Normal Family of Distributions
This is a crucial family of distributions which is used extensively in Statistics … and in
our course. Be wise; learn it well!
FYI: I’ll write the formula for the pdf’s on the board. You need NOT memorize or use it.
Features of normal distributions: [These you SHOULD know.]

It’s a FAMILY whose members differ in location and/or spread.

Each is bell-shaped.

Mean & median: Where are they located?

Standard deviation: = horizontal distance from peak to “inflection pt”

The pdf never equals 0 but its “virtual range” is 6 (3 each side of the peak).
GROUPS:

On a single graph, sketch 3 overlaid pdf’s for the normal distributions with
1. = 0 &  =1 (called the standard normal Z-distribution)
2. = 2 &  =1
3. = 1 &  =1

On a single graph, sketch 3 overlaid pdf’s for the normal distributions with
4. = 0 &  =1 (called the standard normal Z-distribution)
5. = 0 &  =2
6. = 0 &  = 1/2
Standardized Z-scores:

It turns out that every normal distribution calculation can be related (by shifting
and stretching/squishing) to the standard normal z-distribution which has = 0 &
 =1.

The formula to convert from “raw scores” x to/from standardized z-scores is just
like what you saw in chapter 2:
o z = (x  ) / 
Computation of probabilities (i.e. areas!) for normal distributions:
1. Always DTP (Draw The Picture) first!
a) Draw your normal distribution, centered at your mean.
b) Determine the interval(s) of interest.
c) Shade the area above it and below your normal distribution.
d) Be alert to ways you might use the symmetry of normal distributions to
help you save time/effort here.
e) Calculate the z-scores for the endpoint(s) of your interval(s).
2. Area calculation: You have several options. Be able to do these:
a) Calculate by hand using text Table A (back flyleaf of text). [I’ll give you a
copy of Table A to use in subsequent exams.]

The row headings are the ones & tenths places of your z.

The column headings are the hundredth place of your z.

The table entries equal the cumulative probability to the left of
your z-value.
b) Use a graphing-type calculator. NORMCDF(a,b) will give you the area
under the standard normal Z-distribution between z=a and z=b.
Examples to practice:
Text Example 8.21: college women’s heights X (in.) with = 65 &  =2.7.
 P(X<62.3)
 P(X<59.6)
 P(X<56.9)
 P(X>67.7)
 P(X>70.4)
 P(X>73.1)
 P(62.3 < X < 67.7) …recognize that answer?!
 P(59.6 < X < 70.4) …recognize that answer?!
 P(56.9 < X < 73.1) …recognize that answer?!
Calculation of Percentiles (Jeopardy for Normal Distributions)
 Answer: .05 is the area to the left of that z-value.
 Question: What is z= _____?
The method to calculate percentiles is:
1. Always DTP (Draw The Picture) first!
a) Draw your normal distribution, centered at your mean.
b) Determine the area of interest and shade it.
c) Determine the area below your endpoint(s).
d) Calculate the z-scores for the endpoint(s) of your interval(s).
 Table A: Look for desired value IN the table and then read off the row
& column headings to get the ones/tenths & hundredth places for your
z-score.
 Graphing calculator: INVNORM(area) will give you the z-score
which has the specified area to the left of it.
2. Percentile calculation:
a. Solve z = (x  ) /  for x in terms of z, , &  (all now known).
b. Answer: percentile is x=  + z.
deviations above/below the mean]
[correct number z of standard
Examples to practice:
Text Example 8.21: college women’s heights X (in.) with = 65 &  =2.7.





.5’th percentile
first percentile
2.5’th percentile
5’th percentile
10’th percentile





90’th percentile
95’th percentile
97.5’th percentile
99’th percentile
99.5’th percentile
 What interval, centered at the mean, has probability = .90?
 What interval, centered at the mean, has probability = .95?
 What interval, centered at the mean, has probability = .99?
NORMAL APPROXIMATION TO BINOMIAL DISTRIBUTIONS
We’ll look just briefly at this topic, mainly to set up chapter 9 and some methods of
inference that follow.
the approximation:
 If


X is a Binomial Random Variable with n trials and success probability p
n is “large” [Use the following rules of thumb.]
o (mean # successes) np  10
o (mean # failures) n(1-p)  10
 then
 the distribution of X is approximately normal (bell-shaped)

the probability P(Xk)  P(Zz*) where
o
z* 
k


k  np
np(1  p)
is the standardized score for k
note: We’ll bypass the continuity correction on page 296. Its use makes the normal
approximation more accurate but it’s not needed for the use we’ll make of the normal
approximation.
EXAMPLE:
multiple Choice Exam with 100 questions and 5 choices for each
Let X=# correct responses.
If a student guesses at random,
 what distribution does X have?
o Name the family:
o Specify the member of the family:
 Calculate the expected of X.
 Calculate the standard deviation of X
 Calculate the EXACT probability that X25.
 Calculate the APPROXIMATE probability that X25.
 If a student got 25 questions correct, would that be strong enough evidence to
convince you that he/she was NOT guessing at random?