Data Display and Graphical Methods

• Motivation: Use creative displays to extract information about properties of a set.
  • The stem and leaf plots provide the viewer a look at symmetry of the data.
  • Normal probability plots and quantile plots are used to check normal distribution.
• Characterize statistical analysis as the process of drawing conclusion about system variability.
• Statistics provide single measures, whereas a graphical display adds additional information in terms of a picture.
• Box-and-whisker plot encloses the interquartile range of the data in a box that has median displayed within.
• A graphical tool to get an idea about the center, variability and degree of asymmetry of a sample.
• Interquartile range: between the $75^{th}$ percentile (upper quartile) and the $25^{th}$ percentile (lower quartile).
• Box plot provides the viewer information about outliers which represent rare event.

Figure 1: Box-and-Whisker plot.

• Nicotine content was measured in a random sample of 40 cigarettes. The data is displayed in the table.
• Mild outliers: 0.72, 0.85, and 2.55

  
  

  Table 1: Nicotine Data for Example 8.3.

  1.09	0.85	1.86	1.82	1.40	1.92	1.24	1.90
  1.79	1.64	2.31	1.58	1.68	2.46	2.09	1.79
  2.03	1.51	1.88	1.75	2.28	1.70	1.64	2.08
  1.63	1.74	2.17	0.72	1.67	2.37	1.47	2.55
  1.69	1.37	1.75	1.97	2.11	1.85	1.93	1.69

  
  

  Figure 2: Box-and-Whisker plot for nicotine data.

  
  • Sample size $n=40$.
  • Sort the sample.
  • $25^{th}$ percentile: $\left(\frac{25*n}{100} \right) ^{th}$ element in the sorted list.
  • $q(0.25)=X_{sorted}(10)=1.63$
  • $q(0.50)=X_{sorted}(20)=1.75$
  • $q(0.75)=X_{sorted}(30)=1.97$

• Interquartile range: $q(0.75) -q(0.25)=1.97 -1.63 =0.34$
• The whiskers are drawn at a distance of 1.5 times the interquartile range from the $25^{th}$ and $75^{th}$ percentiles.
• 1.63-1.5*0.34 & 1.977+1.5*0.34
• Anything outside that range is shown as an outlier.
• Another graphical tool: Stem-and-leaf plot.
  1. Split each observation into 2 parts: stem and leaf.
     • Stem can be the digit preceding the decimal,
     • Leaf can be the digit after the decimal.
  2. Make a table: List the stem values as rows. Add each leaf value with a specific stem value to that row.
• Gives an idea about what stem values occur more frequently.

Figure 3: Stem-and-leaf plot for the nicotine data.

• Example 8.4: Consider the following data, consisting of 30 samples measuring the thickness of paint can ears.

  Figure 4: Data for Example 8.4.

  

  Figure 5: Box-and-whisker plot for thickness of paint can ``ears''.

  

• Quantile plot
  • Compare samples of data
  • Draw distinctions
  • Depict cumulative distribution function
• Definition 8.8:
  
• Sample median: q(0.5); $75^{th}$ percentile: q(0.75); $25^{th}$ percentile: q(0.25).
• A quantile plot simply plots the data values on the vertical axis against an empirical assessment of the fraction of observations exceeded by the data value.

• Let $f_i$ be the $i^{th}$ observation when they are sorted low to high.
• Then $f_i$ is the $(i/n)^{th}$ quantile where $n$ is the size of the sample.
• So we plot $f_i$ vs $(i/n)$. For theoretical purposes this fraction is computed as
  $$f_i=\frac{i-\frac{3}{8}}{n+\frac{1}{4}}$$

  Plotting position formula

  $$f_i=\frac{i-a}{n+1-2a}$$
  for some $a$

• where $i$ is the order of the observations when they are ranked from low to high.
• In other words, if we denote the ranked observations as
  $$y_{(1)} \leq y_{(2)} \leq \ldots \leq y_{(n-1)} \leq y_{(n)}$$
  then the quantile plot depicts a plot of $y_{(i)}$ against $f_i$.

Figure 6: Quantile plot for paint data.

• In Fig. 6, quantile plot shows all observations.
• Large clusters: slopes near zero. e.g.: 36-38
• Sparse data: steeper slopes. e.g.: 28-30

• Dedection of deviations from normality.
• We often assumes that a data set are realizations of independently identically distributed normal random variables.
• Question: Did this sample come from a population with a normal distribution?
• Tool: We can take advantage of what is known about the quantiles of the normal distribution to answer this question.
• The diagnostic plot can often nicely augment a formal goodness-of-fit test on the data.

• Approximation of quantile of normal distribution
  $$q_{\mu,\sigma}(f)=\mu+\sigma\left\lbrace 4.91\left[ f^{0.14}-(1-f)^{0.14}\right] \right\rbrace$$
  $\mu=0$ and $\sigma=1$ for standard normal distribution
  $$q_{0,1}(f)=4.91\left[ f^{0.14}-(1-f)^{0.14}\right]$$

• Definition 8.8:
  
• A nearly straight line relationship suggests that the data came from a normal distribution.
• The intercept on the vertical axis is an estimate of the population mean $\mu$.
• The slope is an estimate of the standard deviation $\sigma$.

Figure 7: Normal quantile-quantile plot for paint data.

• Normal probability plotting.
• The vertical axis contains $f$ plotted on special paper, known as probability paper.
• The scale used results in a straight line when plotted against the ordered values of a normal random variable.
• If the normal distribution adequately describes the data, the plotted points will fall approximately along a straight line.
• Construct a normal quantile-quantile plot and draw conclusions regarding whether or not it is reasonable to assume that the two samples are from the same $N(\mu,\sigma)$ distribution.

Figure 8: Data for Example 8.5.

• Solution:
• Far from a straight line.
• Station 1 reflect a few values in the lower tail of the distribution and several in the upper tail.
• Unlikely!

Figure 9: Standard Normal Quantile, $q_{0,1}(f)$.