Packet 1: Normal Distributions

After completing this material, you should be able to:

• describe what a normal distribution is.

• use the Empirical rule to describe normal distributions.

• calculate a z-score and interpret its meaning.

• find probabilities ( or percentages) of normal curves above a given observation, below a given observation, and

between two observations.

• determine the value of a normal distribution corresponding to any given percentage (or probability).

• determine when observations are unusual based upon probabilities and explain the reasoning.

What exactly is a normal distribution? In the space below, briefly define a normal distribution and draw several normal

curves.

A normal distribution is controlled by two features:

1.

2.

The ___________________ of the distribution controls the

center of the distribution (see Figure 1 below).

The ______________________________________ of the

distribution controls how spread out the curve is, which in

turn controls its scale (see the Figure 2 below).

Notation Alert

(You must remember this notation!!)

Figure 1.

Blue Curve:

Red Curve:

Figure 2.

Blue Curve:

Red Curve:

Figure 3.

Blue Curve:

Red Curve:

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

Afunctionthatrepresents the

distribution of many random variables

asymmetrical bellshaped graph

Mean average

standard devationlstd dev.lme.am µ

mean standard

standard elevation derationt

ame Std same mean

Fsrethde different daimfnfeqrnent offngenter

same spreadgeo spread

ifferent diffetneffted Std is differen

cMeneimered theme

ifferent

oeecenter

difrerf.BR

Packet 1: Normal Distributions page 2

STA 205 Notes Buckley Spring 2017

Example: Last year, the opening price for one share of Google stock has been approximately normally distributed with a

mean price of $665.45 and a standard deviation of $36.31.

Draw and label the normal distribution corresponding to opening prices. Use the distribution to answer the following:

Example: Delta reports that the average age of their pilots is 42.6 years with a standard deviation of 6.3 years. Assume

that the distribution of ages is normally distributed.

Draw and label the normal model which describes the ages of Delta pilots. What percentage of Delta pilots is older than

50? Why can we not use the Empirical Rule to arrive at an exact answer to this question? What is the best we can say?

To help understand how normal distributions work, we can use the

Empirical Rule

(which is also referred to as the

68-95-99.7 Rule – I’ll let you figure out where this name comes from!). This rule states the following:

• ________ % of the values in a normal distribution fall within __________ standard deviation of the mean.

• ________ % of the values in a normal distribution fall within __________ standard deviations of the mean.

• ________ % of the values in a normal distribution fall within __________ standard deviations of the mean.

Between what two values do we expect the opening price of one share of

Google stock to fall 95% of the time?

What percentage of the time was the opening price more than $701.76?

What percentage of the time is an opening price between $629.14 and

$738.07?

68 I

95 2

997 3

Hsamedistribution oneachside

belt curve

xzstd.de

Price falls between from the

938.07559283 mean

34 341

ias 13.512.351.15 16 coffitmhee

is s

rs

so.si 545E3osi 81.51 falls between

1341

titsi

Iiii

Packet 1: Normal Distributions page 3

STA 205 Notes Buckley Spring 2017

Finding Normal Probabilities (also known as “working forwards”)

In the Delta pilot example, we ran into a problem – the age we were interested in wasn’t exactly 1, 2, or 3 standard

deviations from the mean. This meant the Empirical Rule could no longer be used to find the percentage of the curve

below the observation. We need a way to find probabilities (or percentages) associated with any observation.

In order to find probabilities, we first need a way to determine the exact number of standard deviations above or below

the mean an observation falls. This process is known as standardizing the observation.

Let’s return to the pilot example we looked at last class meeting. The age of Delta pilots is normally distributed with a

mean of 42.6 years and a standard deviation of 6.3 years. We were interested in an age of 50 – calculate and interpret

the z-score associated with this quantity.

How will standardizing the observation by calculating its z-score help us find the percentage of pilots older than 50? To

answer that question, let’s fill in the normal model below:

Formula Alert!!

This formula will be given on the

formula sheet.

What information would we like to know?

Using the normal table, what percentage of the curve is below

an age of 50?

M42.6 F6.3

t

DzScore for an

observation OR

i

Standardize 2N age of 50

50 42.6

26.3 1.17 an average age of 50 is

1it abovethe Mean

What 1ofthecurve is above 5012

gragabfescore ZyEfolt

Ztabledoes Not exactly fallon 50

0.8790

901 loot 87.901 12 101

fqgynmZ sfzre.SIyoi23c

oaoBer o

o