Jefferson Valley Bank

1
Waiting Times of Bank Customers at Different
Banks in minutes

• Jefferson Valley Bank
• Bank of Providence

6.5 4.2
6.6 5.4
6.7 5.8
6.8 6.2
7.1 6.7
7.3 7.7
7.4 7.7
7.7 8.5
7.7 9.3
7.7 10.0
2
Waiting Times of Bank Customers at Different
Banks in minutes

• Jefferson Valley Bank
• Bank of Providence

6.5 4.2
6.6 5.4
6.7 5.8
6.8 6.2
7.1 6.7
7.3 7.7
7.4 7.7
7.7 8.5
7.7 9.3
7.7 10.0
Jefferson Valley Bank
Bank of Providence
7.15 7.20 7.7 7.10
7.15 7.20 7.7 7.10
Mean Median Mode Midrange
3
Dotplots of Waiting Times
4
Section 2.5Measures of Variation
5
3 Measures of Variation(Consistency)

• Range

6
3 Measures of Variation(Consistency)

• Standard Deviation Average deviation from the
  mean.
• Variance

7
Sample Standard Deviation Formula
8
Sample Standard Deviation Formula
? (x - x)2

s
n - 1
9
Population Standard Deviation

? (x - µ)
2
?
n
10
Variance
standard deviation squared
s ??

2
Notation
2
11
Variance
Sample Variance
Population Variance
12
Round-off Rulefor measures of variation

• Carry one more decimal place than is present in
  the original set of values.
• Round only the final answer, never in the middle
  of a calculation.

13
Who did better?

• BERTS QUIZZES
• 20, 20, 20, 5, 20

• ERNIES QUIZZES
• 19, 15, 18, 17, 16

14
Who did better?
BERTS QUIZZES 20, 20, 20, 5, 20 Mean 17 Median
20 Mode 20 Midrange 12.5
ERNIES QUIZZES 19, 15, 18, 17, 16 Mean
17 Median 17 No Mode Midrange 17
15

• BERTS QUIZZES
• 20, 20, 20, 5, 20

• ERNIES QUIZZES
• 19, 15, 18, 17, 16

RANGE High - Low
Berts Range 20 5 15
Ernies Range 19 15 4
16
Berts Standard Deviation
17
2
18
2
Sum 180
19
Lets Find Ernies Standard Deviation and Variance
20
Ernies Standard Deviation
21
Usual Sample Values
22
Usual Sample Values

• minimum usual value ? (mean) - 2 (standard
  deviation)
• minimum ?
• maximum usual value ? (mean) 2 (standard
  deviation)
• maximum ?

23
IQs scores have a mean of 100 with a standard
deviation of 15.

• Minimum usual value

100 2(15) 70
Maximum usual value
100 2(15) 130
The USUAL RANGE for IQ scores is from 70 to 130.
Anyone not in this range is considered unusual.
24
The Empirical Rule - applies to bell-shaped
(normal) distributions
x
25
The Empirical Rule (applies to bell-shaped
distributions)
FIGURE 2-15
68 within 1 standard deviation
34
34
x - s
x s
x
26
The Empirical Rule (applies to bell-shaped
distributions)
FIGURE 2-15
95 within 2 standard deviations
68 within 1 standard deviation
34
34
13.5
13.5
x - 2s
x - s
x
x 2s
x s
27
The Empirical Rule (applies to bell-shaped
distributions)
FIGURE 2-15
99.7 of data are within 3 standard deviations of
the mean
95 within 2 standard deviations
68 within 1 standard deviation
34
34
2.4
2.4
0.1
13.5
13.5
x - 3s
x - 2s
x - s
x
x 2s
x 3s
x s
28
IQ scoresMean 100, Standard Deviation 15
100
Find the IQ scores for 1, 2, and 3 standard
deviations above and below the mean
29
IQ scoresMean 100, Standard Deviation 15
68
100
115
85
30
IQ scoresMean 100, Standard Deviation 15
Unusually High IQ
Unusually Low IQ
95
100
115
85
130
70
Usual Range
31
IQ scoresMean 100, Standard Deviation 15
99.7
100
115
85
130
70
145
55
32
Find the Usual Range for the following set of
exam scores
76, 85, 99, 64, 77, 82, 91, 72, 88, 72, 54, 76
Find x and s by running the stats in the
calculator.
Calculate x - 2s and x 2s
33
Standard Deviation from a Frequency Table
Formula 2-6

n ?(f x 2) -?(f x)2
S
n (n - 1)

• Use the class midpoints as the x values
• Calculators can compute the standard deviation
  for frequency table

34
Mean from a frequency table using technology

• P. 67 20
• Enter midpoints in L1, frequency in L2
• Stat - Calc - 1VarStats L1, L2

35
Range Rule of Thumb (RROT)

• The range rule of thumb can be used to get a
  rough estimate of the standard deviation
• s range/4

36
Section 2.6 Measures of Position
37
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.

38
Measures of Position

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

39
Measures of Position z score

• Sample

x - x
z
s
40
Measures of Position z score

• Sample

Population
x - µ
x - x
z
z
?
s
Always round z to two decimal places!
41
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)

42
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

43
FIGURE 2-16
Interpreting Z Scores
Unusual Values
Unusual Values
Ordinary Values
- 3
- 2
- 1
0
1
2
3
Z
44
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?
45
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
46
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

47
Find the z-score interpret
z 15.7 15.8
-0.33
0.3
z 14.1 14.2
-0.40
0.25