Review

1
Review

• Descriptive Statistics
• Qualitative (Graphical)
• Quantitative (Graphical)
• Summation Notation
• Qualitative (Numerical)
• Central Measures (mean, median, mode and modal
  class)
• Shape of the Data
• Measures of Variability

2
Outlier

• A data measurement which is unusually large or
  small compared to the rest of the data.
• Usually from
• Measurement or recording error
• Measurement from a different population
• A rare, chance event.

3
Advantages/Disadvantages Mean

• Disadvantages
• is sensitive to outliers
• Advantages
• always exists
• very common
• nice mathematical properties

4
Advantages/Disadvantages Median

• Disadvantages
• does not take all data into account
• Advantages
• always exists
• easily calculated
• not affected by outliers
• nice mathematical properties

5
Advantages/Disadvantages Mode

• Disadvantages
• does not always exist, there could be just one
  of each data point
• sometimes more than one
• Advantages
• appropriate for qualitative data

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Review

• 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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Review
Skewed to the right The mean is bigger than the
median.
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Review
Skewed to the left The mean is less than the
median.
9
Review
When the mean and median are equal, the data is
symmetric
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Numerical Measures of Variability

• These measure the variability or spread of the
  data.

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Numerical Measures of Variability

• These measure the variability or spread of the
  data.

Relative Frequency
0.5
0.4
0.3
0.2
0.1
1
3
4
5
2
0
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Numerical Measures of Variability

• These measure the variability or spread of the
  data.

Relative Frequency
0.5
0.4
0.3
0.2
0.1
1
3
4
5
2
0
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Numerical Measures of Variability

• These measure the variability or spread of the
  data.

Relative Frequency
0.5
0.4
0.3
0.2
0.1
1
3
4
5
2
7
0
6
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Numerical Measures of Variability

• These measure the variability, spread or relative
  standing of the data.
• Range
• Standard Deviation
• Percentile Ranking
• Z-score

15
Range

• The range of quantitative data is denoted R and
  is given by
• R Maximum Minimum

16
Range

• The range of quantitative data is denoted R and
  is given by
• R Maximum Minimum
• In the previous examples the first two graphs
  have a range of 5 and the third has a range of 7.

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Range

• R Maximum Minimum
• Disadvantages
• Since the range uses only two values in the
  sample it is very sensitive to outliers.
• Give you no idea about how much data is in the
  center of the data.

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What else?

• We want a measure which shows how far away most
  of the data points are from the mean.

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What else?

• We want a measure which shows how far away most
  of the data points are from the mean.
• One option is to keep track of the average
  distance each point is from the mean.

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Mean Deviation

• The Mean Deviation is a measure of dispersion
  which calculates the distance between each data
  point and the mean, and then finds the average of
  these distances.

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Mean Deviation

• Advantages The mean deviation takes into
  account all values in the sample.
• Disadvantages The absolute value signs are very
  cumbersome in mathematical equations.

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Standard Deviation

• The sample variance, denoted by s², is

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Standard Deviation

• The sample variance, denoted by s², is
• The sample standard deviation is
• The sample standard deviation is much more
  commonly used as a measure of variance.

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Example

• Let the following be data from a sample
• 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.
• Find
• a) The range
• b) The standard deviation of this sample.

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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

• a) The range
• b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2


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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

• a) The range
• b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2


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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

• a) The range
• b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2
-1 1 0

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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

• a) The range
• b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2
-1 1 0 -1 2 -1 -2 1 2 -1
1 1 0 1 4 1 4 1 4 1
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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.
2 4 3 2 5 2 1 4 5 2
-1 1 0 -1 2 -1 -2 1 2 -1
1 1 0 1 4 1 4 1 4 1
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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.
2 4 3 2 5 2 1 4 5 2
-1 1 0 -1 2 -1 -2 1 2 -1
1 1 0 1 4 1 4 1 4 1
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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.
2 4 3 2 5 2 1 4 5 2
-1 1 0 -1 2 -1 -2 1 2 -1
1 1 0 1 4 1 4 1 4 1
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Sample 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.
Standard Deviation
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More Standard Deviation

• Like the mean, we are also interested in the
  population variance (i.e. your sample is the
  whole population) and the population standard
  deviation.
• The population variance and standard deviation
  are denoted s and s2 respectively.

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More Standard Deviation

• The population variance and standard deviation
  are denoted s and s2 respectively.
• The formula for population variance is
  slightly different than sample variance

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Example Using Standard Deviation

• 35, 59, 70, 73, 75, 81, 84, 86.
• The mean and standard deviation are 70.4 and
  16.7, respectively.
• We wish to know if any of are data points are
  outliers. That is whether they dont fit with
  the general trend of the rest of the data.
• To find this we calculate the number of standard
  deviations each point is from the mean.

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Example Using Standard Deviation

• To find this we calculate the number of standard
  deviations each point is from the mean.
• To simplify things for now, work out which data
  points are within
• one standard deviation from the mean i.e.
• two standard deviations from the mean i.e.
• three standard deviations from the mean i.e.

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Example Using Standard Deviation

• Here are eight test scores from a previous Stats
  201 class
• 35, 59, 70, 73, 75, 81, 84, 86.
• The mean and standard deviation are 70.4 and
  16.7, respectively. Work out which data points
  are within
• one standard deviation from the mean i.e.
• two standard deviations from the mean i.e.
• three standard deviations from the mean i.e.

38
Example Using Standard Deviation

• Here are eight test scores from a previous Stats
  201 class
• 35, 59, 70, 73, 75, 81, 84, 86.
• The mean and standard deviation are 70.4 and
  16.7, respectively. Work out which data points
  are within
• one standard deviation from the mean i.e.
• 59, 70, 73, 75, 81, 84, 86
• two standard deviations from the mean i.e.
• 59, 70, 73, 75, 81, 84, 86
• c) three standard deviations from the mean i.e.
• 35, 59, 70, 73, 75, 81, 84, 86