Density curves are mathematical models for a distribution. They provide an overall pattern and ignore any irregularities and outliers. Another reason that density curves are preferred over histograms is that histograms are affected by class width.
As we make the class size smaller and increase the number of observations, the histogram starts to look like the smooth density curve.
Properties of denisty curves:
No set of real data is exactly described by a density curve. Even so, it is an approximation that provides us useful details about the distribution.
The mean and median of a symmetric density curve are equal. The mean is larger than the median for a right-skewed density curve. The mean is smaller than the median for a left-skewed density curve.
Because the density curve is an idealized description of a distribution, we distinguish the mean (now represented by μ) and the standard deviation (now represented by σ) in the density curve from the mean and standard deviation (x-bar and s) of the distribution.
The normal distribution can be described by the normal curve (a.k.a. bell-shaped curve). All normal distributions have the same shape. Because we use the normal distribution often, we use N(μ, σ) to represent the normal distribution with mean, μ, and standard deviation, σ. Every normal density curve is described by its mean μ and standard deviation σ. You can explore how the mean and standard deviation affect the normal curve.
In the normal distribution with mean, μ, and standard deviation, σ:
You can explore the 68–95–99.7 rule.
Use the TI-83 program called NRMHST to check if a set of data is normally distributed.
Since all normal distributions have the same properties, we can standardize the mean and standard deviation.
z-scores is a standardized observation. Let x be an observation from a distribution that has
mean, μ, and standard deviation, σ, then the standardized observation is given by:
z-score = (z – μ)/σ.
A z-score tells us how many standard deviations the original observation falls from the mean and in which direction. Positive z-scores tell us that the observation is larger than the mean and negative z-scores tell us that the observation is smaller than the mean.
The standard normal distribution is N(0, 1) where the mean is zero and the standard deviation is 1.
An area under the curve provides the proportion (in percent or decimal form) of the observations in a distribution. Since all normal distributions are the same after standardization, we can find the area under any normal curve from a single table (Table A on pp. 580-1). You MUST standardize before using the table. Also, the table only provides areas to the left of a given z-score. The area between a and b is equal to the proportion of the observations between a and b.
To calculate the area by hand, follow examples 1.16 and 1.17 on pp. 59-60.
However, the TI-83 provides an easy function to determine the areas!!!! Here are the steps. You must remember that the number that the calculator gives you is the area under the normal curve which represents a proportion (decimal form) of the observations satisfying the given conditions.
Even though the calculator provides the proportion, you should also sketch the area. Use the TI-83 program called NAREA to to see a graph of the normal distribution.
We may want to find the observation above which (or below which) a given proportion exists. In other words, given an area, what is the observation that has the given area to the right (or left). This is slightly more difficult to do using Table A. Luckily, the TI-83 calculator provides a built-in function to find the value of the observation!!!! Here are the steps.