Download The Mean and Standard Deviation of Continuous

The Mean and Standard Deviation of
Continuous Random Variables
Remember at the beginning of this course we manually calculated measures of central tendency such as
the mean, median, and mode, we also looked at measures of spread, such as the range, percentiles, and
standard deviation. The calculations for theses questions took a lot of time, but could be solved using
spreadsheet programs or a graphing calculator.
In this section, you will focus on analyzing the mean and standard deviation of continuous probability
distributions. Recall that standard deviation is a measure of spread that tells you how closely the data is
clustered around the middle (usually the mean) of the data set. The larger the standard deviation is, the
larger the variation in the data will be.
Example
A town’s community centre runs English Language Learner (ELL) classes every Tuesday, Wednesday, and
Thursday evening. Every person is enrolled in one of the three classes and is to attend the same class
each week, as space is limited in the classroom.
38
43
55
(a)
(b)
(c)
(d)
54
60
60
65
84
68
57
58
74
66
72
98
42
67
59
62
80
76
65
88
71
71
70
45
67
93
56
Group these test scores and display them in a frequency polygon.
Determine the mean and standard deviation of these test scores.
State the test scores that are one standard deviation (1SD) and two standard deviation (2SD)
above the mean, as well as the test scores that are one (-1SD) and two (-2SD) standard
deviations below the mean.
How many standard deviations are test scores of 60 and 90 away from the mean?
Solution
(a)
Test Scores
0-9
10 - 19
20 – 29
30 – 39
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 - 100
Frequency
-1
00
90
89
80
–
79
70
–
69
60
–
59
50
–
49
–
40
–
30
–
20
39
10
9
8
7
6
5
4
3
2
1
0
29
Frequency
Frequency Polygon of Tuesday Night Test Scores in ELL
Class
Test Scores
(b) Mean is 65.5, Standard Deviation 14.1 the values have been rounded to one decimal place.
(c) A test score 1SD away from the mean = ______ - _____ = ______
A test score 2SD away from the mean = ______ - __________ = ______
A test score -1SD away from the mean = ______ - _____ = ______
A test score -2SD away from the mean = ______ - __________ = ______
(d) The number of standard deviations that a score is away from the mean is also known as a z-score.
x – the test score
μ – the mean
δ – the standard deviation
z – the number of standard deviations away from the mean (z – score)
x = μ +(δ x z)
z
x

Example
Here are the test scores of the students in Wednesday evening’s class:
28
57
57
44
56
70
55
32
78
64
52
60
88
55
83
49
61
45
66
57
50
61
74
58
35
48
33
46
62
50
(a) Group theses scores and display them in the same frequency polygon used for the Tuesday evening
class.
(b) Determine the mean and standard deviation of these test scores.
(c) State the test scores that are one standard deviation (1SD) and two standard deviation (2SD)
above the mean, as well as the test scores that are one (-1SD) and two (-2SD) standard
deviations below the mean.
(d) How many standard deviations are test scores of 60 and 90 away from the mean?
Solution
(a)
Test Scores
0-9
10 - 19
20 – 29
30 – 39
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 - 100
Frequency
Test Scores
-1
00
90
89
80
–
79
70
–
69
60
–
59
50
–
49
40
–
39
–
30
–
29
10
9
8
7
6
5
4
3
2
1
0
20
Frequency
Frequency Polygon of Tuesday Night Test Scores in ELL
Class
(b) Mean is 65.5, Standard Deviation 17.1 the values have been rounded to one decimal place.
(c) A test score 1SD away from the mean = ______ - _____ = ______
A test score 2SD away from the mean = ______ - __________ = ______
A test score -1SD away from the mean = ______ - _____ = ______
A test score -2SD away from the mean = ______ - __________ = ______
(d) Score of 90:
Score of 60:
Example
Here are the test scores of the students in Thursday evening’s class:
69
48
58
62
84
73
95
85
82
92
65
49
39
68
65
94
46
55
39
75
57
58
59
64
75
77
28
82
78
45
(a) Group theses scores and display them in the same frequency polygon used for the Tuesday evening
class.
(b) Determine the mean and standard deviation of these test scores.
(c) State the test scores that are one standard deviation (1SD) and two standard deviation (2SD)
above the mean, as well as the test scores that are one (-1SD) and two (-2SD) standard
deviations below the mean.
(d) How many standard deviations are test scores of 60 and 90 away from the mean?
Solution
(a)
Test Scores
0-9
10 - 19
20 – 29
30 – 39
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
90 - 100
Frequency
-1
00
90
89
80
–
79
70
–
69
60
–
59
50
–
49
–
40
–
30
–
20
39
10
9
8
7
6
5
4
3
2
1
0
29
Frequency
Frequency Polygon of Tuesday Night Test Scores in ELL
Class
(b) Mean is 65.5, Standard Deviation 17.1 the values
have been rounded to one decimal place.
Test Scores
(c) A test score 1SD away from the mean = ______ - _____ = ______
A test score 2SD away from the mean = ______ - __________ = ______
A test score -1SD away from the mean = ______ - _____ = ______
A test score -2SD away from the mean = ______ - __________ = ______
(d) Score of 90:
Score of 60: