Descriptive and Inferential Statistics

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Descriptive and Inferential Statistics

• Descriptive statistics Mathematical methods
  (such as mean, median, standard deviation) that
  summarize and interpret some of the properties of
  a set of data (sample) but do not infer the
  properties of the population from which the
  sample was drawn.
• Mathematical methods (such as hypothesis
  development) that employ probability theory for
  deducing (inferring) the properties of a
  population from the analysis of the properties of
  a set of data (sample) drawn from it.

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Did it happen by chance?

• How do you know if something caused or correlates
  with something else?
• The appropriate Statistic will tell you
• If there is a difference from some expected value
• If the difference is statistically significant or
  merely due to random chance

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Descriptive Statistics

• Types of descriptive statistics
• Calculation for interval data

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Types of descriptive statistics

• Statistic is a quantitative index that describes
  performance of a sample or samples
• Parameter is a quantitative index describing the
  performance of a population
• Measures of central tendency are used to
  determine the typical or average value among a
  group of values
• Measures of variability indicate how spread out
  the values are

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Graphing data

• Provides a quick view of the what your data is
  telling you.
• There are various types of graphs which are used
  in statistics including bar graphs, histograms,
  scatter plots, pie charts, frequency polygons etc.

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Example group of test scores
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Frequency Polygon and Pie Chart
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Sample Bar Graph
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Sample Histogram
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Sample Scatter Plot
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Frequency Distributions
Frequency distributions are like frequency
polygons however, instead of straight lines, a
frequency distribution uses a smooth curve to
connect the points and, similar to a graph, is
plotted on two axes.
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J Shaped Curve
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Bimodal Curve with Two Peaks
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Positively Skewed Bell Curve
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Negatively Skewed Bell Curve
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Symmetric Bell Curve/Normal Distribution
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What is the Normal Distribution ?

• Where did it come from and why is it so special?
• As shown by Galton (19th century guy), just
  about anything you measure turns out to be
  normally distributed, at least approximately so.
• That is, usually most of the observations cluster
  around the mean, with progressively fewer
  observations out towards the extremes

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Sample Histogram
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Just about any histogram can be converted into a
line graph
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Which can be used to plot a normal distribution
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But how do we get from the normal to the standard
normal?
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Measures of central tendency

• Mean arithmetic average of a set of values and
  most frequently used measure of central tendency
• Median- midpoint of values if they are ordered
  from high to low
• Mode value that occurs most frequently

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Mean, Median and Mode

• Sample Numbers
• 7 26 54 82 32 26 51
• Find the mean, median and mode

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Mean

• Finding the Mean X
• Total up the numbers
• Divide the total by the n (number of values)

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Mean

• Finding the Mean X
• Total up the numbers (287)
• Divide the total by n (number of values) (287 /
  7 39.71)

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Median

• Finding the Median
• Swerve the car to the left till you hit something
  in the middle of the road
• The median is the middle value when numbers are
  arranged in order
• Arrange numbers from highest to lowest
• Find the middle number (odd number of values)
• 7 26 26 32 51 54 82

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Median

• Median trick question
• Find the median now!
• 7 26 54 82 32 26 51 36

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Mean, Median and Mode

• 7 26 26 32 36 51 54 82

Split the Difference Between the Two
middle numbers
Median is 34
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Mode

• The mode is the value that occurs most frequently
• Arranging the numbers in order helps here
• 7 26 26 32 51 54 82

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Mode

• What is this called?
• 7 26 26 32 51 51 54 82

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Answer
Bimodal!
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Measures of variability

• Range Difference between the highest and lowest
  values (high value -low value range)
• Variance S2
• Standard Deviation S
• variation of values about the mean

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Measures of variation range

• Range highest value-lowest value
• Bank waiting time values
• Values of 4, 7, 7 the range is 7-4 or 3
• With values of 1, 3, 14, the range is 14-1 or 13

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Other key measures of variation

• S2 Variance
• S Standard Deviation

(Triloa, Elementary Statistics, 9th Ed, 2004)
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Measures of variation standard deviation
x
6 6 6
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Measures of variation variance
x
6 6 6
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The Z statistic will allow you to standardize a
normal distribution
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Lets compare some of the great running backs of
NFL history
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Lets compute z-scores..
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To derive the Stardard Normal Curve
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Bringing us back to the concept of Six Sigma