Download 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