AP Statistics
2.2 Normal Distributions
Objectives: * Identify the main properties of the Normal Curve as a particular density curve.
*List three reasons why normal distributions are important in statistics.
*Explain the 68-95-99.7 rules (the Empirical rule)
*Explain the notation N(,)
*Define the Standard Normal Distribution
*Use a table of values for the standard Normal curve (Table A on your Formula Sheet)
to compute the proportion of observations that are (a) less than a given z-score, (b)
greater than a given z-score, or (c) between two given z-scores.
*Use a table of values for the standard Normal curve to find the proportion of
observations in any region given any Normal distribution (i.e. Given raw data than zscores)
*Use a table of values for the standard Normal curve to find a value with a given
proportion of observations above or below it (inverse Normal)
Normal Distributions
 A single-peaked, symmetric, “bell-shaped” density curve.
 Changing  (mean) without changing  moves the ENTIRE curve
along the horizontal axis without changing its SPREAD.
  and  completely determine the shape of the curve.
  can be estimated by the inflection points (changes in the
steepness of the curve – done by eye). pg. 134
Three Reasons for using a Normal curve:
1. Good description for real data.
2. Good approximation to the results of “chance outcome”.
3. Statistical Inference procedures are based on Normal distributions
Notation for a Normal Distribution: N(,)
Note: Normal is a SPECIAL name for a distribution (that is why we capitalize the word),
 Symmetric does not always imply Normal
The 68-95-99.7 Rule
In the Normal Distribution with mean  and the standard deviation 

Approximately 68% of the observations fall within ± 1 of 

Approximately 95% of the observations fall within ±2 of 

Approximately 99.7% of the observations fall within ±3 of 
68-95-97.5 rule for Normal distributions
Two Normal curves with the same mean, but
different standard deviations.
Standard Normal Distribution
 The standard Normal distribution, N(0,1) with mean 0 and standard deviation 1.
 If a variable x has any Normal Distribution, N(,), then the standardized variable:
has the standard Normal distribution.
The Standard Normal Table
 To find the proportion of observations that lie in a given interval we must perform
Normal calculations.
1. Table A: (inside front cover of your textbook or in the formulas I gave you)
 Find the value of z
 Sketch the Normal distribution and shade the area you are looking to find the
proportion of (HELPS A LOT!)
 Look up in the table:
o The table will tell you the proportion of the observations to the left of your z
value.
o Subtract from 1 to find the proportion to the right of your z value
o To find the proportion of a given interval, you need to look up the z value for
both the upper and lower bounds of your interval and subtract (UPPER z
table value – LOWER z table value)
CAUTION and AP Exam Notes:
!! A common mistake is to look up a z-value in Table A and report the entry. ALWAYS make sure you
now if you are looking for the area to the LEFT or to the RIGHT. A good way to ensure this is to
DRAW A PICTURE!!
Exercise 2.25 (pg. 137)
Exercise 2.29 (pg. 142)
HW: pg. 137; 2.32, 2.33, 2.35