Quantitative Data (Graphical)

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Quantitative Data (Graphical)
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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, 30-40, 40-50,
  50-60.

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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, 30-40, 40-50,
  50-60.
• Histograms, stem and leaf plots and dot plots are
  other common methods of displaying quantitative
  data.

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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.

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NBA Payroll
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Stem and Leaf, and Dot Plots

• Notice in the histogram on the previous page we
  lose some information. That is we dont know
  exactly what each team is paying in salary just
  how many are paying in the range of 1.885 million
  dollars.

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Stem and Leaf, and Dot Plots

• Notice in the histogram on the previous page we
  lose some information. That is we dont 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.

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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.

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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.

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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 .

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Histogram of Exam Scores
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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
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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
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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
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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 2
9
10
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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 2 9
9
10
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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 0
8 8 2 9
9
10
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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 0
8 2 8 9
9
10
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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 9
4
5
6 3 4 7
7 0 5 6
8 1 2 4 5 6 8 9 9
9 0 6 6
10 0
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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
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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
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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
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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
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
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Quantitative (in contrast to graphical) methods

• Measures of central tendency
• Mean
• Median
• Mode
• Measures of dispersion
• Range
• Standard deviation
• Mode
• Median Stndard deviatin

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Summation Notation

• Here is a typical (small) data set
• 2 7 1 3 2

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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

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Summation Notation

• So we can talk about a general data set we let
• In general for a sample of n points of data we
  call them, in order

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Summation Notation

• In general for a sample of n points of data we
  call them, in order
• When we wish to sum (add them up), we use the
  notation
• This is called summation notation.

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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

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Descriptive Statistics

• Qualitative Variables
• Graphical Methods
• Quantitative Variables
• Graphical Methods
• Numerical Methods

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Numerical descriptive measures

• Two types of measures we look for
• Ones which tell us about the central tendency of
  measurements
• Ones which tell us about the variability or
  spread of the data.

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Numerical Measures of Central Tendency

• Three Measures
• a) Mean
• b) Median
• c) Mode
• Problem

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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.

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Mean

• Sample the size of the sample is usually denoted
  with n, and the mean of the sample (sample mean)
  is denoted with
• Population the size of the population is usually
  denoted N and the population mean is denoted µ.

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Mean

• The mean is given by

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Example

• Given the sample
• Find the mean.

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Example

• Given the sample
• Find the mean.

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Example

• Given the sample
• Find the mean.

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Example

• Given the sample
• Find the mean.

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Example

• However, given the sample
• we find the mean is quite different from 3.125.

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Example

• However, given the sample
• we find the mean is quite different from 3.125.
• This is not a good indication of the center of
  the sample.

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Mean

• Usually the sample mean 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

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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.

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Median

• The sample median is denoted M.
• If n is even, take the average of the two middle
  numbers.

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Examples

• Find the median in the following two data sets

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Examples

• Find the median in the following two data sets
• In both cases we found M3.5.
• The median is sometimes a better estimate of the
  population mean µ than the sample mean because
  it puts less emphasis on outliers.

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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/Ske
  wDistribution/

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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.
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What the median and mean tell you
Skewed to the right The mean is bigger than the
median.
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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.
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What the median and mean tell you
Skewed to the left The mean is less than the
median.
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What the median and mean tell you
When the mean and median are equal, the data is
symmetric
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Mode

• The mode is the measurement which occurs most
  frequently

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Mode

• The mode is the measurement which occurs most
  frequently
• mode 4
• mode 4, 1

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Mode

• When dealing with histograms or qualitative data,
  the measurement with the highest frequency is
  called the modal class.