Chapter 3: Descriptive Analysis and Presentation of Bivariate Data

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Chapter 3 Descriptive Analysis and Presentation
of Bivariate Data
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Chapter Goals

• To be able to present bivariate data in tabular
  and graphic form.
• To gain an understanding of the distinction
  between the basic purposes of correlation
  analysis and regression analysis.
• To become familiar with the ideas of descriptive
  presentation.

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3.1 Bivariate Data

• Bivariate Data Consists of the values of two
  different response variables that are obtained
  from the same population of interest.
• Three combinations of variable types
• 1. Both variables are qualitative (attribute).
• 2. One variable is qualitative (attribute) and
  the other is quantitative (numerical).
• 3. Both variables are quantitative (both
  numerical).

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• Two Qualitative Variables
• When bivariate data results from two qualitative
  (attribute or categorical) variables, the data is
  often arranged on a cross-tabulation or
  contingency table.
• Example A survey was conducted to investigate
  the relationship between preferences for
  television, radio, or newspaper for national
  news, and gender. The results are given in the
  table below.

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• This table may be extended to display the
  marginal totals (or marginals). The total of the
  marginal totals is the grand total.
• Contingency tables often show percentages
  (relative frequencies). These percentages are
  based on the entire sample or on the subsample
  (row or column) classifications.

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• Percentages based on the grand total (entire
  sample)
• The previous contingency table may be converted
  to percentages of the grand total by dividing
  each frequency by the grand total and multiplying
  by 100.
• For example, 175 becomes 13.3

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These same statistics (numerical values
describing sample results) can be shown in a
(side-by-side) bar graph.
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• Percentages based on row (column) totals
• The entries in a contingency table may also be
  expressed as percentages of the row (column)
  totals by dividing each row (column) entry by
  that rows (columns) total and multiplying by
  100. The entries in the contingency table below
  are expressed as percentages of the column
  totals.
• These statistics may also be displayed in a
  side-by-side bar graph.

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• One Qualitative and One Quantitative Variable
• 1. When bivariate data results from one
  qualitative and one quantitative variable, the
  quantitative values are viewed as separate
  samples.
• 2. Each set is identified by levels of the
  qualitative variable.
• 3. Each sample is described using summary
  statistics, and the results are displayed for
  side-by-side comparison.
• 4. Statistics for comparison measures of central
  tendency, measures of variation, 5-number
  summary.
• 5. Graphs for comparison dotplot, boxplot.

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• Example A random sample of households from three
  different parts of the country was obtained and
  their electric bill for June was recorded. The
  data is given in the table below.
• The part of the country is a qualitative variable
  with three levels of response. The electric bill
  is a quantitative variable. The electric bills
  may be compared with numerical and graphical
  techniques.

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• Comparison using dotplots
• . . . . . . . .
• ------------------------------------
  ---------------Northeast
• .
• ... ..
• ------------------------------------
  ---------------Midwest
• 
  .
• . . .
  . . . .
• ------------------------------------
  ---------------West
• 24.0 32.0 40.0 48.0
  56.0 64.0
• The electric bills in the Northeast tend to be
  more spread out than those in the Midwest. The
  bills in the West tend to be higher than both
  those in the Northeast and Midwest.

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• Comparison using Box-and-Whisker plots

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• Two Quantitative Variables
• 1. Expressed as ordered pairs (x, y)
• 2. x input variable, independent variable.
• y output variable, dependent variable.
• Scatter Diagram A plot of all the ordered pairs
  of bivariate data on a coordinate axis system.
  The input variable x is plotted on the horizontal
  axis, and the output variable y is plotted on the
  vertical axis.
• Note Use scales so that the range of the
  y-values is equal to or slightly less than the
  range of the x-values. This creates a window
  that is approximately square.

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• Example In a study involving childrens fear
  related to being hospitalized, the age and the
  score each child made on the Child Medical Fear
  Scale (CMFS) are given in the table below.
• Construct a scatter diagram for this data.

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Scatter diagram age input variable, CMFS
output variable
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3.2 Linear Correlation

• Measure the strength of a linear relationship
  between two variables.
• As x increases, no definite shift in y no
  correlation.
• As x increase, a definite shift in y
  correlation.
• Positive correlation x increases, y increases.
• Negative correlation x increases, y decreases.
• If the ordered pairs follow a straight-line path
  linear correlation.

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Example no correlation. As x increases, there
is no definite shift in y.
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• Example positive correlation.
• As x increases, y also increases.

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• Example negative correlation.
• As x increases, y decreases.

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• Note
• 1. Perfect positive correlation all the points
  lie along a line with positive slope.
• 2. Perfect negative correlation all the points
  lie along a line with negative slope.
• 3. If the points lie along a horizontal or
  vertical line no correlation.
• 4. If the points exhibit some other nonlinear
  pattern no linear relationship, no correlation.
• 5. Need some way to measure correlation.

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• Coefficient of linear correlation r, measures
  the strength of the linear relationship between
  two variables.
• Pearsons product moment formula
• Note
• 1.
• 2. r 1 perfect positive correlation
• 3. r -1 perfect negative correlation

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• Alternate formula for r

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• Example The table below presents the weight (in
  thousands of pounds) x and the gasoline mileage
  (miles per gallon) y for ten different
  automobiles. Find the linear correlation
  coefficient.

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• To complete the calculation for r

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• Note
• 1. r is usually rounded to the nearest hundredth.
• 2. r close to 0 little or no linear correlation.
• 3. As the magnitude of r increases, towards -1 or
  1, there is an increasingly stronger linear
  correlation between the two variables.
• 4. Method of estimating r based on the scatter
  diagram.
• Window should be approximately square.
• Useful for checking calculations.

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3.3 Linear Regression

• Regression analysis finds the equation of the
  line that best describes the relationship between
  two variables.
• One use of this equation to make predictions.

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• Models or prediction equations
• Some examples of various possible relationships.
• Linear
• Quadratic
• Exponential
• Logarithmic
• Note What would a scatter diagram look like to
  suggest each relationship?

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• Method of least squares
• Equation of the best-fitting line
• Predicted value
• Least squares criterion
• Find the constants b0 and b1 such that the sum
• is as small as possible.

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• Observed and predicted values of y

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• The equation of the line of best fit
• Determined by
• b0 slope
• b1 y-intercept
• Values that satisfy the least squares criterion

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• Example A recent article measured the job
  satisfaction of subjects with a 14-question
  survey. The data below represents the job
  satisfaction scores, y, and the salaries, x, for
  a sample of similar individuals.
• 1. Draw a scatter diagram for this data.
• 2. Find the equation of the line of best fit.

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Preliminary calculations needed to find b1 and b0
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• Finding b1 and b0

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• Scatter diagram

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• Note
• 1. Keep at least three extra decimal places while
  doing the calculations to ensure an accurate
  answer.
• 2. When rounding off the calculated values of b0
  and b1, always keep at least two significant
  digits in the final answer.
• 3. The slope b1 represents the predicted change
  in y per unit increase in x.
• 4. The y-intercept is the value of y where the
  line of best fit intersects the y-axis.
• 5. The line of best fit will always pass through
  the point

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• Making predictions
• 1. One of the main purposes for obtaining a
  regression equation is for making predictions.
• 2. For a given value of x, we can predict a value
  of y,
• 3. The regression equation should be used to make
  predictions only about the population from
  which the sample was drawn.
• 4. The regression equation should be used only
  to cover the sample domain on the input
  variable. You can estimate values outside the
  domain interval, but use caution and use values
  close to the domain interval.
• 5. Use current data. A sample taken in 1987
  should not be used to make predictions in 1999.