step 7:Add the

product to the lower score.

Let’s PracticeMeasures of Position for

Ungrouped Data

Solve the following:Find the (a) 𝑸𝟑, (b) 𝑫𝟖 and (c) 𝑷𝟖𝟓.

1. Consider the samples 5, 7, 10, 11.

2. Consider the sample 9, 11, 8, 6 and 10.

3. Consider the samples: 14, 9, 18, 13, 16, 12, 15 and 15.

4. Consider the samples: 12.8, 16.4, 21.6, 20.0, 10.4, 19.1, 18.1, 14.6, 16.6, 11.1, 17.2, 14.0, 13.0, 13.6, 15.3, 16.7, 15.1, 14.4, 19.2, 19.3.

Lesson 5.2Measures of Position for

Grouped Data

Consider the Given Frequency

Distribution:Class Intervals f

62 – 68 8

69 – 75 5

76 – 82 9

83 – 89 11

90 – 96 12

97 – 103 10

104 – 110 5

111 – 117 5

n = 65

From the given frequency distribution, determine:(a)𝑸𝟐;(b)𝑫𝟕;(c)𝑷𝟒𝟓;(d)𝑸𝟑;(e)𝑫𝟑;(f)𝑷𝟗𝟓.

Steps in Solving forMeasures of Position for

Grouped Data

Step 1:Record the cumulative frequencies

Consider the Given Frequency

Distribution:Class Intervals f f≤

62 – 68 8 8

69 – 75 5 13

76 – 82 9 22

83 – 89 11 33

90 – 96 12 45

97 – 103 10 55

104 – 110 5 60

111 – 117 5 65

n = 65

Step 2:Determine the

𝒑𝒏

𝒒and

identify the class

interval where 𝒑𝒏

𝒒falls.

Where:•𝒑 = 𝒅𝒆𝒔𝒊𝒓𝒆𝒅 𝒑𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆•𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒔𝒄𝒐𝒓𝒆𝒔•𝒒 = 𝒒𝒖𝒂𝒏𝒕𝒊𝒍𝒆 (𝟒, 𝟏𝟎, 𝟏𝟎𝟎)

Quartile:𝒑𝒏

𝟒

Decile:𝒑𝒏

𝟏𝟎

Percentile:𝒑𝒏

𝟏𝟎𝟎

Step 3:Determine the

Lower Real Limit

Step 4:Use the formula

pth Quartile:

𝑸𝒑 = 𝑳𝑳𝑹 +

𝒑𝒏𝟒− 𝒇 ≤

𝒇𝒘

pth Decile:

𝑫𝒑 = 𝑳𝑳𝑹 +

𝒑𝒏𝟏𝟎− 𝒇 ≤

𝒇𝒘

pth Percentile:

𝑷𝒑 = 𝑳𝑳𝑹 +

𝒑𝒏𝟏𝟎𝟎

− 𝒇 ≤

𝒇𝒘

Final Answer:

𝑸𝟐 = 𝟖𝟗. 𝟏𝟖(𝑴𝒆𝒅𝒊𝒂𝒏)

Final Answer:

𝑫𝟕 = 𝟗𝟔. 𝟖𝟓

Final Answer:

𝑷𝟒𝟓 = 𝟖𝟕. 𝟏𝟏

Final Answer:

𝑸𝟑 = 𝟖𝟎. 𝟓𝟔

Final Answer:

𝑫𝟑 = 𝟗𝟔. 𝟖𝟓

Final Answer:

𝑷𝟗𝟓 = 𝟔𝟏. 𝟕𝟓

Let’s PracticeMeasures of Position for

Ungrouped Data

Consider the Given Frequency

Distribution: From the given frequency distribution, determine:(a)𝑸𝟏;(b)𝑫𝟖;(c)𝑷𝟖𝟓;(d)𝑫𝟑;(e)𝑷𝟐;(f)𝑷𝟔𝟎.

Class Interval

f f≤

7 – 9 2

10 – 12 10

13 – 15 12

16 – 18 9

19 – 21 7

n = 40

Consider the Given Frequency

Distribution: From the given frequency distribution, determine:(a)𝑸𝟑;(b)𝑫𝟓;(c)𝑷𝟔𝟓;(d)𝑫𝟐;(e)𝑷𝟒𝟎;(f)𝑷𝟏𝟓.

Class Interval F f≤

152 – 165 4

138 – 151 7

124 – 137 4

110 – 123 9

96 – 109 13

82 – 95 8

68 – 81 9

54 – 67 11

n = 65

Consider the Given Frequency

Distribution: From the given frequency distribution, determine:(a)𝑸𝟐;(b)𝑫𝟖;(c)𝑷𝟗𝟎;(d)𝑫𝟕;(e)𝑷𝟒𝟓;(f)𝑷𝟏𝟎.

Class Interval F f≤

96 – 100 4

91 – 95 4

86 – 90 6

81 – 85 3

76 – 80 7

71 – 75 3

66 – 70 2

61 – 65 1

n = 30

Lesson 5.3:Other Measures

of Central Tendency

Three Additional Measures of

Central Tendency:

•The Trimean

•The Geometric Mean

•The Trimmed Mean

The TrimeanOther Measures of

Central Tendency

Trimean•The trimean is the weighted average of

the 25th percentile, the 50th percentile, and the

75th percentile.

