Here's what you'll find in this section:Normal Distribution, Sampling Distribution of a Statistic
The normal distribution (also called the Gaussian distribution or the famous ``bell-shaped'' curve) is the most important and widely used distribution in statistics. The normal distribution is characterized by two parameters, and , namely, the mean and variance of the population having the normal distribution. We will use the notation to denote a random variable which is distributed normally with mean and variance .
For the normal distribution, the mean, median, and mode are all equal to value where the curve is highest. The mean is sometimes called a location parameter because increasing (decreasing) it shifts the entire normal curve right (left). Likewise, is called a scale parameter because increasing (decreasing) it causes the curve to be wider (narrower). See the ``Z, t, Chi-square, F'' concept lab for examples. The value of is the distance from to the inflection points of the normal curve. (The inflection points are the locations where the curve changes from turning downward to turning upward.) Note that the total area under any continuous curve is equal to one.
In the figure below, we have drawn the standard normal (or Z) curve, that is, the normal curve for and . We have also placed on the plot the percent of the area under the curve between -3 and -2 (2.15%), between -2 and -1 (13.59%), and so on. Note how the curve is symmetric about 0, how around 68% of the area is between -1 and 1, and approximately 95.44% between -2 and 2. (Recall the empirical rule from Week 2.)
The plot of any normal curve will look basically the same as this figure except that the numbers on the horizontal axis instead of being centered at 0 will be centered at and the other numbers will be at multiples of away from . For example, the 1 would be replaced at , the 2 by , the -3 by and so on. The IQ histogram back in Week 1 gives an example of this where the curve is centered at = 100 and the labels on the horizontal axis are at multiples of = 15 away from 100.
Since there are an infinite number of normal distributions (there is one for any choice of and ), we will standardize all our normal distribution problems to a standard normal distribution. The notation we will use is . So for any ,
Our method for finding probabilities for normal distribution problems will be as follows:
EXAMPLE:\ Let X be the blood platelet count (measured in thousands per cc of blood) in humans, and suppose . What is the probability that an individual has a platelet count between 185.4 and 360.2? It is a good idea to always draw a picture. The solution is as follows. We have
is given by
which, in turn is P(-1.96<Z<1.96), which is
We might also ask what the value of X is equivalent to the 99% percentile (99% of all individuals will have a platelet count below this value). To solve this we reverse the process used above:
which means that
Thus, someone with a blood-platelet count of 376.72 has a higher count than 99% of the population.
The webmaster and author of this Math Help site is Graeme McRae.