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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

teams

ofNumber

dollars of millionsin 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 don’t 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 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.

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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

Frequency

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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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88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

Dot Plot of Exam Scores

30 40 50 60 70 80 90 100

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88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

Dot Plot of Exam Scores

30 40 50 60 70 80 90 100

24

88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

Dot Plot of Exam Scores

30 40 50 60 70 80 90 100

25

88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

Dot Plot of Exam Scores

30 40 50 60 70 80 90 100

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88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

Dot Plot of Exam Scores

30 40 50 60 70 80 90 100

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 32

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Summation Notation

Here is a typical (small) data set:

2 7 1 32

So we can talk about a general data set we let:,21 x ,72 x ,13 x ,34 x 25 x

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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:

,21 x ,72 x ,13 x ,34 x 25 x

nxxxxx ,...,,,, 4321

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Summation NotationIn 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.

nxxxxx ,...,,,, 4321

n

n

ii xxxxxx

...43211

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Summation NotationIn 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

iixx

1

n

iixx

1

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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:

1) Ones which tell us about the central tendency of measurements

2) 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.

.x

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Mean

The mean is given by

n

xx

n

xx

n

ii

OR 1

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Example

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3Given the sample:

Find the mean.

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Example

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3Given the sample:

Find the mean.

n

xx

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Example

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3Given the sample:

Find the mean.

8

41442613

n

xx

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Example

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3Given the sample:

Find the mean.

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41442613

n

xx

125.3

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However, given the sample:

we find the mean is quite different from 3.125.

Example

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3

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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.

Example

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3

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401442613

n

xx

625.7

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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

x

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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.

10000 ,41 ,21 ,01 ,4 ,4 ,1

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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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ExamplesFind the median in the following two data sets:

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3 a)

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3 b)

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ExamplesFind the median in the following two data sets:

In both cases we found M=3.5.

The median is sometimes a better estimate of the population mean µ than the sample mean because it puts less emphasis on outliers.

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3 a)

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3 b)

x

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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/SkewDistribution/

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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.

xM

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What the median and mean tell you

Skewed to the right: The mean is bigger than the median.

xM

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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.

x M

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What the median and mean tell you

Skewed to the left: The mean is less than the median.

x M

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What the median and mean tell you

When the mean and median are equal, the data is symmetric

Mx

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Mode

The mode is the measurement which occurs most frequently

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3 a)

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3 b)

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Mode

The mode is the measurement which occurs most frequently

a) mode= 4

b) mode= 4, 1

4 ,1 ,4 ,4 ,2 ,6 ,1 ,3 a)

40 ,1 ,4 ,4 ,2 ,6 ,1 ,3 b)

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Mode

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