LSP 121 Intro to LSP 121 Normal Distributions

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’

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

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

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%

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%

** 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, true) yields 0. 158655 =NORMDIST(17000, 16500, true) yields 0. 841345 Subtract: 0. 841345 – 0. 158655 yields 0. 682689 Or use the graph two slides back

Another Example How many paid less than $16, 000? =NORMDIST(16000, 16500, 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)

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 50 th percentile.

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 61 st percentile

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

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’

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.

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 86 th percentile, what was the student’s exact numeric score? Answer: x = 88. 9