• The overall pattern of a distribution can often be described compactly by a density curve. A density curve has total area 1 underneath it. Areas under a density curve give proportions of observations for the distribution.
• The mean μ (balance point), the median (equal-areas point), and the quartiles can be approximately located by eye on a density curve. The standard deviation σ cannot be located by eye on most density curves. The mean and median are equal for symmetric density curves, but the mean of a skewed curve is located farther toward the long tail than is the median.
• The Normal distributions are described by bell-shaped, symmetric, unimodal density curves. The mean μ and standard deviation σ completely specify the Normal distribution N(μ, σ). The mean is the center of symmetry, and σ is the distance from μ to the change-of-curvature points on either side. All Normal distributions satisfy the 68–95–99.7 rule.
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• To standardize any observation x, subtract the mean of the distribution and then divide by the standard deviation. The resulting z-score z = (x − μ)/σ says how many standard deviations x lies from the distribution mean. All Normal distributions are the same when measurements are transformed to the standardized scale.
• If X has the N(μ, σ) distribution, then the standardized variable Z = (X − μ)/σ has the standard Normal distribution N(0, 1). Proportions for any Normal distribution can be calculated by software or from the standard Normal table (Table A), which gives the cumulative proportions of Z < z for many values of z.
• The adequacy of a Normal model for describing a distribution of data is best assessed by a Normal quantile plot, which is available in most statistical software packages. A pattern on such a plot that deviates substantially from a straight line indicates that the data are not Normal.