Section 2.3
Summer 2013 - Math 1040
June 3, 2013
(1040)
M 1040 - 2.3
June 3, 2013
1 / 10
Roadmap
Today we work with quantitative data. Remember that sigma, Σ, is the
shorthand symbol for addition. There is arithmetic used in today’s lesson:
addition of many numbers, multiplication, and division.
Describe means, medians, and modes.
Describe outliers and effects of them on centers.
Weighted means.
(1040)
M 1040 - 2.3
June 3, 2013
2 / 10
Mean
The mean of a data set is the sum of the entries divided by the sample
size.
sample mean: x̄ =
Σx
n
or
x̄ =
(1040)
x1 + x2 + · · · + xn
n
M 1040 - 2.3
June 3, 2013
3 / 10
Mean
The mean of a data set is the sum of the entries divided by the sample
size.
sample mean: x̄ =
Σx
n
or
x̄ =
x1 + x2 + · · · + xn
n
To find a mean,
1
Add all values together.
2
Divide to total by the sample size n.
(1040)
M 1040 - 2.3
June 3, 2013
3 / 10
Mean
The mean of a data set is the sum of the entries divided by the sample
size.
sample mean: x̄ =
Σx
n
or
x̄ =
x1 + x2 + · · · + xn
n
To find a mean,
1
Add all values together.
2
Divide to total by the sample size n.
The mean may be reasonably rounded. The units of measurement of a
mean is the same one of the data.
(1040)
M 1040 - 2.3
June 3, 2013
3 / 10
Mean
The data set gives the age in years of twenty students.
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The sum (20 + 20 + · · · + 24 + 65) = 475. The sample size n = 20. Then
x̄ =
(1040)
475
= 23.75 years.
20
M 1040 - 2.3
June 3, 2013
4 / 10
Median
The median of a data set is the middle value of the ordered data. The
median will separate the data into two half. When the sample size is odd,
the median is a data entry value. When the sample size is even, the
median is the average of the middle values.
(1040)
M 1040 - 2.3
June 3, 2013
5 / 10
Median
The median of a data set is the middle value of the ordered data. The
median will separate the data into two half. When the sample size is odd,
the median is a data entry value. When the sample size is even, the
median is the average of the middle values.
20
22
(1040)
20
22
20
22
20
23
20
23
20
23
M 1040 - 2.3
21
23
21
24
21
24
21
65
June 3, 2013
5 / 10
Median
The median of a data set is the middle value of the ordered data. The
median will separate the data into two half. When the sample size is odd,
the median is a data entry value. When the sample size is even, the
median is the average of the middle values.
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The median is the average of the values 21 and 22, which is 21.5 years.
(1040)
M 1040 - 2.3
June 3, 2013
5 / 10
Mode
The mode is the most frequent entry in a data set.
(1040)
M 1040 - 2.3
June 3, 2013
6 / 10
Mode
The mode is the most frequent entry in a data set.
20
22
(1040)
20
22
20
22
20
23
20
23
20
23
M 1040 - 2.3
21
23
21
24
21
24
21
65
June 3, 2013
6 / 10
Mode
The mode is the most frequent entry in a data set.
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mode is 20 years. (Six occurances.)
(1040)
M 1040 - 2.3
June 3, 2013
6 / 10
Mode
The mode is the most frequent entry in a data set.
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mode is 20 years. (Six occurances.)
Party
Democrat
Republican
Other
(1040)
Frequency f
34
56
21
M 1040 - 2.3
June 3, 2013
6 / 10
Mode
The mode is the most frequent entry in a data set.
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mode is 20 years. (Six occurances.)
Party
Democrat
Republican
Other
Frequency f
34
56
21
The mode is Republican.
(1040)
M 1040 - 2.3
June 3, 2013
6 / 10
Outliers
20
22
(1040)
20
22
20
22
20
23
20
23
20
23
M 1040 - 2.3
21
23
21
24
21
24
21
65
June 3, 2013
7 / 10
Outliers
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mean of this set is 23.75 years.
Removing the unusual number 65 gives Σx = 410 and
x̄ = 410 ÷ 19 ≈ 21.6 years.
(1040)
M 1040 - 2.3
June 3, 2013
7 / 10
Outliers
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mean of this set is 23.75 years.
Removing the unusual number 65 gives Σx = 410 and
x̄ = 410 ÷ 19 ≈ 21.6 years.
The median of this set is 21.5 years.
Removing the entry 65 gives 21 years.
(1040)
M 1040 - 2.3
June 3, 2013
7 / 10
Outliers
20
22
20
22
20
22
20
23
20
23
20
23
21
23
21
24
21
24
21
65
The mean of this set is 23.75 years.
Removing the unusual number 65 gives Σx = 410 and
x̄ = 410 ÷ 19 ≈ 21.6 years.
The median of this set is 21.5 years.
Removing the entry 65 gives 21 years.
The entry is an outlier. These values are far removed from the other
entries. They have a big influence on means, but little influence of
medians.
(1040)
M 1040 - 2.3
June 3, 2013
7 / 10
Weighted Means
Weighted means are means of data sets with entries having various
weights. For example, a course grade may be weighted for different
percents for homework, projects, exams, etc. The general form is
Σ(x · w )
Σw
with w as the weight for each entry.
x̄ =
(1040)
M 1040 - 2.3
June 3, 2013
8 / 10
Weighted Means
Salaries for 8 employees with MBAs and 17 with BAs are given below.
Degree
MBA
BA
(1040)
Salary x
$ 45,500
$ 32,000
Weight w
8
17
M 1040 - 2.3
x ·w
$ 364,000
$ 544,000
June 3, 2013
9 / 10
Weighted Means
Salaries for 8 employees with MBAs and 17 with BAs are given below.
Degree
MBA
BA
Salary x
$ 45,500
$ 32,000
Weight w
8
17
x ·w
$ 364,000
$ 544,000
Σw = 25, and Σ(x · w ) = 908, 000. The weighted mean is
x̄ =
(1040)
$908, 000
= $36, 320.
25
M 1040 - 2.3
June 3, 2013
9 / 10
Assignements
Assignment:
1
Read pages 65 - 71.
2
Try page 71 exercises 1-57 odd.
Vocabulary: Mean, median, mode, outliers, weighted mean.
Understand: How to compute a mean. How to find a median and a
mode. Know how they compare, and how they are effected by outliers.
Compute a weighted mean.
(1040)
M 1040 - 2.3
June 3, 2013
10 / 10