Measures of Position

1
Section 2.6 Measures of Position
2
Who did better?

• Sue scored 68 on an exam in which the class mean
  was 54 with a standard deviation of 7
• John scored 32 on an exam in which the class mean
  was 28 with a standard deviation of 1.5.

3
Measures of Position

• z Score (or standard score)
• the number of standard deviations that a given
  value x is above or below the mean

4
Measures of Position z score

• Sample

x - x
z
s
5
Measures of Position z score

• Sample

Population
x - µ
x - x
z
z
?
s
Always round z to two decimal places!
6
Finding z-scores

• Mean heights of women
• ? 63.6 ? 2.5
• Mean heights of men
• ? 69.0 ? 2.8
• Find Michael Jordons z score (78 tall)
• Find Rebecca Lobos z score (76 tall)
• Find Mugsy Bogues z score (63 tall)

7
Finding x when z is known

• Mean heights of women
• ? 63.6 ? 2.5
• Mean heights of men
• ? 69.0 ? 2.8
• Find the height of a man whose z score is -0.35
• Find the height of a woman whose z score is 1.67
• Find the height of a man whose z score is 2.11

8
FIGURE 2-16
Interpreting Z Scores
Unusual Values
Unusual Values
Ordinary Values
- 3
- 2
- 1
0
1
2
3
Z
9
IQs are normally distributed with a mean of 100
and a standard deviation of 15. Find the z-score
for a person with an IQ of 140. Is this person
unusually intelligent?
10
IQs are normally distributed with a mean of 100
and a standard deviation of 15. Find the z-score
for a person with an IQ of 140. Is this person
unusually intelligent?
z 140 - 100
2.67
15
11
Find the z-score interpret

• Find the z-score raw score 15.7 mean 15.8
  s 0.3
• Find the z-score raw score 14.1 mean
  14.2 s .25

12
Find the z-score interpret
z 15.7 15.8
-0.33
0.3
z 14.1 14.2
-0.40
0.25
13
Measures of Position
Quartiles, Deciles, Percentiles
14
Quartiles
15
Quartiles
Q1, Q2, Q3
16
Quartiles
Q1, Q2, Q3 divides ranked scores into four
equal parts
17
Quartiles
Q1, Q2, Q3 divides ranked scores into four
equal parts
25
25
25
25
Q3
Q2
Q1
18
Quartiles
Q1, Q2, Q3 divides ranked scores into four
equal parts
25
25
25
25
Q3
Q2
Q1
(minimum)
(maximum)
(median)
19
Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked
data into ten equal parts
20
Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked
data into ten equal parts
21
Percentiles

• 99 Percentiles

22
Quartiles

• Q1 P25
• Q2 P50
• Q3 P75

23
Quartiles
Deciles

• Q1 P25
• Q2 P50
• Q3 P75

24
Finding the Percentile of a Given Score
25
Finding the Percentile of a Given Score
number of scores less than x
Percentile of score x
100
total number of scores

• (table 2-12 p. 89)
• Find percentile rank of 112
• Find percentile rank of 222
• Find percentile rank of 284

26
Find the percentile for the given score
Data 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 7, 8, 9,
9, 9, 9, 10, 11, 11, 11, 12, 14, 16, 19
4
9
16
27
Finding the Score Given a Percentile
28
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
29
Finding the Value of the kth Percentile
Start
Sort the data. (Arrange the data in order of
lowest to highest.)
Compute L n where n number
of values k percentile in question
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.
Is L a whole number ?
Yes
No
Change L by rounding it up to the next larger
whole number.
The value of Pk is the Lth value, counting from
the lowest
30
Finding Pk

• (table p. 89)
• P25
• P60
• D7
• Q3

31
Data 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 7, 8, 9,
9, 9, 9, 10, 11, 11, 11, 12, 14, 16, 19
Find the score corresponding to the percentile
P65
P30
D2
32
Other Measures of Position

• Interquartile Range (IQR)
• (Q3 - Q1)