The normal distribution is very important in statistical inference. We should realize, however, that it is not a natural law that we encounter each time we analyze a continuous random variable. The normal distribution is a theoretical or ideal, distribution. No set of measurements conforms exactly to its specifications. Many sets of measurements, however, are approximately normally distributed. In such cases, the normal distribution is quite useful when we try to answer practical questions regarding these data.
In particular, whenever a set of measurements is approximately normally distributed, we can find the probability of occurrence of values within any specific interval, just as we can with the standard normal distribution. We can do this because we can easily transform any normal distribution with a known mean JJL and standard deviation CT to the standard normal distribution. Once we have made this transformation, we can use a table of standard normal areas to find relevant probabilities.
We can transform a normal distribution to the standard normal distribution using the formula z=(x- (JL)/CT. This transforms any value of x in an original distribution with mean) x and standard deviation CT to the corresponding value of z in the standard normal distribution.
The Normal Approximation to the Binomial
The normal distribution gives a good approximation to the binomial distribution when n is large and p is not too close to 0 or 1. This enables us to calculate probabilities for large binomial samples for which binomial tables are not available. A good rule of thumb is that the normal approximation to the binomial is appropriate when np and n(l - p) are both greater than 5. To normally distributed, we can make more powerful probability statements than we could fusing Chebyshev's theorem.
The normal distribution is completely determined by its parameters u, and cr. That is, each different value of JJL or o~ specifies a different normal distribution.
The Standard Normal Distribution
The normal distribution is really a family of distributions in which one member is distinguished from another on the basis of the values of |x and a. In other words, as already indicated, there is a different normal distribution for each different value of either |x or a.
The most important member of this family of distributions is the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We usually use the letter z for the random variable that results from the standard normal distribution. The probability that z lies between any two points on the z axis is determined by the area bounded by perpendiculars erected at each of these points, the curve, and the horizontal axis. We find areas under the curve of a continuous distribution by integrating the function between two values of the variable. There are tables that give the results of integrations in which we might be interested. The table of the standard normal distribution may be presented in many different forms.
Applications of the Normal Distribution
The normal distribution is very important in statistical inference. We should realize, however, that it is not a natural law that we encounter each time we analyze a continuous random variable. The normal distribution is a theoretical or ideal, distribution. No set of measurements conforms exactly to its specifications. Many sets of measurements, however, are approximately normally distributed. In such cases, the normal distribution is quite useful when we try to answer practical questions regarding these data.
In particular, whenever a set of measurements is approximately normally distributed, we can find the probability of occurrence of values within any specific interval, just as we can with the standard normal distribution. We can do this because we can easily transform any normal distribution with a known mean ju, and standard deviation a to the standard normal distribution.
Once we have made this transformation, we can use a table of standard normal areas to find relevant probabilities.
We can transform a normal distribution to the standard normal distribution using the formula z = (x- (x)/a. This transforms any value of x in an original distribution with mean u- and standard deviation CT to the corresponding value of z in the standard normal distribution.
The Normal Approximation to the Binomial
The normal distribution gives a good approximation to the binomial distribution when n is large and p is not too close to 0 or 1. This enables us to calculate probabilities for large binomial samples for which binomial tables are not available. We convert values of the original variable to values of z to find the probabilities of interest.
The Continuity Correction. The normal distribution is continuous and the binomial is discrete. Therefore we get better results if we make an adjustment to account for this when we use the approximation. The need for such an adjustment, called the continuity correction, is evident when we compare a histogram constructed from binomial data with a superimposed smooth curve.
Applications Of The Normal Distribution
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