# The Normal Distribution

## The Standard Normal Distribution

The standardized normal distribution is a type of normal distribution, with a mean of 0 and standard deviation of 1. It represents a distribution of standardized scores, called , as opposed to raw scores (the actual data values). A  indicates the number of standard deviation a score falls above or below the mean. Z-scores allow for comparison of scores, occurring in different data sets, with different means and standard deviations. It would not make sense to compare apples and oranges. Likewise, it does not make sense to compare scores from two different samples that have different means and standard deviations. Z-scores can be looked up in a Z-Table of Standard Normal Distribution, in order to find the area under the standard normal curve, between a score and the mean, between two scores, or above or below a score. The standard normal distribution allows us to interpret standardized scores and provides us with one table that we may use, in order to compute areas under the normal curve, for an infinite number of data sets, no matter what the mean or standard deviation.
A z-score is calculated as 
 
  . The score itself can be found by using algebra and solving for x. Multiplying both sides of the equation by σ gives: 
 
  . Adding μ to both sides of the equation gives 
 
  .
Suppose we have a data set with a mean of 5 and standard deviation of 2. We want to determine the number of standard deviations the score of 11 falls above the mean. We can find this answer (or z-score) by writing
or
we can solve for z.
We have determined that the score of 11 falls 3 standard deviations above the mean of 5.
With a standard normal distribution, we indicate the distribution by writing Z ~ N(0, 1) which shows the normal distribution has a mean of 0 and standard deviation of 1. This notation simply indicates that a standard normal distribution is being used.

### Z-Scores

As described previously, if X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is
The z-score tells you how many standard deviations the value x is above, to the right of, or below, to the left of, the mean, μ. Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.
When determining the z-score for an x-value, for a normal distribution, with a given mean and standard deviation, the notation above for a normal distribution, will be given.

The Empirical Rule If X is a random variable and has a normal distribution with mean µ and standard deviation σ, then the Empirical Rule states the following:
1. About 68 percent of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean).
2. About  95 percent of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
3. About 99.7 percent of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
4. The z-scores for +1σ and –1σ are +1 and –1, respectively.
5. The z-scores for +2σ  and –2σ are +2 and –2, respectively.
6. The z-scores for +3σ  and –3σ are +3 and –3, respectively.
So, in other words, this is that about 68 percent of the values lie between z-scores of –1 and 1, about 95% of the values lie between z-scores of –2 and 2, and about 99.7 percent of the values lie between z-scores of -3 and 3. These facts can be checked, by looking up the mean to z area in a z-table for each positive z-score and multiplying by 2.
The empirical rule is also known as the 68–95–99.7 rule.

### References

CollegeBoard. (2012). 2012 College-bound seniors: Total group profile report. Retrieved from http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf
Joyce, C. A., Janssen, S., & Liu, M. L. (2010). The world almanac and book of facts, 2010. New York, NY: World Almanac Books.
London School of Hygiene and Tropical Medicine. (2009). Calculation of z-scores. Retrieved from http://conflict.lshtm.ac.uk/page_125.htm
National Center for Education Statistics. (2009). ACT score averages and standard deviations, by sex and race/ethnicity, and percentage of ACT test takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009. Retrieved from http://nces.ed.gov/programs/digest/d09/tables/dt09_147.asp
NBA.com. (2013). NBA Media Ventures. Retrieved from http://www.nba.com
StatCrunch. (2010). Blood pressure of males and females. Retrieved from http://www.statcrunch.com/5.0/viewreport.php?reportid=11960
The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/
Wikipedia. (2013). List of stadiums by capacity - Wikipedia. Retrieved from https://en.wikipedia.org/wiki/List_of_stadiums_by_capacity

### Chapter Review

A z-score is a standardized value. Its distribution is the standard normal, Z ~ N(0, 1). The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution N(µ, σ), then z tells you how many standard deviations x is abovegreater thanor belowless thanµ.

### Formula Review


Z ~ N(0, 1)
z = a standardized value (z-score)
mean = 0, standard deviation = 1
To find the kth percentile of X when the z-score is known,k = μ + (z)σ
z-score: z = 
 
Z = the random variable for z-scores
Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.

### Homework

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.