Chapter 16 Highlights of What We’ve Seen So Far

16.1 Key Graphical Descriptive Summaries for Quantitative Data

  • Histograms and their variants, including smooth density curves, and normal density functions based on the sample mean and sample standard deviation
  • Boxplots and the like, including ridgeline plots and violin plots, that show more of the distribution in a compact format that is especially useful for comparisons
  • Normal QQ Plots which are plots of the ordered data (technically, the order statistics) against certain quantiles of the Normal distribution - show curves to indicate skew, and “S” shaped arcs to indicate seriously heavy- or light-tailed distributions compared to the Normal.

16.2 Key Numerical Descriptive Summaries for Quantitative Data

  • Measures of Location (including Central Tendency), such as the mean, median, quantiles and even the mode.
  • Measures of Spread, including the range, IQR (which measures the variability of points near the center of the distribution), standard deviation (which is less appropriate as a summary measure if the data show substantial skew or heavy-tailedness), variance, standard error, median absolute deviation (which is less affected by outlying values in the tails of the distribution than a standard deviation).
  • I’ll mention the coefficient of variation (ratio of the standard deviation to the mean, expressed as a percentage, note that this is only appropriate for variables that take only positive values.)
  • One Key Measure of Shape is nonparametric skew (skew1), which can be used to help confirm plot-based decisions about data shape.

16.3 The Empirical Rule - Interpreting a Standard Deviation

If the data are approximately Normally distributed, then the mean and median will be very similar, and there will be minimal skew and no large outlier problem.

Should this be the case, the mean and standard deviation describe the distribution well, and the Empirical Rule will hold reasonably well.

If the data are (approximately) Normally distributed, then

  • About 68% of the data will fall within one standard deviation of the mean
  • Approximately 95% of the data will fall within two standard deviations of the mean
  • Approximately 99.7% of the data will fall within three standard deviations of the mean.

16.4 Identifying “Outliers” Using Fences and/or Z Scores

  • Distributions can be symmetric, but still not Normally distributed, if they are either outlier-prone (heavy-tailed) or light-tailed.
  • Outliers can have an important impact on other descriptive measures.
  • John Tukey described fences which separated non-outlier from outlier values in a distribution. Generally, the fences are set 1.5 IQR away from the 25th and 75th percentiles in a boxplot.
  • Or, we can use Z scores to highlight the relationship between values and what we might expect if the data were normally distributed.
  • The Z score for an individual value is that value minus the data’s mean, all divided by the data’s standard deviation.
  • If the data are normally distributed, we’d expect all but 5% of its observations to have Z scores between -2 and +2, for example.

16.5 Summarizing Bivariate Associations: Scatterplots and Regression Lines

  • The most important tools are various scatterplots, often accompanied by regression lines estimated by the method of least squares, and by (loess) smooths which permit local polynomial functions to display curved relationships.
  • In a multivariate setting, we will occasionally consider plots in the form of a scatterplot matrix to enable simultaneous comparisons of multiple two-way associations.
  • We fit linear models to our data using the lm function, and we evaluate the models in terms of their ability to effectively predict an outcome given a predictor, and through R-square, which is interpreted as the proportion of variation in the outcome accounted for by the model.

16.6 Summarizing Bivariate Associations With Correlations

  • Correlation coefficients, of which by far the most commonly used is the Pearson correlation, which is a unitless (scale-free) measure of bivariate linear association for the variables X and Y, symbolized by r, and ranging from -1 to +1. The Pearson correlation is a function of the slope of the least squares regression line, divided by the product of the standard deviations of X and Y.
  • Also relevant to us is the Spearman rank correlation coefficient, which is obtained by using the usual formula for a Pearson correlation, but on the ranks (1 = minimum, n = maximum, with average ranks are applied to the ties) of the X and Y values. This approach (running a correlation of the orderings of the data) substantially reduces the effect of outliers. The result still ranges from -1 to +1, with 0 indicating no monotone association.