The normal distribution

Importance of the distribution This is probably the most important distribution for statistical applications Many quantities are approximately normally distributed. Even when the individual observations do not follow this distribution, according to the Central Limit Theorem, sums and averages of the random variables are approximately normal.

Examples of data with approximate normal distributions Height Weight IQ scores Standardized test scores Body temperature Repeated measurements of same quantity

Definition A continuous rv X is said to have a normal distribution with parameters and (where and ) if the pdf of X for is

Graph of the distribution

Properties of the distribution The density curve is symmetric about its mean (which is also the median ) The value of is the distance from the mean to the inflection points of the curve (the points at which the curve changes from turning downward to turning upward).

Graph of the distribution for different values of sigma

Standard normal distribution When the mean is 0 and the variance is 1, the normal distribution is Z is the typical notation for the standard normal distribution.

Computing probabilities for the standard normal distribution An elementary antiderivative of the density function doesn’t exist. However, the distribution is easy to deal with because probabilities have been computed using numerical techniques. Table A.3 gives , the area in the standard normal distribution to the left of z.

Values of the standard normal table Probability values lie in the center of the table.

Example calculations using table

Percentiles of the standard normal distribution For any p between 0 and 1, Table A.3 can be used to obtain the (100p)th percentile of the standard normal distribution. We may not find the exact probability p in the table. For example, the 99th percentile is approximately 2.33.

Notation for percentiles As notation, denotes the value for which of the area under the z curve lies to the right of . Since of the area under the curve lies to the right of , is the th percentile of the distribution.

Standard normal percentiles and critical values 90 95 97.5 99 99.5 99.9 (tail area) .1 .05 .025 .01 .005 .001 1.28 1.645 1.96 2.33 2.58 3.08

Nonstandard normal distribution If X has a normal distribution with mean and standard deviation , then has a standard normal distribution. By “standardizing”, any probability for X can be expressed as a probability for Z.

Breakdown voltage of a diode The breakdown voltage of a randomly chosen diode is know to be normally distributed. What is the probability that the diode’s breakdown voltage is within one standard deviation of its mean?

Diode voltage (continued) The probability that X is within 2 standard deviations of its mean is The probability that X is within 3 standard deviations of its mean is

Empirical rule If the probability distribution of a variable is (approximately) normal, then Roughly 68% of the values are within 1 s.d. of the mean. Roughly 95% of the values are within 2 s.d.’s of the mean. Roughly 99.7% of the values are within 3 s.d.’s of the mean.

Percentiles of an arbitrary normal distribution To compute the (100p)th percentile of a normal distribution with mean and standard deviation , take the (100p)th percentile of the standard normal distribution, multiply it by , and add . ( , )

An example The amount of distilled water dispensed by a certain machine is normally distributed with mean 64 oz. and standard deviation .78 oz. What container size c will ensure that overflow occurs only .5% of the time? We want . Then (in ounces)