The Normal Distribution The “Bell Curve” The “Normal Curve”

Probability You can think of the normal distribution as a probability distribution. Scores close to the mean are very likely and, therefore, very common, while scores far away from the mean are unlikely and, therefore, not very common.

Probability Think of the distribution of IQ scores. Scores right around 100 (the mean) are very common, but the probability of a randomly selected individual scoring very high or very low is improbable (but not impossible).

Calculating Probability The area under parts of the curve can be used to determine the probability of getting a score that falls under that part of the curve. There are formulas for calculating the area, but you can use a z-score and a table (Table A, p, 401) to determine the probability. With common values, you can also determine the probability by looking at a normal curve and doing the math.

For Example What is the probability that someone will score between 85 and 115 on the IQ test? Using the diagram, we can see that it is about 68%.

For Example We can use basic math to determine the probability of someone scoring below 115. 50% of people will score below 100, and 34% of people will score between 100 and 115. 50 + 34 = 84, so about 84% of people will score below 115 (i.e., there is an 84% chance that a randomly selected person will score below 115).

You Try What is the probability of a randomly selected person scoring higher than 85? 68% 95% 16% 84%

You Try What is the probability of a randomly selected person scoring between 55 and 85? 16% 95% 68% 2%

Properties of the Normal Curve The graph we were working with is not very precise, and often you are interested in values that are not listed on normal distribution curves, like the one below.

Properties of the Normal Curve What do you notice about the x-axis on this normal curve? That’s right, it has z-scores, or standard deviation units. All of the rules from chapter 6 apply here. For example, the mean has a z-score of 0, and a z-score of 1 is one standard deviation above the mean, etc.

Properties of the Normal Curve Even though the normal curve extends out to ∞ in both directions, you will rarely deal with scores more than 3 standard deviations (-3 or +3) from the mean, so it is almost always shown divided into 6 standard deviation units: -3 to +3.

Properties of the Normal Curve The normal distribution is symmetrical, 50% of the scores falling on one side and 50% on the other. Use this property to your advantage when doing the math.

Properties of the Normal Curve The normal curve can be flatter or taller depending on the distribution of scores. The larger the standard deviation, the flatter the curve.

Important Numbers: 68, 95, 99

z-scores Remember our formulas from chapter 6. They are going to come in handy when working with normal curves.

z-scores Which raw score (X) has a z-score of 1 in this example?

z-scores We can see that the mean is 100 and the standard deviation is 15, so 115 has a z-score of 1. We can use this information to find z-scores for other raw scores as well.

z-scores Once you have calculated your z-score for an X of interest, you can look the z-score up in the table and find out the area under the curve associated with that z-score. For example, the z-score for a score of 130 on the IQ test is 2. You want to know the probability of an individual scoring 130 or lower on the test. You go to your table (p. 104) and look for 2 in the z column. Uh oh. It doesn’t say area below z, it says area beyond z. What to do? That’s right: 1 – area beyond z = area below z. (Remember, the total area under the curve adds up to 1 or 100%)

Area Below Score = Percentile Rank Actually, what we just found, the percentage of scores less than or equal to the score of interest is the percentile rank. Remember, a percentile is a score at or below which a given percentage of scores lie.

You Try What is the probability of an individual getting a score of 105 or lower? 59.93% 90.82% 62.93% 40.82% So we know that a score of 105 is the 63rd percentile.

You Try What is the probability of an individual getting a score of 73 or lower (remember, symmetry)? 96.41% 3.59% 2.1% 46.41% So we know that a score of 73 is the 3.6th percentile.

TIPS ALWAYS draw a little normal curve when you are solving this type of problem. Shade the area you are trying to determine, then use the table and any math necessary to determine the area. If you are looking for the area below a negative z-score, find the area above the same positive z-score. It will always be the same because the normal curve is symmetrical. Also, use common sense. If you are trying to find a tiny area under the tail and you end up with an answer of 98.2%, you probably forgot to subtract from 1.

Area Above Score Finding the area above the score should be easy now, especially since that is the value that is listed in your table. But what about negative z-values? There are two ways of finding the area. One, find the area below the positive z-score. Two, find the area between the positive z-score and the mean (it’s in the table) and then add 50%.

Area Above Score What is the probability of selecting an individual with a height of 35 cm or greater? If we convert to z, then look in the table under 1.5, we see that the area between the mean and z is 43.32. Then we add 50 and get 93.32%.

You Try What is the probability of selecting an individual with a height of 40 cm or greater? 34.13% 15.87% 82.45% 84.13% Notice that 100 – 15.87 = 84.13 and 34.13 + 50 = 84.13

Area Between Two Scores First, convert both scores to z-scores. When the two z-scores are on opposite sides of the mean (i.e., one is negative and one is positive), you can just find the area between that score and the mean for each score, easy. When they are both on the same side of the mean, you have to do a little more work.

Area Between Two Scores (both scores on the same side of the mean) Find the area between the mean and z for both scores. Then subtract the smaller area from the larger area.

You Try What is the probability of selecting an individual who weighs between 250 and 300 lbs? .99% 99.1% 49.99% 2.1% What if we took a sample of 1000 people from the U.S. population and found that 100 people in our sample weighed between 250 and 300 lbs? Either our sample is biased or our population data are inaccurate. Mean = 180 s = 30

From z-score to Raw Score You may want to know the raw score for a certain percentile or want to know the raw score values for given z-score values on a normal distribution. You already know how to convert z-scores to raw scores: So all you have to do is find the z-score(s).

From z-score to Raw Score For example, if you want to find the raw score at the 20th percentile, first find the z-score at the 20th percentile by locating the z-score with 20% beyond z, then make it negative. Convert that z-score to a raw score and you are done. If your mean is 50 and your standard deviation is 5: (-.84)(5) + 50 = 45.8

Using Probability One common use for the percentages we are finding is estimating the number of people expected to fall above or below a score or between two scores. The math is easy: Just multiply the probability by your sample or population N. For example, if I want to estimate the number of people who will get a score of 100 or less on the IQ test in my sample of 1000 people. I multiply the probability, .5, by 1000 and get 500. What

Using Probability You can also find out what scores are so unlikely that they will occur less than a certain percent of the time. For example, a common percentage that we will see frequently later on is 5%. To find the scores that have a 5% chance of occurring, find the values that cut 2.5% off of each tail. If the mean is 61, and the standard deviation is 3, what would the two raw scores be?

Activity Calculate the probability of a randomly selected individual weighing between 100 and 120 lbs. If we have a sample of 500, how many people would we expect to fall within that interval? Find the raw score at the 80th percentile (the score at or below which 80% of the scores should fall.) Mean = 180 s = 30

Examples of Normal Distribution Height, Weight, Body Temperature ACT Scores, IQ Scores, GRE Scores Sample Means drawn from a single population

Homework Study for Chapter 8 Quiz Read Chapter 9 Do Chapter 8 HW