# intro to lsp 121 normal distributions lsp 121. welcome to lsp 121 quantitative reasoning and...

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• Intro to LSP 121 Normal DistributionsLSP 121

• Welcome to LSP 121Quantitative Reasoning and Technological Literacy IIContinuation of concepts from LSP 120Topics we feel you will need to make it through college and into a careerNormal distributionsDescriptive statistics and correlationProbability and riskDatabasesAlgorithmsIf you feel you know this material, take the testSee Syllabus under Prerequisites

• What is a Normal Distribution?Very common, very special type of distributionMost data values are clustered near the mean (a single peak)Distribution is symmetricTapering tales as you move away from the meanLooks like a bell curve

• The 68-95-99.7 RuleAbout 68% (68.3%), or just over 2/3, of the data points fall within 1 standard deviation (+ or -) of the meanAbout 95% (95.4%) of the data points fall within 2 standard deviations of the meanAbout 99.7% of the data points fall within 3 standard deviations of the mean

• Pop-Quiz

How many percent lie between mean -1 standard deviation and mean + 1 standard deviation?68%

How many percent lie between mean + 1 stdev and mean +3 stdev?15.85%

How many percent lie greater than mean + 3 stdev? 0.15%

• ExampleIn the real world, SAT exams typically produce normal distributions with a mean of 500 and a standard deviation of 100.Thus, 68% of the students score between 400 and 60095% of the students score between 300 and 70099.7% score between 200 and 800What if someone scored 720 on the SAT? What percentage of students scored less than or equal to 720?Use Excels NORMDIST functionIn a cell type: =NORMDIST(X, mean, stdev, true)For our problem: =NORMDIST(720, 500, 100, TRUE)Answer = 0.986097, or 98.6097%What percentage scored greater than 720?

• ** Another ExampleA survey finds that prices paid for two-year-old Ford Explorers are normally distributed with a mean of \$16,500 and a standard deviation of \$500. Consider a sample of 10,000 people who bought two-year-old Ford Explorers.How many people paid between \$16,000 and \$17,000?=NORMDIST(16000,16500,500,true) yields 0.158655=NORMDIST(17000, 16500, 500, true) yields 0.841345Subtract: 0.841345 0.158655 yields 0.682689Or use the graph two slides back

• Another ExampleHow many paid less than \$16,000?=NORMDIST(16000, 16500, 500, true) yields 0.158655, or 15.8655 % Or use the graph

What is another way of saying What percentage of values are less than or equal to some value X? (see next slide)

• PercentilesThe nth percentile of a data set is the smallest value in the set with the property that n% of the data values are less than or equal to it.In a normal distribution, a z score of 0 is the mean. At the mean, 50% (or 0.50) of all the values are less than or equal to the mean. The mean is the 50th percentile.

• ExampleCholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.In what percentile is a man with a cholesterol level of 190?

Using Excels normdist function:=normdist(190,178,41,true) returns 0.61, or 61st percentile

• Standard ScoresThe standard score is the number of standard deviations a value lies above or below the mean. aka: Standard score, z-score, z

The standard score of the mean is z=0Recall that mean is a better word for average

Example: The standard score of a data value 1.5 standard deviations above the mean is z=1.5Example: What is the standard score for a student who scores 300 on an exam with a mean of 400, standard deviation of 100?This student scored exactly 1 SD below the mean, so: z = -1

• Standard ScoresThe standard score of a data value 2.4 standard deviations below the mean is z = -2.4

In general:z = (data value mean) / standard deviationthe data value is typically called x

• ExampleThe Stanford-Binet IQ test is designed so that scores are normally distributed with a mean of 100 and a standard deviation of 16. What are the z-scores for IQ scores of 95 and 125?z = (95 - 100) / 16 = -0.31z = (125 - 100) / 16 = 1.56 Thus, an IQ score of 125 lies 1.56 standard deviations above the mean.

• Inverse Normal Distribution FunctionWhat if you know the mean, standard deviation, and percentile, and want to know the actual value (X)?Recall: z = (x mean) / standard deviationYou can also use Excels NORMINVKnow how to use BOTH. On an exam, youll use the formula.

Example: If a set of exam scores has a mean of 76, a standard deviation of 12, and one score is at the 86th percentile, what was the students exact numeric score? Answer: x = 88.9

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