1Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSSection 2-6 Measures of Position

2Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Position

3Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

z Score (or standard score)

the number of standard deviations that a given value x is above or

below the mean

Measures of Position

4Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sample

z = x - xs

Measures of Positionz score

5Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sample

z = x - xs

Population

z = x - µ

Measures of Positionz score

6Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sample

z = x - xs

Population

z = x - µ

Round to 2 decimal places

Measures of Positionz score

7Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

- 3 - 2 - 1 0 1 2 3

Z

Unusual Values

Unusual Values

OrdinaryValues

Interpreting Z Scores

FIGURE 2-16

8Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Position

Quartiles, Deciles,

Percentiles

9Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Quartiles

10Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Q1, Q2, Q3

Quartiles

11Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Q1, Q2, Q3

divides ranked scores into four equal parts

Quartiles

12Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Q1, Q2, Q3 divides ranked scores into four equal parts

Quartiles

25% 25% 25% 25%

Q3Q2Q1

13Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Q1, Q2, Q3 divides ranked scores into four equal parts

Quartiles

25% 25% 25% 25%

Q3Q2Q1(minimum) (maximum)

(median)

14Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

D1, D2, D3, D4, D5, D6, D7, D8, D9

divides ranked data into ten equal parts

Deciles

15Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

D1, D2, D3, D4, D5, D6, D7, D8, D9

divides ranked data into ten equal parts

Deciles

10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

D1 D2 D3 D4 D5 D6 D7 D8 D9

16Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

99 Percentiles

Percentiles

17Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Quartiles, Deciles, Percentiles

Fractiles

18Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Quartiles, Deciles, Percentiles

Fractiles

(Quantiles)partitions data into approximately equal parts

19Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Percentile of a Given Score

20Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Percentile of a Given Score

Percentile of score x = • 100number of scores less than x

total number of scores

21Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Score Given a Percentile

22Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Score Given a Percentile

n total number of values in the data set

k percentile being used

L locator that gives the position of a value

Pk kth percentile

L = • nk100

23Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Value of the

kth Percentile

Sort the data.

(Arrange the data in

order of lowest to

highest.)

The value of the kth percentile

is midway between the Lth value

and the next value in the

sorted set of data. Find Pk by

adding the L th value and the

next value and dividing the

total by 2.

Start

Compute

L = n where

n = number of values

k = percentile in question

)( k100

Change L by rounding

it up to the next

larger whole number.

The value of Pk is the

Lth value, counting from the lowest

Is L a whole

number?

Yes

No

Figure 2-17

24Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Quartiles

Q1 = P25

Q2 = P50

Q3 = P75

25Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Quartiles

Q1 = P25

Q2 = P50

Q3 = P75

D1 = P10

D2 = P20

D3 = P30

• • •

D9 = P90

Deciles

26Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Interquartile Range (or IQR): Q3 - Q1

27Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Interquartile Range (or IQR): Q3 - Q1

Semi-interquartile Range:2

Q3 - Q1

28Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Interquartile Range (or IQR): Q3 - Q1

Semi-interquartile Range:

Midquartile: 2

2

Q3 - Q1

Q1 + Q3

29Chapter 2. Section 2-6. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Interquartile Range (or IQR): Q3 - Q1

Semi-interquartile Range:

Midquartile:

10 - 90 Percentile Range: P90 - P10

2

2

Q3 - Q1

Q1 + Q3