1.9 Comparisons of the Distributions for One Quantitative Variable

Compare multiple quantitative one-variable graphical representations.

• Graphical representations of a quantitative variable can be used to compare important features between two or more distributions of the same quantitative variable. Histograms, back-to-back stem-and-leaf plots, and dotplots may be used to compare center, variability, shape, outliers, clusters, or gaps in two or more distributions. Boxplots may be used to compare center, variability, outliers, and skewness (or symmetry).

Compare multiple quantitative one-variable graphical representations of summary statistics.

• A comparison of graphical representations for two or more distributions can include any of the numerical summaries (e.g., mean, standard deviation, etc.).

Justify a claim using multiple quantitative one-variable graphical representations.

• Multiple quantitative one-variable graphical representations may reveal information that can be used to justify claims about the variable in context.

Calculate z-scores with population parameters.

• A standardized score measures the number of standard deviations a data value falls above or below the mean.
• A z-score is calculated as xi - μ σ , where xi is the data value, µ is the population mean, and is the population standard deviation. A z-scor σ e measures how many standard deviations a data value is above (positive z-score) or below (negative z-score) the mean. When the population mean and standard deviation are unknown, the sample mean and standard deviation may be used to determine a z-score.

Compare z-scores as measures of relative position for distributions.

• z-scores may be used to compare relative positions of individual values within a distribution or between distributions.