Classification of Variables

1
Classification of Variables

• Discrete Numerical Variable
• A variable that produces a response that comes
  from a counting process.

2
Classification of Variables

• Continuous Numerical Variable
• A variable that produces a response that is the
  outcome of a measurement process.

3
Classification of Variables

• Categorical Variables
• Variables that produce responses that belong to
  groups (sometimes called classes) or categories.

4
Measurement Levels

• Nominal and Ordinal Levels of Measurement refer
  to data obtained from categorical questions.
• A nominal scale indicates assignments to groups
  or classes.
• Ordinal data indicate rank ordering of items.

5
Frequency Distributions

• A frequency distribution is a table used to
  organize data. The left column (called classes
  or groups) includes numerical intervals on a
  variable being studied. The right column is a
  list of the frequencies, or number of
  observations, for each class. Intervals are
  normally of equal size, must cover the range of
  the sample observations, and be non-overlapping.

6
Construction of a Frequency Distribution

• Rule 1 Intervals (classes) must be inclusive
  and non-overlapping
• Rule 2 Determine k, the number of classes
• Rule 3 Intervals should be the same width, w
  the width is determined by the following
• Both k and w should be rounded upward, possibly
  to the next largest integer.

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Construction of a Frequency Distribution

• Quick Guide to Number of Classes for a Frequency
  Distribution
• Sample Size Number of Classes
• Fewer than 50 5 6 classes
• 50 to 100 6 8 classes
• over 100 8 10 classes

8
Cumulative Frequency Distributions

• A cumulative frequency distribution contains the
  number of observations whose values are less than
  the upper limit of each interval. It is
  constructed by adding the frequencies of all
  frequency distribution intervals up to and
  including the present interval.

9
Relative Cumulative Frequency Distributions

• A relative cumulative frequency distribution
  converts all cumulative frequencies to cumulative
  percentages

10
Histograms and Ogives

• A histogram is a bar graph that consists of
  vertical bars constructed on a horizontal line
  that is marked off with intervals for the
  variable being displayed. The intervals
  correspond to those in a frequency distribution
  table. The height of each bar is proportional to
  the number of observations in that interval.

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Histograms and Ogives

• An ogive, sometimes called a cumulative line
  graph, is a line that connects points that are
  the cumulative percentage of observations below
  the upper limit of each class in a cumulative
  frequency distribution.

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Histogram and Ogive for Example 2.1
13
Stem-and-Leaf Display

• A stem-and-leaf display is an exploratory data
  analysis graph that is an alternative to the
  histogram. Data are grouped according to their
  leading digits (called the stem) while listing
  the final digits (called leaves) separately for
  each member of a class. The leaves are displayed
  individually in ascending order after each of the
  stems.

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Stem-and-Leaf Display
Stem-and-Leaf Display for Gilottis Deli Example
15
Tables- Bar and Pie Charts -
Frequency and Relative Frequency Distribution for
Top Company Employers Example
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Tables- Bar and Pie Charts -
Figure 2.9 Bar Chart for Top Company Employers
Example
17
Tables- Bar and Pie Charts -
Figure 2.10 Pie Chart for Top Company Employers
Example
18
Pareto Diagrams

• A Pareto diagram is a bar chart that displays the
  frequency of defect causes. The bar at the left
  indicates the most frequent cause and bars to the
  right indicate causes in decreasing frequency. A
  Pareto diagram is use to separate the vital few
  from the trivial many.

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Line Charts

• A line chart, also called a time plot, is a
  series of data plotted at various time intervals.
  Measuring time along the horizontal axis and the
  numerical quantity of interest along the vertical
  axis yields a point on the graph for each
  observation. Joining points adjacent in time by
  straight lines produces a time plot.

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Line Charts
21
Parameters and Statistics

• A statistic is a descriptive measure computed
  from a sample of data.
• A parameter is a descriptive measure computed
  from an entire population of data.

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Measures of Central Tendency- Arithmetic Mean -

• The arithmetic mean of a set of data is the sum
  of the data values divided by the number of
  observations.

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Sample Mean

• If the data set is from a sample, then the sample
  mean, , is

24
Population Mean

• If the data set is from a population, then the
  population mean, ? , is

25
Measures of Central Tendency- Median -

• An ordered array is an arrangement of data in
  either ascending or descending order. Once the
  data are arranged in ascending order, the median
  is the value such that 50 of the observations
  are smaller and 50 of the observations are
  larger.

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Measures of Central Tendency- Median -

• If the sample size n is an odd number, the
  median, Xm, is the middle observation. If the
  sample size n is an even number, the median, Xm,
  is the average of the two middle observations.
  The median will be located in the 0.50(n1)th
  ordered position.

