THE NORMAL DISTRIBUTIONS Chapter 2

Z-Scores and Density Curves

A Question Last year, Eunice had Mr. Allen for math and received a 87% in the class, while Irene had Mr. Merlo for the same math class and received a 80%. It has been mathematically proven that Mr. Merlo is a much harder teacher. In fact, his class average was 15% lower than Mr. Allen last year. Who is smarter? Why can you argue that Irene is smarter? What extra piece of information might prove that Eunice is actually smarter?

The Standardized Value The Standardized Value (z-score) is a measure of the number of standard deviations a piece of data is away from the mean in a normal distribution. If a test or other measure has been standardized, z- scores can be used to determine whether or not individuals are better.

A More Detailed Question Last year, Eunice had Mr. Allen for math and received a 87% in the class, while Irene had Mr. Merlo for the same math class and received a 80%. It has been mathematically proven that Mr. Merlo is a much harder teacher. In fact, Mr. Allen’s class average was 15 points higher than Mr. Merlo’s 70% average. If we know that Mr. Allen’s class had a standard deviation of 2% and Mr. Merlo’s class had a standard deviation of 10%, Who is smarter?

Density Curves A density curve is what you get when you collect a lot of data and you get a fluid shaped graph. It has an area of exactly 1 underneath it.  That’s because it represents 100% of your data.  The median cuts the area in half.  The mean is the balance point.

Two Different Density Curves

What Is the Most Common Density Curve?

Normal Distribution This is the standard bell-shaped curve. The mean and median are always the same in a normal distribution. Although different normal distributions are similar, they might have different shapes.  Some are “taller” or “wider” than others.  What determines how “tall” or “wide” a normal distribution is?  The standard deviation.

One Up, One Down Although the shape may change, the proportion of the data between the two standard deviations remains the same.  68% of the outcomes are between one standard deviation above and below the average.  Notice one standard deviation away is at the inflection point.

The Empirical Rule The Empirical Rule ( ) Rule tells you the proportion of the data that is in the middle when you move standard deviations away from the mean.

AND WHAT THE AP GRADERS ARE LOOKING FOR Standard Normal Calculations

Finding a Probability If a population is known to have a normal distribution of ages with an average of 16 and a standard deviation of 1.2, what is the probability that a randomly chosen individual will be older than 18? N(µ, σ)  N(16, 1.2) P(x>18) = P(z>(18-16)/1.2) = P(z>1.67) = =.0475

Know how to find the probability of an event occuring Using the same information from the previous slide, what proportion of the population is between the ages of 16 and 17? N(16, 1.2) P(16 < X < 17) = P((16-16)/1.2)< Z < (17-16)/1.2) = P(0 < Z < 0.83) = – 0.5 =0.2967

Know what a percentile is and how to a value at a certain percentile Using the same information from the previous two slides, what age does an individual have to be in order to be above the 35 th percentile? N(16, 1.2) P(Z < -0.39) = = (x – 16)/ = x – 16 X = 15.53

Calculator Normalcdf(lowerbound, upperbound, avg, s.d.) Example Find P(x>18)=normalcdf(18, , 16, 1.2) InvNorm(percent behind, avg, s.d.) Example P(x < ___) = 0.35  InvNorm(0.35, 16, 1.2)

Know how to find a population average if you know the probability of an event and s.d. In a certain baseball league 20% of the individuals have more than 60 RBIs. If the standard deviation of all the players’ RBIs is 15 and the distribution is known to be approximately normal, what is the average number of RBIs in this league? The league average is 47.4 RBIs

Know how to find the middle n% of a normal distribution Looking at a distribution that is N(10, 2), what interval contains the 20% of the population with the shortest interval? Solution In any normal distribution, the n% with the will be in the middle, because that is where your largest percent of data is. So, this question is really just, “where is the middle 20%?”

Solution Continued Since we’re looking for the middle 20% of a N(10, 2), we will look for the z-score that have 40% above and 40% below. A similar method using z=0.25 would give us an x value of So, the smallest interval contain 20% of the data is between 9.5 and 10.