Methods for Describing Sets of Data

1
Chapter 2

• Methods for Describing Sets of Data

2
Objectives

• Describe Data using Graphs
• Describe Data using Charts

3
Describing Qualitative Data

• Qualitative data are nonnumeric in nature
• Best described by using Classes
• 2 descriptive measures
• class frequency number of data points in a
  class
• class relative class frequency
• frequency total number of data points in
  data set
• class percentage class relative freq. x 100

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Describing Qualitative Data Displaying
Descriptive Measures

• Summary Table

Class Frequency
Class percentage class relative frequency x 100
5
Describing Qualitative Data Qualitative Data
Displays

• Bar Graph

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Describing Qualitative Data Qualitative Data
Displays

• Pie chart

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Describing Qualitative Data Qualitative Data
Displays

• Pareto Diagram

8
Graphical Methods for Describing Quantitative Data

• The Data

9
Graphical Methods for Describing Quantitative Data

• For describing, summarizing, and detecting
  patterns in such data, we can use three graphical
  methods
• dot plots
• stem-and-leaf displays
• histograms

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Graphical Methods for Describing Quantitative Data

• Dot Plot

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Graphical Methods for Describing Quantitative Data

• Stem-and-Leaf Display

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Graphical Methods for Describing Quantitative Data

• Histogram

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Graphical Methods for Describing Quantitative Data

• More on Histograms

Number of Observations in Data Set Number of Classes
Less than 25 5-6
25-50 7-14
More than 50 15-20
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Summation Notation

• Used to simplify summation instructions
• Each observation in a data set is identified by a
  subscript
• x1, x2, x3, x4, x5, . xn
• Notation used to sum the above numbers together
  is

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Summation Notation

• Data set of 1, 2, 3, 4
• Are these the same? and

16
Numerical Measures of Central Tendency

• Central Tendency tendency of data to center
  about certain numerical values
• 3 commonly used measures of Central Tendency
• Mean
• Median
• Mode

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Numerical Measures of Central Tendency

• The Mean
• Arithmetic average of the elements of the data
  set
• Sample mean denoted by
• Population mean denoted by
• Calculated as and

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Numerical Measures of Central Tendency

• The Median
• Middle number when observations are arranged in
  order
• Median denoted by m
• Identified as the observation if n is odd, and
  the mean of the and observations if n is even

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Numerical Measures of Central Tendency

• The Mode
• The most frequently occurring value in the data
  set
• Data set can be multi-modal have more than one
  mode
• Data displayed in a histogram will have a modal
  class the class with the largest frequency

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Numerical Measures of Central Tendency

• The Data set 1 3 5 6 8 8 9 11 12
• Mean
• Median is the or 5th observation, 8
• Mode is 8

21
Numerical Measures of Variability

• Variability the spread of the data across
  possible values
• 3 commonly used measures of Variability
• Range
• Variance
• Standard Deviation

22
Numerical Measures of Variability

• The Range
• Largest measurement minus the smallest
  measurement
• Loses sensitivity when data sets are large
• These 2 distributionshave the same range.
• How much does therange tell you about the data
  variability?

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Numerical Measures of Variability

• The Sample Variance (s2)
• The sum of the squared deviations from the mean
  divided by (n-1). Expressed as units squared
• Why square the deviations? The sum of the
  deviations from the mean is zero

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Numerical Measures of Variability

• The Sample Standard Deviation (s)
• The positive square root of the sample variance
• Expressed in the original units of measurement

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Numerical Measures of Variability

• Samples and Populations - Notation

Sample Population
Variance s2
Standard Deviation s
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Numerical Measures of Relative Standing

• Descriptive measures of relationship of a
  measurement to the rest of the data
• Common measures
• percentile ranking
• z-score

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Numerical Measures of Relative Standing

• Percentile rankings make use of the pth
  percentile
• The median is an example of percentiles.
• Median is the 50th percentile 50 of
  observations lie above it, and 50 lie below it
• For any p, the pth percentile has p of the
  measures lying below it, and (100-p) above it

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Numerical Measures of Relative Standing

• z-score the distance between a measurement x
  and the mean, expressed in standard units
• Use of standard units allows comparison across
  data sets

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Numerical Measures of Relative Standing

• More on z-scores
• Z-scores follow the empirical rule for mounded
  distributions

30
Methods for Detecting Outliers

• Outlier an observation that is unusually large
  or small relative to the data values being
  described
• Causes
• Invalid measurement
• Misclassified measurement
• A rare (chance) event
• 2 detection methods
• Box Plots
• z-scores

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Methods for Detecting Outliers

• Box Plots
• based on quartiles, values that divide the
  dataset into 4 groups
• Lower Quartile QL 25th percentile
• Middle Quartile - median
• Upper Quartile QU 75th percentile
• Interquartile Range (IQR) QU - QL

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Methods for Detecting Outliers

• Box Plots
• Not on plot inner and outer fences, which
  determine potential outliers

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Methods for Detecting Outliers

• Rules of thumb
• Box Plots
• measurements between inner and outer fences are
  suspect
• measurements beyond outer fences are highly
  suspect
• Z-scores
• Scores of ?3 in mounded distributions (?2 in
  highly skewed distributions) are considered
  outliers