Formula:

•𝑻𝒓𝒊𝒎𝒆𝒂𝒏 = 𝑷𝟐𝟓+𝟐𝑷𝟓𝟎+𝑷𝟕𝟓

𝟒

Letting:

•𝑷𝟐𝟓 be the 25th percentile

•𝑷𝟓𝟎 be the 50th

•𝑷𝟕𝟓 be the 75th percentile

Example: The ages of passengers in a

plane going to Barcelona.

37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 20

19 19 18 18 18 18 16 15

14 14 14 12 12 9 6

Step 1:

Using the formula for getting the location,

determine 𝑷𝟐𝟓, 𝑷𝟓𝟎 𝒂𝒏𝒅 𝑷𝟕𝟓.

𝑷𝟐𝟓37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 20

19 19 18 18 18 18 16 1514 14 14 12 12 9 6

𝑷𝟓𝟎37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 2019 19 18 18 18 18 16 15

14 14 14 12 12 9 6

𝑷𝟕𝟓37 33 33 32 29 28 28 2322 22 22 21 21 21 20 20

19 19 18 18 18 18 16 15

14 14 14 12 12 9 6

Table 1. The Percentiles of the ages of

passengers of a plane going to Barcelona.

Percentile Value

25 15

50 20

75 23

Therefore, using the formula:

•𝑻𝒓𝒊𝒎𝒆𝒂𝒏 = 𝟏𝟗. 𝟓

The Geometric MeanOther Measures of

Central Tendency

Geometric Mean

•The geometric mean is computed by multiplying all the numbers together and then taking the nth

root of the product.

Example:

•Consider the set of data: 1, 10 and 100.

Determine the geometric mean.

Formula:

•𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄 𝑴𝒆𝒂𝒏 = 𝒊=𝟏𝒏 𝒙𝒊

𝟏

𝑵

Π•“Pi”, it is the 16th

letter of the Greek alphabet. It means to

multiply.

Example: The ages of passengers in a

plane going to Barcelona.

37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 20

19 19 18 18 18 18 16 15

14 14 14 12 12 9 6

Therefore, using the formula:

•𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄 𝑴𝒆𝒂𝒏 = 𝟏𝟗. 𝟏𝟎

Note:

•The geometric mean only makes sense if all

the numbers are positive.

What is the application of

geometric mean in real life?

•In business world, the geometric mean is an

appropriate measure to use for averaging rates.

Example:

•For example, consider a stock portfolio that began with a

value of $1,000 and had annual returns of 13%, 22%,

12%, -5%, and 13%.

Annual Rate of Return•In finance, it is a profit on an investment. It comprises any

change in value and interest or dividends or other such cash

flows which the investor receives from the investment.

Table 2. The annual returns of

a stock portfolio

Year Return Value

1 13% $1,130

2 22% $1,379

3 12% $1,544

4 -5% $1,467

5 -13% $1,276

Question:

•How to compute the average annual

rate of return?

Answer:

•To compute the geometric mean of

the returns.

How:•Instead of using the

percentages, each return is represented as a multiplier

indicating how much higher the value is after the year.

Then:•1.13 for a 13% return

•1.22 for a 22% return

•1.12 for a 12% return

•0.95 for a -5% return

•0.87 for a -13%

Therefore:

•The geometric mean of these multipliers is 1.05. Therefore, the average

annual rate of return is 5%.

Comparing:•The table 3 shows how a portfolio gaining 5% a year

would end up with the same value ($1,276) as

shown in table 2.

Table 3. The annual returns of

a stock portfolio

Year Return Value

1 5% $1,050

2 5% $1,103

3 5% $1,158

4 5% $1,216

5 5% $1,276

The Trimmed MeanOther Measures of

Central Tendency

Trimmed Mean•To compute a trimmed

mean, you remove some of the higher and lower scores and compute the mean of

the remaining scores.

10% trimmed mean•A mean trimmed 10% is a mean computed with 10% of the scores trimmed off: 5% from the bottom and

5% from the top.

50% trimmed mean•A mean trimmed 50% is

computed by trimming the upper 25% of the scores and the lower 25% of the scores and computing the mean of

the remaining scores.

Do you know?•The trimmed mean is

similar to the median which, in essence, trims the upper 49+% and the lower

49+% of the scores.

Do you know?

•The trimmed mean is a hybrid of the

mean and median.

Example:•Compute the mean trimmed 20% for the

ages of passengers of a plane going to

Barcelona.

Remove the lower 10% of the

scores:

37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 20

19 19 18 18 18 18 16 15

14 14 14 12 12 9 6

Remove the upper 10% of the

scores:

37 33 33 32 29 28 28 23

22 22 22 21 21 21 20 20

19 19 18 18 18 18 16 15

14 14 14 12 12 9 6

Using the Arithmetic Mean formula:

𝑻𝒓𝒊𝒎𝒎𝒆𝒅𝑴𝒆𝒂𝒏 = 𝟐𝟎. 𝟏𝟔

Let’s PracticeOther Measures of

Central Tendency

Solve the following problem:

(1 – 25) Given: 45, 67, 74, 48, 32, 10, 8, 61, 11, 12, 11, 14, 15, 11, 15, 21, 5, 55, 24, 30, 31

Determine the following:

(1)The trimean

(2)The geometric mean

(3)The (a) 30% trimmed mean, (b) 50% trimmed mean and (c) 80% trimmed mean

Solve the following problem:(26 – 30) Would it make sense to takethe geometric mean of these numbers:-9, -6, -4, -2, 0, 3, 5? Justify youranswer.

(31-35) What is (1) the trimean and (2)the geometric mean of: 2, 4, 5, 6, 6, 6,6, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12?