Download The Normal Distribution (Gaussian Distribution)

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Central limit theorem wikipedia, lookup

Transcript
The Normal Distribution
(Gaussian Distribution)
Honors Analysis
Learning Target: I can analyze data using the normal distribution.
German mathematician
Influenced statistics, algebra,
number theory, geometry,
physics.
 Child prodigy!
 Constructed heptadecagon
 Triangular numbers
 Proved Fundamental
Theorem of Algebra
 Influenced development
of statistics, including
Normal Distribution
(Gaussian Distribution)


Carl Friedrich Gauss (1777-1855)
Imagine you took a test in two different
classes.

In the first class, you made a 93%. The
class mean was a 96%, and the standard
deviation was 3%.

In the second class, you made a 78%.
The class mean was a 74%, and the
standard deviation was 2%.
Which test performance was better?
𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝜇 = 𝑚𝑒𝑎𝑛
Normal Distribution
(Gaussian Distribution)
(Approximately)
68% within 1 std
dev. of mean
 95% within 2 std.
deviations of
mean
 99.7% fall within
3 standard
deviations of
mean

68-95-99.7 Rule
Calculate the mean (central value on
curve)
 Each region increases or decreases by one
standard deviation from the mean
 Ex: Test score mean: 74% Std. dev: 2%

Labeling a Simple Normal Curve

So what happens if you want to calculate
a percentage for a value that ISN’T on
your normal curve?

Ex: PSAT math test with mean of 48 and a
std. deviation of 3. What percent of
scores are below 50?
Normal distribution with a mean of 0 and
a standard deviation of 1.
 Total area under curve = 1
 Area to left of a given value on the curve
gives the percentile rank – percent of
scores LOWER than a given score.

Standard Normal Distribution

You can convert values to standard
normal distribution form by calculating a
z-score:
𝑋−𝜇
𝑍=
𝜎

Z-Score percentages can be looked up in
a table or on a calculator.
Z-Scores
Example
Solution
Example Part II
Unrepresentative Sample
 Undercoverage (Convenience sample,
voluntary sample)
 Non-response Bias
 Voluntary response Bias
Measurement Error
 Response Bias (Leading questions, social
desirability)
Types of Bias