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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
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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?
** 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.
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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
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)
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.
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
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 86th percentile, what
was the student’s exact numeric score?
 Answer: x = 88.9