We can sometimes describe the overall pattern of a distribution by a density curve. A density curve has total area 1 underneath it. An area under a density curve gives the proportion of observations that fall in a range of values.
A density curve is an idealized description of the overall pattern of a distribution that smooths out the irregularities in the actual data. We write the mean of a density curve as and the standard deviation of a density curve as to distinguish them from the mean and standard deviation of the actual data.
The mean, the median, and the quartiles of a density curve can be located by eye. The mean is the balance point of the curve. The median divides the area under the curve in half. The quartiles and the median divide the area under the curve into quarters. The standard deviation cannot be located by eye on most density curves.
The mean and median are equal for symmetric density curves. The mean of a skewed curve is located farther toward the long tail than is the median.
The Normal distributions are described by a special family of bell-shaped, symmetric density curves, called Normal curves. The mean and standard deviation completely specify a Normal distribution . The mean is the center of the curve, and is the distance from to the change-of-curvature points on either side.
To standardize any observation , subtract the mean of the distribution and then divide by the standard deviation. The resulting z-score
says how many standard deviations lies from the distribution mean.
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All Normal distributions are the same when measurements are transformed to the standardized scale. In particular, all Normal distributions satisfy the 68-95-99.7 rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.
If x has the distribution, then the standardized variable has the standard Normal distribution with mean 0 and standard deviation 1. Table A gives the proportions of standard Normal observations that are less than for many values of . By standardizing, we can use Table A for any Normal distribution.
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.