Ó 2004, S. D. Cochran. All rights reserved.

THE NORMAL DISTRIBUTION 

  1. This chapter is about taking advantage of something we know is true to give us an edge in making decisions--we do this by translation 

  1. We already translate units all the time for convenience. We translate feet into inches to make addition easier, e.g. 1 1/4 feet + 1 1/12 feet = 16 inches + 13 inches = 29 inches = 2 5/12 feet 

  2. We say URSA instead of the formal name (University Records System Access) to communicate 

  3. We are now going to learn about a distribution (the z-distribution) that has some very handy attributes and we are going to learn how to translate our observations, scores, distributions, into that scale and back again 

  4. Standard scales convey important information in addition to the value observed. Example: How frequently do you go to the student store on a 1 to 7 scale? How many days per week is that? Does knowing the standard units of days give us more information? 

  1. The normal distribution, also called the z-distribution 

1. Drawing the normal distribution

  1. You know the formula for a straight line: y = bx + c; The normal curve is also a line but it is defined by: 

 

  1. The result is what is called a bell-shaped curve 

  1. Important properties of this line and the shape it creates 

  1. Symmetric

  2. Mean = median = mode

  3. Mean is 0, S.D. is 1

  4. Area under the curve is 100%--at 0 height is about 40%--draw two triangles and calculate area (1/2*40*± 3) as an approximation of the curve 

  5. We know at any point on the x-axis what the density or area under the curve is 

  1.  ± 1 S.D. = 68%

  2.  ± 2 S.D. = 95%

  3.  ± 3 S.D. = 99% 

  1. Using a standard normal table 

  1. The book gives many examples and lots of practice--good idea

  2. Remember area under the curve is 100%; 50% of the curve lies to either side of the mean

  3. So the strategy is to:

  1. Draw the curve.

  2. Detail the area you want to estimate

  3. The table in the book gives the area in percentages bounded on the positive and negative side by the z-value (which is equivalent in units like this: 1 z = 1 S.D.) 

  1. Examples:

  1. What percent of the standard normal curve lies below 1.75 standard deviations? (For 1.75 z, the area = 91.99%, 91.99/2 + 50 = 95.99%) 

  2. What standard score cuts off 9.7% of the distribution to the left of score? (We know 100%-(2*9.7)%=80.6%, so we find z = 1.30, and we want the negative side so z = -1.30, or 1.3 standard deviations to the left)

  1. Standard units 

  1. Definition: z is a distribution that is scaled in standard units where the unit is standard deviations from the mean in a normal distribution. (Just like calendar days are a distribution where the scale is approximately the time it takes for earth to make one complete orbit around the sun) 

  2. The formula for z is: 

 

  1. And we can relate z directly to the normal distribution and so we know the percentile of the score 

    2.

  2. Example of translating into z 

I.Q. scores have a normal distribution, approximately, with a mean of 100 and a standard deviation of 16. If Max has an I.Q. of 115, how much smarter is he than everybody else? 

First we translate the I.Q. score out of I.Q. units into z-units by using the equation. (115-100)/16 = .94. From Table A we find that going out from that z-score on both sides cuts off 65.79 percent of people. We divide that in half and add 50% to get the correct %. So we can say, Max, with his I.Q. of 115 is smarter than about 83% of people, or 17% of people are smarter than Max. 

  1. Example of translating percentile into a score 

We can also go the other way. Let's say your teacher does not want you to know your I.Q. exactly because she thinks you'll end up with a swelled head. So instead, she tells you that you are smarter than 94% of people. Well, you know that I.Q. scores have a mean of 100 and a S.D. of 16 so you do the math. First you look in the table to find out what z score cuts off 94% of the distribution, 1.55. Then you put it in the equation (1.55 = (X - 100)/16) and find that your I.Q. is 125. So much for secrets. 

You can also use your percentile score on the SAT to estimate your IQ except for what? (Not everyone takes the SAT....) 

  1. If your data are not normally distributed, then using the normal distribution as an approximation will give you the wrong answer 

  1. How do you know? Use common sense. Think about your distribution and whether or not it should a bell-shaped curve 

Examples: Number of pieces of pizza eaten by your friends at dinner? Most eat one or two, but there's always one way out there eating 6. Income. Number of children. Housing prices.  

  1. If you calculate a z-score and the conclusion is nonsensical (like a negative number of pizza slices eaten) this is a flag that your distribution is not normal 

  2. If your data are not normally distributed, you might think about using the median and the interquartile range to report your data.  

  1. Definition of a percentile: a number, y, is in the nth percentile for the data if n% of the data are less than or equal to y. 

  2. Quartiles are the 25th, 50th, and 75th percentiles. The 50th percentile is the median 

  3. The interquartile range (IQR) is that range of the distribution that contains the 25th to 75th percentiles