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Measures of Central Tendency- Mode -

• The mode, if one exists, is the most frequently
  occurring observation in the sample or
  population.

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Shape of the Distribution

• The shape of the distribution is said to be
  symmetric if the observations are balanced, or
  evenly distributed, about the mean. In a
  symmetric distribution the mean and median are
  equal.

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Shape of the Distribution

• A distribution is skewed if the observations are
  not symmetrically distributed above and below the
  mean. A positively skewed (or skewed to the
  right) distribution has a tail that extends to
  the right in the direction of positive values. A
  negatively skewed (or skewed to the left)
  distribution has a tail that extends to the left
  in the direction of negative values.

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Shapes of the Distribution
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Measures of Central Tendency - Geometric Mean -

• The Geometric Mean is the nth root of the product
  of n numbers
• The Geometric Mean is used to obtain mean growth
  over several periods given compounded growth from
  each period.

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Measures of Variability- The Range -

• The range is in a set of data is the difference
  between the largest and smallest observations

33
Measures of Variability- Sample Variance -

• The sample variance, s2, is the sum of the
  squared differences between each observation and
  the sample mean divided by the sample size minus
  1.

34
Measures of Variability- Short-cut Formulas for
s2

• Short-cut formulas for the sample variance, s2,
  are

35
Measures of Variability- Population Variance -

• The population variance, ?2, is the sum of the
  squared differences between each observation and
  the population mean divided by the population
  size, N.

36
Measures of Variability- Sample Standard
Deviation -

• The sample standard deviation, s, is the positive
  square root of the variance, and is defined as

37
Measures of Variability- Population Standard
Deviation-

• The population standard deviation, ?, is

38
The Empirical Rule(the 68, 95, or almost all
rule)

• For a set of data with a mound-shaped histogram,
  the Empirical Rule is
• approximately 68 of the observations are
  contained with a distance of one standard
  deviation around the mean ?? 1?
• approximately 95 of the observations are
  contained with a distance of 2 standard
  deviations around the mean ?? 2?
• almost all of the observations are contained with
  a distance of three standard deviation around the
  mean ?? 3?

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Coefficient of Variation

• The Coefficient of Variation, CV, is a measure of
  relative dispersion that expresses the standard
  deviation as a percentage of the mean (provided
  the mean is positive).
• The sample coefficient of variation is

40
Coefficient of Variation

• The population coefficient of variation is

41
Percentiles and Quartiles

• Data must first be in ascending order.
  Percentiles separate large ordered data sets into
  100ths. The Pth percentile is a number such that
  P percent of all the observations are at or below
  that number.
• Quartiles are descriptive measures that separate
  large ordered data sets into four quarters.

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Percentiles and Quartiles

• The first quartile, Q1, is another name for the
  25th percentile. The first quartile divides the
  ordered data such that 25 of the observations
  are at or below this value. Q1 is located in the
  .25(n1)st position when the data is in ascending
  order. That is,

43
Percentiles and Quartiles

• The third quartile, Q3, is another name for the
  75th percentile. The first quartile divides the
  ordered data such that 75 of the observations
  are at or below this value. Q3 is located in the
  .75(n1)st position when the data is in ascending
  order. That is,

44
Interquartile Range

• The Interquartile Range (IQR) measures the spread
  in the middle 50 of the data that is the
  difference between the observations at the 25th
  and the 75th percentiles

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Five-Number Summary

• The Five-Number Summary refers to the five
  descriptive measures minimum, first quartile,
  median, third quartile, and the maximum.

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Box-and-Whisker Plots

• A Box-and-Whisker Plot is a graphical procedure
  that uses the Five-Number summary.
• A Box-and-Whisker Plot consists of
• an inner box that shows the numbers which span
  the range from Q1 Box-and-Whisker Plot to Q3.
• a line drawn through the box at the median.
• The whiskers are lines drawn from Q1 to the
  minimum vale, and from Q3 to the maximum value.

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Box-and-Whisker Plots (Excel)
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Grouped Data Mean

• For a population of N observations the mean is
• Where the data set contains observation values
  m1, m2, . . ., mk occurring with frequencies f1,
  f2, . . . fK respectively

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Grouped Data Mean

• For a sample of n observations, the mean is
• Where the data set contains observation values
  m1, m2, . . ., mk occurring with frequencies f1,
  f2, . . . fK respectively

50
Grouped Data Variance

• For a population of N observations the variance
  is

Where the data set contains observation values
m1, m2, . . ., mk occurring with frequencies f1,
f2, . . . fK respectively
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Grouped Data Variance

• For a sample of n observations, the variance is