Develop and interpret a stem-and-leaf display

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Ch 4 Describing DataDisplaying and Exploring
Data Goals

• Develop and interpret a stem-and-leaf display
• Develop and interpret a
• Dot plot
• Develop and interpret quartiles, deciles, and
  percentiles
• Develop and interpret a
• Box plots
• Compute and understand the
• Coefficient of Variation
• Coefficient of Skewness
• Draw and interpret a scatter diagram
• Set up and interpret a contingency table

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Stem-and-leaf Displays

• Note Advantages of the stem-and-leaf display
  over a frequency distribution
• We do not lose the identity of each observation
• We can see the distribution

Stem-and-leaf display A statistical technique
for displaying a set of data. Each numerical
value is divided into two parts the leading
digits become the stem and the trailing digits
the leaf.
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Stock prices on twelve consecutive days for a
major publicly traded company
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Stem and leaf display of stock prices
Trailing digit(s) along horizontal axis
Compare to
Leading digit(s) along vertical axis
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Stem-and-Leaf Example

• Suppose the seven observations in the 90 up to
  100 class are 96, 94, 93, 94, 95, 96, and 97.
• The stem value is the leading digit or digits, in
  this case 9. The leaves are the trailing digits.
  The stem is placed to the left of a vertical line
  and the leaf values to the right. The values in
  the 90 up to 100 class would appear as
• Then, we sort the values within each stem from
  smallest to largest. Thus, the second row of the
  stem-and-leaf display would appear as follows

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Stem-and-leaf Another Example

• Listed in Table 41 is the number of 30-second
  radio advertising spots purchased by each of the
  45 members of the Greater Buffalo Automobile
  Dealers Association last year. Organize the data
  into a stem-and-leaf display. Around what values
  do the number of advertising spots tend to
  cluster? What is the fewest number of spots
  purchased by a dealer? The largest number
  purchased?

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Stem-and-leaf Another Example
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Dot Plots

• A dot plot groups the data as little as possible
  and the identity of an individual observation is
  not lost.
• To develop a dot plot, each observation is simply
  displayed as a dot along a horizontal number line
  indicating the possible values of the data.
• If there are identical observations or the
  observations are too close to be shown
  individually, the dots are piled on top of each
  other.

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Dot Plot

• Dot plots
• Report the details of each observation
• Are useful for comparing two or more data sets

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Percentage of men participating In the labor
force for the 50 states.
Percentage of women participating In the labor
force for the 50 states.
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Quartiles, Deciles, Percentiles(Measures Of
Dispersion)

• Quartiles divide a set of data into four equal
  parts (three points)
• Each interval contains 1/4 of the scores
• Deciles divide a set of data into ten equal parts
  (nine points)
• Each interval contains 1/10 of the scores
• Percentiles divide a set of data into 100 equal
  parts (99 points)
• Each interval contains 1/100 of the scores
• GPA in the 43rd percentile means that 43 of the
  students have a GPA lower and 57 of the students
  have a GPA higher

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Location Of A Percentile In An Ordered Array
If there are an even number of observation, your
Lp may be between two numbers In this case, you
must estimate the number that you will report as
the percentile If Lp 4.25, and the distance
between the two numbers is 37, Lower number
.25(37) percentile
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Compute And Interpret Quartiles

• Quartiles
• Quartiles divide a set of data into four equal
  parts
• Value Q1, Value Q2 (Median), Value Q3 , are the
  three marking points that divide the data into
  four parts
• 25 of the values occur below Value Q1
• 50 of the values occur below Value Q2
• 75 of the values occur below Value Q3
• Value Q1 is the median of the lower half of the
  data
• Value Q2 is the median of all the data
• Value Q3 is the median of the upper half of the
  data

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Example 2 Using twelve stock prices, we can find
the median, 25th, and 75th percentiles as follows
Quartile 3
Median
Quartile 1
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75th percentile Price at 9.75 observation 88
.75(91-88) 90.25
96 92 91 88 86 85 84 83 82 79 78 69
12 11 10 9 8 7 6 5 4 3 2 1
Q4
Q3
50th percentile Median Price at 6.50 observation
85 .5(85-84) 84.50
Q2
25th percentile Price at 3.25 observation 79
.25(82-79) 79.75
Q1
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A box plot is a graphical display, based on
quartiles, that helps to picture a set of data.

