More Examples of Boxplots

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More Examples of Boxplots
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Quantile Plots

• Quantile plots visually portray the quantiles, or
  percentiles (which are quantiles times 100) of
  the distribution of sample data.
• Percentiles of importance e.g. median, quartiles
  are easily discerned from a quantile plot.
• With experience, the spread, skewness, as well as
  bimodal character, can be examined.
• Quantile plots have 3 advantages
• 1. Arbitrary categories are not required, as with
  histograms and S and Ls.
• 2. All the data are displayed, unlike boxplots.
• 3. Every point has a distinct position, without
  overlap.

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Construction of a Quantile Plot

• Data of size n are ranked from smallest to
  largest. The smallest data value is assigned a
  rank i 1.
• The data values are plotted along the x-axis
  usually.
• The y-axis is the plotting position, which is a
  function of the rank i and size n. The plotting
  position is

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• As sample sizes increase, the quantile plot will
  more closely mimic the underlying population cdf.
• Example calculation n 55
• i qi pi
• 1 994.3 0.01
• 2 1263.1 0.03
• . . .
• . . .
• 54 7270.1 0.97
• 55 7730.7 0.99
• Plot pi versus qi.
• Note Other plotting positions are sometimes used
  e.g. Hazen, Weibull, Blom, Gringorten, etc.
  Cunnanes is a compromise formula.

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Probability Plots

• One variation of the quantile plot is the
  probability plot.
• Probability plots are used to determine how well
  data fit a theoretical distribution, such as the
  normal, lognormal, or Gumbel.
• By expressing the theoretical distribution as a
  straight line, departures from the distribution
  are more easily perceived. This is what occurs
  with a probability plot.
• Probability plots are thus plots of the quantiles
  of sample data versus the quantiles of the
  standardized theoretical distribution.
• For a quantile plot, plot qi vs. pi.

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• For a normal probability plot, plot qi vs. zi
  where zis are the normal quantiles for a given
  pi.
• The zis can be obtained from a table of
  standard normal distribution or
• For comparison it is useful to plot a reference
  straight line on the plot. The solid line on the
  plot below is the normal distribution which has
  the same mean and standard deviation as do the
  sample data.
• The equation of the line can be obtained by
  linear regression between qi and zi. Linearity
  is easily checked by degree of correlation
  between qi and zi. A table is available to test
  for this.
• The mean is at zi 0, the standard deviation is
  the slope of the line.

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• Commercially printed probability paper is often
  used for probability plots (by hand). This paper
  retransformed the linear scale for zi back into a
  nonlinear scale of pi.
• Also, probits scale paper are available probit
  zi 5.
• Probability paper can be easily constructed for
  distributions other than the normal and
  lognormal. E.g. Gumbel, Weibull, etc.
• Normal probability plots can easily be done on
  Minitab. It will also test whether the data are
  normal.
• Minitab can also do other types of probability
  plots e.g. Weibull, lognormal, logistic, etc.

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Deviations From a Linear Pattern

• Indicates data are left or right skewed.
• Indicates outliers present.
• Indicates heavy or thick tailed data.

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Use of Probability Plots for Obtaining Summary
Statistics for Censored Data - a Robust Method.

• Set all less-thans to slightly different values
  all below the reporting limit.
• Develop a linear regression equation between qi
  and zi using only above-limit observations.
• Estimates for the below-limit data are
  extrapolated using the regression equation from
  zi for the below-limit data.
• Extrapolated estimates are used together with
  above-limit data to obtain summary statistics of
  interest for whole data set.

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Other useful graphical procedures

• X-Y Scatter plots. For looking at whether there
  is a relationship or correlation between X and Y.
  This plot is very useful in correlation and
  regression analysis.
• Bubble plots Similar to the simple X-Y scatter
  plot, but the size of the bubble now represents
  the magnitude of a third variable. E.g. X
  Eastings, Y Northings, and Z size of bubble
  (mean concentration of a chemical). Bubble plots
  are not available in Minitab.

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Summary

• 1. Reasons for using graphics.
• 2. Advantages and disadvantages of
• Histograms
• Stem and leaf plots
• Dotplots
• Boxplots
• Quantile plots
• Normal probability plots - construction and
  testing
• Most useful aids are x-y plots, boxplots, and
  normal probability plots