Download document

Quantitative Data (Graphical)
1
Quantitative Data (Graphical)
• This is numerical data
• We may describe quantitative data using the
same methods as qualitative by breaking our
numerical data into classes. That is 20-30, 3040, 40-50, 50-60.
2
Quantitative Data (Graphical)
• This is numerical data
• We may describe quantitative data using the
same methods as qualitative by breaking our
numerical data into classes. That is 20-30, 3040, 40-50, 50-60.
• Histograms, stem and leaf plots and dot plots
are other common methods of displaying
quantitative data.
3
Histograms
• A histogram is a bar graph where you use
intervals for your data class.
• The following histogram summarizes the
NBA payroll. You should note that the are
adjacent to one another.
4
NBA Payroll
Number of
teams
Payroll in millions of dollars
5
Stem and Leaf, and Dot Plots
• Notice in the histogram on the previous
page we lose some information. That is we
don’t know exactly what each team is
paying in salary just how many are paying
in the range of 1.885 million dollars.
6
Stem and Leaf, and Dot Plots
• Notice in the histogram on the previous
page we lose some information. That is we
don’t know exactly what each team is
paying in salary just how many are paying
in the range of 1.885 million dollars.
• A stem and leaf plot is a graphical device
which uses numbers so that no information
is lost.
7
Stem and Leaf, and Dot Plots
• A stem and leaf plot is a graphical device
which uses numbers so that no information
is lost.
• The technique separates each data point into
two numbers, the stem (the leading digit)
and the leaves.
8
Stem and Leaf, and Dot Plots
• The technique separates each data point into
two numbers, the stem (the leading digit) and
the leaves.
• In a dot plot we start with a number line of
all possible values for the data. Each data
point is represented with a dot above the
appropriate number. If a number appears
more than once in your data you build a
tower of dots above that point.
9
Example
• Here is a list of exam scores:
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
Construct a histogram (with interval size 10
starting at 24), a stem and leaf diagram and
a dot plot .
10
Histogram of Exam Scores
Frequency
11
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
12
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
9
10
13
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
8
9
10
14
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
82
9
10
15
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
829
9
10
16
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
0
829
9
10
17
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
5
6
7
8
0
289
9
10
18
Stem and Leaf Plot of Exam
Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
3
4
49
5
6
347
7
8
056
12456899
9
066
10
0
19
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
20
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
21
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
22
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
23
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
24
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
25
Dot Plot of Exam Scores
88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90,
96, 76, 34, 81, 64, 75, 84, 89, 96
30
40
50
60
70
80
90
100
26
Quantitative (in contrast to
graphical) methods
• Measures of central tendency
• Mean
• Median
• Mode
• Measures of dispersion
• Range
• Standard deviation
27
Summation Notation
Here is a typical (small) data set:
2
7
1
3
2
28
Summation Notation
Here is a typical (small) data set:
2
7
1
3
2
So we can talk about a general data set we let:
x1  2, x2  7, x3  1, x4  3, x5  2
29
Summation Notation
So we can talk about a general data set we let:
x1  2, x2  7, x3  1, x4  3, x5  2
In general for a sample of n points of data we
call them, in order:
x1 , x2 , x3 , x4 ,..., xn
30
Summation Notation
In general for a sample of n points of data we
call them, in order:
x1 , x2 , x3 , x4 ,..., xn
When we wish to sum (add them up), we use
the notation:
n
x
i 1
i
 x1  x2  x3  x4  ...  xn
This is called summation notation.
31
Summation Notation
In statistics, sometimes the i is not included in
the sum since it is implied that we are
summing over all points in our data set. That
is you may see the following:
n
x x
i 1
x
n
2
i
  xi
i 1
2
32
Descriptive Statistics
• Qualitative Variables
– Graphical Methods
• Quantitative Variables
– Graphical Methods
– Numerical Methods
33
Numerical descriptive measures
Two types of measures we look for:
1) Ones which tell us about the central
tendency of measurements
2) Ones which tell us about the variability or
spread of the data.
34
Numerical Measures of Central
Tendency
Three Measures
a) Mean
b) Median
c) Mode
Problem
35
Mean
The mean of a data set is the average or
expected value of the readings in the data.
Problem: I wish to talk about the mean of the
population and the mean of the sample
separately. Therefore we need to
introduce two different notations.
36
Mean
Sample: the size of the sample is usually
denoted with n, and the mean of the
sample (sample mean) is denoted with x .
Population: the size of the population is
usually denoted N and the population
mean is denoted µ.
37
Mean
The mean is given by
n
x
x
i 1
n
i
OR
x

x
n
38
Example
Given the sample:
3, 1, 6, 2, 4, 4, 1, 4
Find the mean.
39
Example
Given the sample:
3, 1, 6, 2, 4, 4, 1, 4
Find the mean.
x

x
n
40
Example
Given the sample:
3, 1, 6, 2, 4, 4, 1, 4
Find the mean.
x 3 1 6  2  4  4 1 4

x

n
8
41
Example
Given the sample:
3, 1, 6, 2, 4, 4, 1, 4
Find the mean.
x 3 1 6  2  4  4 1 4

x

n
8
 3.125
42
Example
However, given the sample:
3, 1, 6, 2, 4, 4, 1, 40
we find the mean is quite different from 3.125.
43
Example
However, given the sample:
3, 1, 6, 2, 4, 4, 1, 40
we find the mean is quite different from 3.125.
x 3  1  6  2  4  4  1  40

x

n
 7.625
8
This is not a good indication of the center of
the sample.
44
Mean
Usually the sample mean x is used to
estimate the population mean µ.
The accuracy of this estimate tends to be
effected by:
– The size of the sample
– Variability or spread of the data
45
Median
The median of a quantitative data set is the
middle number in the set.
For example in the following data the median
is 10.
1, 4, 4, 10, 12, 14, 10000
46
Median
The sample median is denoted M.
If n is even, take the average of the two
middle numbers.
47
Examples
Find the median in the following two data
sets:
a) 3, 1, 6, 2, 4, 4, 1, 4
b) 3, 1, 6, 2, 4, 4, 1, 40
48
Examples
Find the median in the following two data
sets:
a) 3, 1, 6, 2, 4, 4, 1, 4
b) 3, 1, 6, 2, 4, 4, 1, 40
In both cases we found M=3.5.
The median is sometimes a better estimate of
the population mean µ than the sample mean x
49
because it puts less emphasis on outliers.
What the median and mean tell
you
A data set is skewed if one tail of the
distribution has more extreme observations
than the other.
http://www.shodor.org/interactivate/activities/
SkewDistribution/
50
What the median and mean tell
you
This data set is skewed to the right. Notice
the mean is to the right of the median.
M
x
51
What the median and mean tell
you
Skewed to the right: The mean is bigger than
the median.
M
x
52
What the median and mean tell
you
This data set is skewed to the left. Notice the
mean is to the left of the median.
x
M
53
What the median and mean tell
you
Skewed to the left: The mean is less than the
median.
x
M
54
What the median and mean tell
you
When the mean and median are equal, the
data is symmetric
x M
55
Mode
The mode is the measurement which occurs
most frequently
a) 3, 1, 6, 2, 4, 4, 1, 4
b) 3, 1, 6, 2, 4, 4, 1, 40
56
Mode
The mode is the measurement which occurs
most frequently
a) 3, 1, 6, 2, 4, 4, 1, 4
b) 3, 1, 6, 2, 4, 4, 1, 40
a) mode= 4
b) mode= 4, 1
57
Mode
When dealing with histograms or qualitative
data, the measurement with the highest
frequency is called the modal class.
58