The Normal

Distribution

MDM4U Unit 6 Lesson 2

Normal Distributions

� Many data sets display similar

characteristics

� The normal distribution is a way of

describing a certain kind of "ideal"

data set

� Although no real-world data is perfect,

a surprising amount of natural

phenomena are approximately

"normal"

Properties of the "Bell

Curve"

� Symmetrical

� no skew

� mean, median, mode all equal

� Mound / Bell Shaped

� peaks in the

middle, slopes

down towards

the sides0.02

0.04

0.06

0.08

0.10

Dens

ity

10 15 20 25 30 35 40 45 50 55

sDensity = x 30 sd, ,( )normalDensity

data Histogram

Why is Normal good?

� The Normal Distribution is so well

behaved that we can draw a curve

that almost matches it

� This makes it very easy to measure

how tall the histogram bars are

� The height of the bars are given by

the curve that matches it

� This allows us to find almost exactly

how much data is in each part of the

distribution

Where are we on the curve?

σ+x σ2+xσ2−x σ−x x

the mean

Where are we on the curve?

σ+x σ2+xσ2−x σ−x x

the mean

one standarddeviation above x

one standarddeviation below x

Where are we on the curve?

σ+x σ2+xσ2−x σ−x x

the mean

one standarddeviation above x

one standarddeviation below x

2 standarddeviations above

x

2 standarddeviations below

x

Area under the curve

σ+x σ2+xσ2−x σ−x x

34%34%

13.5% 13.5%2.25% 2.25%

68%

95%

σ3−x σ3+x

More Properties

� Approximately 68% of the data is

within one standard deviation of the

mean

� Approximately 95% of the data is

within two standard deviations of the

mean

� Approximately 99.7% of the data is

within three standard deviations of the

mean

Notation

� If we want to say "this data is

approximated by the standard

distribution"...

� We should also state what the mean

and standard deviation are

)σ,N(~X 2x

)σ,N(~X 2x

our data(call it X)

"is approximated by" the normal distribution

with this mean

and this standarddeviation or variance

Notation Example

� The data is normal, and has a mean

of 3 and a standard deviation of 2

� The data is normal, has a mean of

5.4, and a standard deviation of 3

)2,N(3~X 2

)9,N(5.4~Xbe careful - if there is no square, then the second number is the variance,

and you need to take the square root to get the standard deviation....

Problem Example

Julie is an engineer who is designing roller coasters. Her roller coaster must have mass restrictions that are suitable for 95% of the population.

The average adult in North America has a mass of 71.8kg with a standard deviation of 13.6kg.

What range of mass should her ride accommodate?

Problem Example

Julie is an engineer who is designing roller coasters. Her roller coaster must have mass restrictions that are suitable for 95% of the population.

The average adult in North America has a mass of 71.8kg with a standard deviation of 13.6kg.

What range of mass should her ride accommodate?

1. Assume that the masses are normally distributed.

2. 95% of the data willfall within two standarddeviations

Consequently, the range will be between 71.8 - 2(13.6) = 44.6kg and 71.6 + 2(13.6) = 99 kg

Problem Example

All That Glitters, a sparkly cosmetic powder, is machine-packaged in a process that puts approximately 50 g of powder in each package. The actual masses have a normal distribution with:

)0.6,N(50.5~X 2

The manufacturers want to ensure that each package contains at least 49.5 g of powder. What percent of packages do not contain this much powder?

Problem Example

The manufacturers want to ensure that each package contains at least 49.5 g of powder. What percent of packages do not contain this much powder?

This answer falls between 1 and 2 standard deviations below the mean.

The Standard Normal Curve

� For the standard normal curve

� the mean is equal to 0

� the standard deviation is equal to 1

2/2

π2

1)( xexf

−=

0=x

1=σ

2σ2/)(

2

2

πσ2

1)( xxexf

−−=

sub in

Calculating z-scores

σ

xxz

−=

number of standarddeviations is x away

from the mean

The standard deviation

The meanThe data

The deviation

The z-score is thedeviation divided by the

standard deviation

Calculating z-scores

This means 49.5 g is a mass thatfalls exactly 1.67 standard deviationsbelow the mean.

Standard Normal Curve

Tables (pg. 606 and 607)

This table has negative x values This table has positive x values

Finding P values in the

Standard Normal Table

x = -1.67

P(x ≤ -1.67) = 0.0475

1. Locate the number and its

first decimal place in the left

column

2. Go across to get more precision

3. The number in the cell is the area

up to x

Finding P values in the

Standard Normal Table

x = -1.67

P(x ≤ -1.67) = 0.0475

This means that 4.75% of

the packages of powder

will have a mass of 49.5 g

or below.

Assigned Work

pg. 430

# 1 – 3, 7, 9