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The Normal Distribution

The Normal Distribution. Today, we will revisit the Empirical Rule and other characteristics of the normal distribution. The Empirical Rule. The Empirical Rule: 68% of the data under a normal curve lies within one standard deviation of the mean.

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The Normal Distribution

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  1. The Normal Distribution Today, we will revisit the Empirical Rule and other characteristics of the normal distribution

  2. The Empirical Rule The Empirical Rule: 68% of the data under a normal curve lies within one standard deviation of the mean. 95% of the data under a normal curve lies within two standard deviations of the mean. 99.7% of the data under a normal curve lies within one standard deviation of the mean. 68% 95% 99.7%

  3. The Empirical Rule Here is a another way to look at it: 34% 34% 2.35 2.35 13.5% 13.5%

  4. The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, between what two numbers does 95% of the data lie. Answer: Between 2 standard deviations from the mean. 50 + 2*5 = 60 50 – 2*5 = 40

  5. The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, what percent of the data lies between 45 and 50 Answer: 68% of the data lies within 1 sd of the mean. The percent between 45 and 50 is half of that: 68/2 = 34%

  6. The Empirical Rule The Empirical Rule: A normal population has a mean of 50 and a standard deviation of 5, what percent of the data lies between 40 and 45 Answer: ½ of 95% = 47.5% of the data lies between 40 and 50. ½ of 68% = 34% of the data lies between 45 and 50. 47.5 – 34 = 13.5% of the data between 40 and 45

  7. The Empirical Rule The Empirical Rule: Mr. Gillam gives a test. The mean is 50 and the standard deviation is 5, what percent of the students scored between 40 and 45? Answer: ½ of 95% = 47.5% of the data lies between 40 and 50. ½ of 68% = 34% of the data lies between 45 and 50. 47.5 – 34 = 13.5% of the data between 40 and 45 13.5% of the students score between 40 and 45

  8. The Empirical Rule Remember the Empirical Rule only applies to the normal distribution. 68% 95% 99.7%

  9. The Normal Distribution The Normal Distribution and z-scores The empirical rule provides areas, which can be interpreted as probabilities, under the normal curve between 1, 2 and three standard deviations. This only allows us to solve problems for integer z-scores. To solve other problems, we look up values on a table, or use our calculator.

  10. The Normal Distribution Mr. Gillam gives a test. The mean is 50 and the standard deviation is 5. what percent of the students scored less than 53? We can use our calculator to find Normalcdf(-e99,.6)=.7257  about 73% of the students scored below 53. The Normal Distribution and z-scores

  11. The Normal Distribution You can also look up the area to the left of a particular z value on the normal curve from a table: Area to the left of Z = 0.6 is .7257

  12. The Normal Distribution Mr. Gillam gives a test. The mean is 50 and the standard deviation is 5, what percent of the students scored between 53 and 58? We can use our calculator to find Normalcdf(.6,1.6)=.2195 About 22% of the students score between 53 and 58.

  13. Using the table in this situation requires subtraction. We subtract the area to the left of z = 0.6 from the area to the left of 1.6. The Normal Distribution A2 = .9452 A1 = .7257 Area between =.9452-.7257 = .2195

  14. A related problem is to find the data point associated with a particular area: Mr. Gillam gives a test. The mean is 50 and the standard deviation is 5. What grade has 75% of the other grades below it? The Normal Distribution Use InvNorm(.75) on your calculator to find the Z-score associated with the 75th percentile. InvNorm gives the z-score with the corresponding area to the left. InvNorm(.75) = .6745 Plug this into the formula for z and solve to find X = 53.38

  15. To find the associated z-score using a table, read it backwards The Normal Distribution Z-score with .75 of the data to it’s left is .67

  16. Two other handy things to know: • The total area under the curve is 100% • Half of the curve has area 50% • P(-1<x<.6) = Area(-1,.6) calculator • =Area(-e99,.6)-Area(-e99,-1) left area table • =Area(-1,0)+Area(0,.6) middle area table The Normal Distribution

  17. Two other handy things to know: • The total area under the curve is 100% • Half of the curve has area 50% • Example: Find z The Normal Distribution We are looking for the area that corresponds to .3051 + .50 =.8051 InvNorm(.8051) = .86 Z = 0.86 OR you could read the area table Backwards. .3051

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