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Mastering Normal Distributions in Quantitative Reasoning

Understand normal distributions, descriptive statistics, & correlations for college & career success. Learn about probability, risk, databases, & algorithms. Put your knowledge to the test!

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Mastering Normal Distributions in Quantitative Reasoning

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  1. LSP 121 Intro to LSP 121 Normal Distributions

  2. Welcome to LSP 121 • Quantitative Reasoning and Technological Literacy II • Continuation of concepts from LSP 120 • Topics we feel you will need to make it through college and into a career • Normal distributions • Descriptive statistics and correlation • Probability and risk • Databases • Algorithms • If you feel you know this material, take the test • See Syllabus under ‘Prerequisites’

  3. What is a Normal Distribution? • Very common, very special type of distribution • Most data values are clustered near the mean (a single peak) • Distribution is symmetric • Tapering tales as you move away from the mean • Looks like a bell curve

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

  5. 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%

  6. Example • In 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 600 • 95% of the students score between 300 and 700 • 99.7% score between 200 and 800 • What if someone scored 720 on the SAT? What percentage of students scored less than or equal to 720? • Use Excel’s NORMDIST function • In 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?

  7. ** Another Example • A 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 0.841345 • Subtract: 0.841345 – 0.158655 yields 0.682689 • Or use the graph two slides back

  8. Another Example • How 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)

  9. Percentiles • The 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.

  10. Example • Cholesterol 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 Excel’s normdist function: =normdist(190,178,41,true) returns 0.61, or 61st percentile

  11. Standard Scores • The 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=0 • Recall 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=1.5 • Example: 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

  12. Standard Scores • The standard score of a data value 2.4 standard deviations below the mean is z = -2.4 • In general: z = (data value – mean) / standard deviation the data value is typically called ‘x’

  13. Example • The 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 = -0.31 z = (125 - 100) / 16 = 1.56 Thus, an IQ score of 125 lies 1.56 standard deviations above the mean.

  14. Inverse Normal Distribution Function • What if you know the mean, standard deviation, and percentile, and want to know the actual value (“X”)? • Recall: z = (x – mean) / standard deviation • You can also use Excel’s NORMINV • Know how to use BOTH. On an exam, you’ll 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 student’s exact numeric score? • Answer: x = 88.9

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