• Five pieces of data are needed to construct a box
  plot
• Minimum Value
• First Quartile
• Median
• Third Quartile
• Maximum Value

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Boxplot - Example
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Boxplot Example
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Outliers

• Outlier gt Q3 1.5(Q3 Q1)
• Outlier lt 0

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Coefficient Of Variation

• Coefficient Of Variation converts the standard
  deviation to standard deviations per unit of
  mean
• Use Coefficient Of Variation to compare
• Data in different units
• Data in the same units, but the means are far
  apart
• s Standard Deviation
• Xbar Sample Mean

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Example 1 of Coefficient of Variance
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Skewness - Formulas for Computing

• The coefficient of skewness can range from -3 up
  to 3.
• Measures the lack of symmetry in a distribution
• A value near -3, such as -2.57, indicates
  considerable negative skewness.
• A value such as 1.63 indicates moderate positive
  skewness.
• A value of 0, which will occur when the mean and
  median are equal, indicates the distribution is
  symmetrical and that there is no skewness
  present.

In Excel use the SKEW function
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Commonly Observed Shapes
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Skewness An Example

• Following are the earnings per share for a sample
  of 15 software companies for the year 2005. The
  earnings per share are arranged from smallest to
  largest.
• Compute the mean, median, and standard deviation.
  Find the coefficient of skewness using Pearsons
  estimate. What is your conclusion regarding the
  shape of the distribution?

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Skewness An Example Using Pearsons Coefficient
The skew is moderately positive. This means that
a few large values are pulling the mean up, above
the median and mode.
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Describing Relationship between Two Variables

• One graphical technique we use to show the
  relationship between variables is called a
  scatter diagram.
• To draw a scatter diagram we need two variables.
  We scale one variable along the horizontal axis
  (X-axis) of a graph and the other variable along
  the vertical axis (Y-axis).

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Describing Relationship between Two Variables
Scatter Diagram Examples
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Scatter Diagram

• Independent Variable (X)
• The Independent Variable provides the basis for
  estimation
• It is the predictor variable
• Dependent Variable
• The Dependent Variable is the variable being
  predicted or estimated
• Scatter Diagrams
• Visual portrayal of the relationship between two
  variables
• A chart that portrays the relationship between
  the two variables
• X axis (properly labeled name and units)
• Y axis (properly labeled name and units)
• Scatter diagram requires both variables to be at
  least interval scale

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Describing Relationship between Two Variables
Scatter Diagram Excel Example

• In the Introduction to Chapter 2 we presented
  data from AutoUSA. In this case the information
  concerned the prices of 80 vehicles sold last
  month at the Whitner Autoplex lot in Raytown,
  Missouri. The data shown include the selling
  price of the vehicle as well as the age of the
  purchaser.
• Is there a relationship between the selling price
  of a vehicle and the age of the purchaser? Would
  it be reasonable to conclude that the more
  expensive vehicles are purchased by older buyers?

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Describing Relationship between Two Variables
Scatter Diagram Excel Example
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Contingency Tables

• A scatter diagram requires that both of the
  variables be at least interval scale.
• What if we wish to study the relationship between
  two variables when one or both are nominal or
  ordinal scale? In this case we tally the results
  in a contingency table.

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A contingency table is used to classify
observations according to two identifiable
characteristics.
Contingency tables are used when one or both
variables are nominally or ordinally scaled.
A contingency table is a cross tabulation that
simultaneously summarizes two variables of
interest.
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Contingency Tables Example 1
Weight Loss 45 adults, all 60 pounds overweight,
are randomly assigned to three weight loss
programs. Twenty weeks into the program, a
researcher gathers data on weight loss and
divides the loss into three categories less
than 20 pounds, 20 up to 40 pounds, 40 or more
pounds. Here are the results.
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Contingency Tables Example 1
Weight Loss Plan Less than 20 pounds 20 up to 40 pounds 40 pounds or more
Plan 1 4 8 3
Plan 2 2 12 1
Plan 3 12 2 1
Compare the weight loss under the three plans.
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Contingency Tables Example 2

• A manufacturer of preassembled windows produced
  50 windows yesterday. This morning the quality
  assurance inspector reviewed each window for all
  quality aspects. Each was classified as
  acceptable or unacceptable and by the shift on
  which it was produced. Thus we reported two
  variables on a single item. The two variables are
  shift and quality. The results are reported in
  the following table.