Transcript 3.3
Chapter 3
Section 3
Measures of Central Tendency and
Dispersion from Grouped Data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 1 of 19
Chapter 3 – Section 3
● Learning objectives
1
The mean from grouped data
2 The weighted mean
3
The variance and standard deviation for grouped data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 2 of 19
Chapter 3 – Section 3
● Data may come in groups rather than
individually
● The values may have been summarized in
frequency distributions
Ranges of ages (20 – 29, 30 – 39, ...)
Ranges of incomes ($10,000 – $19,999, $20,000 –
$39,999, $40,000 – $79,999, ...)
● The exact values for the mean, variance, and
standard deviation cannot be calculated
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 3 of 19
Chapter 3 – Section 3
● Learning objectives
1
The mean from grouped data
2 The weighted mean
3
The variance and standard deviation for grouped data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 4 of 19
Chapter 3 – Section 3
● To compute the mean for grouped data
Assume that, within each class, the mean of the data
is equal to the class midpoint
Use the class midpoint in the formula for the mean
The number of times the class midpoint value is used
is equal to the frequency of the class
● If 6 values are in the interval [ 8, 10 ] , then we
proceed as if all 6 values are equal to 9 (the
midpoint of [ 8, 10 ]
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 5 of 19
Chapter 3 – Section 3
● As an example, for the following frequency table,
0 – 1.9
2 – 3.9
4 – 5.9
6 – 7.9
Midpoint
1
3
5
7
Frequency
3
7
6
1
Class
we calculate the mean as if
The value 1 occurred 3 times
The value 3 occurred 7 times
The value 5 occurred 6 times
The value 7 occurred 1 time
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 6 of 19
Chapter 3 – Section 3
0 – 1.9
2 – 3.9
4 – 5.9
6 – 7.9
Midpoint
1
3
5
7
Frequency
3
7
6
1
Class
● The calculation for the mean would be
1 1 1 3 3 3 3 3 3 3 5 5 5 5 5 5 7
17
or
(1 3) (3 7) (5 6) (7 1)
17
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 7 of 19
Chapter 3 – Section 3
● Evaluating this formula
(1 3) (3 7) (5 6) (7 1)
61
3. 6
3 7 6 1
17
● The mean is about 3.6
● In mathematical notation
xi fi
fi
● This would be μ for the population mean and x
for the sample mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 8 of 19
Chapter 3 – Section 3
● Learning objectives
1
The mean from grouped data
2 The weighted mean
3
The variance and standard deviation for grouped data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 9 of 19
Chapter 3 – Section 3
● Sometimes not all data values are equally
important
● To compute a grade point average (GPA), a
grade in a 4 credit class is worth more than a
grade in a 1 credit class
● The weights wi quantify the relative importance
of the different values
● Higher weights correspond to more important
values
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 10 of 19
Chapter 3 – Section 3
● As an example, the following grades
Course
Statistics
French Literature
Biochemistry
Badminton
Credits
3
3
Grade
A
B
5
1
B
D
would yield a GPA (on a 4 point scale) of
(3 4) (3 3) (5 3) (1 1)
37
3.08
3 3 5 1
12
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 11 of 19
Chapter 3 – Section 3
● In mathematical notation, if wi is the weight
corresponding to the data value xi, then the
weighted mean is
w i xi
xw
wi
● This formula looks similar to one for the mean
for grouped data, and the concepts are similar
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 12 of 19
Chapter 3 – Section 3
● Learning objectives
1
The mean from grouped data
2 The weighted mean
3
The variance and standard deviation for grouped data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 13 of 19
Chapter 3 – Section 3
● To compute the variance for grouped data
Assume again that, within each class, the mean of the
data is equal to the class midpoint
Use the class midpoint in the formula for the variance
The number of times the class midpoint value is used
is equal to the frequency of the class
● If 6 values are in the interval [ 8, 10 ] , then we
assume that all 6 values are equal to 9 (the
midpoint of [ 8, 10 ]
● The same approach as for the mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 14 of 19
Chapter 3 – Section 3
● As an example, for the following frequency table,
0 – 1.9
2 – 3.9
4 – 5.9
6 – 7.9
Midpoint
1
3
5
7
Frequency
3
7
6
1
Class
we calculate the variance as if
The value 1 occurred 3 times
The value 3 occurred 7 times
The value 5 occurred 6 times
The value 7 occurred 1 time
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 15 of 19
Chapter 3 – Section 3
0 – 1.9
2 – 3.9
4 – 5.9
6 – 7.9
Midpoint
1
3
5
7
Frequency
3
7
6
1
Class
● From our previous example, the mean is 3.6
● Just as for the mean, the calculation for the
variance would then be
((1 3.6)2 3) ((3 3.6)2 7) ((5 3.6)2 6) ((7 3.6)2 1)
17
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 16 of 19
Chapter 3 – Section 3
● Evaluating this formula
((1 3.6)2 3) ((3 3.6)2 7) ((5 3.6)2 6) ((7 3.6)2 1)
17
46.1
2.7
17
● The variance is about 2.7
● The standard deviation would be about
2.7 1.6
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 17 of 19
Chapter 3 – Section 3
● In mathematical notation
● The population variance would be
2
( xi ) fi
fi
2
● The sample variance would be
2
(
x
x
)
fi
2
i
s
( fi ) 1
● The standard deviations would be the
corresponding square roots
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 18 of 19
Summary: Chapter 3 – Section 3
● The mean for grouped data
Use the class midpoints
Obtain an approximation for the mean
● The variance and standard deviation for grouped
data
Use the class midpoints
Obtain an approximation for the variance and
standard deviation
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 3 Section 3 – Slide 19 of 19