Transcript The Normal Distribution (Gaussian Distribution)

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