Griffiths D. Head First Statistics

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using the normal distribution

As with any other continuous probability distribution, you find

probabilities by calculating the area under the curve of the

distribution. The curve gives the probability density, and the

probability is given by the area between particular ranges. If, for

instance, you wanted to find the probability that a variable X lies

between a and b, you’d need to find the area under the curve between

points a and b.

If the probability density

decreases the further you get

from μ, when does it reach 0?

No matter how far you go out on the graph, the

probability density never equals 0.

The probability density gets closer and closer to 0, but never quite

reaches it. If you looked at the probability density curve a very long

way from μ, you’d find that the curve just skims above 0.

Another way of looking at this is that events become more and more

unlikely to occur, but there’s always a tiny chance they might.

Sound complicated? Don’t worry, it’s easier than you might think.

Working out the area under the normal curve would be difficult if you had to

do it all by yourself, but fortunately you have a helping hand in the form of

probability tables. All you need to do is work out the range of the area you

want to find, and then look up the corresponding probability in the table.

So how do we find normal probabilities?

a b

The shaded area gives the

probability between that X is

between a and b.

342 Chapter 8

Grab your distribution and range

Standardize it

Look up the probabilities

There are a few steps you need to take in order to find normal

probabilities. We’ll guide you through the process, but for now here’s

a roadmap of where we’re headed.

If the normal distribution applies to

your situation, see if you can find what

the mean and standard deviation are.

You’ll need these before you can find

your probabilities. You also need to

figure out what area you need to find.

Don’t worry about this for now;

we’ll show you how to do this

really soon.

Once you’ve transformed your

normal curve, you can look

up probabilities using handy

probability tables. Job done!

Three steps to calculating normal probabilities

finding normal probabilities

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using the normal distribution

Step 1: Determine your distribution

The first thing we need to do is determine the distribution of the data.

Julie has been given the mean and standard deviation of the heights of eligible

men in Statsville. The mean is 71 inches, and the variance is 20.25 inches. This

means that if X represents the heights of the men, X ~ N(71, 20.25).

We also need to know which range of values will give us the right probability

area. In this case, we need to find the probability that Julie’s blind date will be

sufficiently tall.

That’s easy. Julie wants her date to be taller

than her, so we can work out probabilities based

on her height.

Julie is 64 inches tall, so we’ll find the probability that her date is taller.

Here’s a sketch:

The probability of Julie’s

date being over 64 inches

tall is given by this area

here: P(X > 64).

= 20.25

X ~ N(71, 20.25)

μ = 71

This is shorthand for “The variable

X follows a normal distribution, and

has a mean of 71 and a variance of

20.25.”

This is the graph of

X ~ N(71, 20.25).

Here’s where X = 64 inches.

344 Chapter 8

Step 2: Standardize to N(0, 1)

The next step is to standardize our variable X so that the mean becomes 0

and the standard deviation 1. This gives us a standardized normal variable

Z where Z ~ N(0, 1).

Is that your idea of fun?

Why would I want to do that?

Probability tables only give probabilities for N(0, 1).

Probability tables focus on giving the probabilities for N(0, 1) distributions, as

it would be impossible to produce probability tables for every single normal

distribution curve. There are an infinite number of possible values for μ and

, and as the normal curve uses these as parameters to indicate the center

and spread of the curve, there are also an infinite number of possible normal

distribution curves.

There’s an infinite number of possible

values for μ and σ

, so there are an

infinite number of possible curves for

the normal distribution

Being able to use a standard normal distribution means that we can use the

same set of probability tables for all possible values of μ and σ

. There’s just

one question—how do we convert out normal distribution into a standard

form?

How do you think we might be able to standardize our normal distribution?

0.2

0.4

0.6

0.8

1.0

-3 -2 -1

1 2

µ = 0; σ

= 0.2

µ = 0, σ

= 5.0

µ = -2, σ

= 0.5

standardizing normal variables

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using the normal distribution

To standardize, first move the mean…

Let’s start off by transforming our normal distribution so that the mean

becomes 0 rather than 71. To do this, we move the curve to the left by 71.

Move to the left by 71

This gives us a new distribution of

X - 71 ~ N(0, 20.25)

…then squash the width

We also need to adjust the variance. To do this, we “squash” our distribution

by dividing by the standard deviation. We know the variance is 20.25, so the

standard deviation is 4.5.

Recall that the standard

deviation is the square root of

the variance.

“Squash” the distribution

by dividing by the

standard deviation.

Doing this gives us X - 71 ~ N(0, 1)

4.5

or Z ~ N(0, 1) where

Z = X - 71

4.5

Z = X - μ

Look familiar? This is the standard score we encountered when

we first looked at the standard deviation in Chapter 3. In general,

you can find the standard score for any normal variable X using

μ = 0

μ = 71

μ = 0

σ = 1

The mean of X

X is the variable

we’re trying to find

probabilities for.

The standard deviation

of X

346 Chapter 8

Standard Score

To find the standard score

of a value, use

Z = X - μ

Vital StatisticsVital Statistics

So far we’ve looked at how our probability distribution can be standardized to

get from X ~ N(μ, σ

) to Z ~ N(0, 1). What we’re most interested in is actual

probabilities. What we need to do is take the range of values we want to find

probabilities for, and find the standard score of the limit of this range. Then

we can look up the probability for our standard score using normal probability

tables.

In our situation, we want to find the probability that Julie’s date is taller than

her. Since Julie is 64 inches tall, we need to find P(X > 64). The limit of this

range is 64, so if we calculate the standard score z of 64, we’ll be able to use

this to find our probability.

z = x - μ

= 64 - 71

4.5

= -1.56 (to 2 decimal places)

Let’s find the standard score of 64.

These areas are the same, but we

don’t know the value of z yet.

We can find it by calculating the

standard score of 64.

We need to convert this... ...to this

N(71, 20.25)

N(0, 1)

So -1.56 is the standard score of 64, using the mean and standard

deviation of the men’s heights in Statsville.

Now that we have this, we can move onto the final step, using

tables to look up the probability.

μ = 0

Now find Z for the specific value you want to find probability for

finding Z

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using the normal distribution

It’s time to standardize. We’ll give you a distribution and value,

and you have to tell us what the standard score is.

Is this the same standard score that we saw before?

A: Yes it is. It has more uses than just the normal distribution, but

it’s particularly useful here as it allows us to use standard normal

probability tables.

Is the probability for my standardized range really the

same as for my original distribution? How does that work?

A: The probabilities work out the same, but using the probability

tables is a lot more convenient.

When we standardize our original normal distribution, everything

keeps the same proportion. The overall area doesn’t grow or shrink,

and as it’s area that gives the probability, the probability stays the

same too.

1. N(10, 4), value 6 2. N(6.3, 9), value 0.3

3. N(2, 4). If the standard score is 0.5, what’s the value?

4. The standard score of value 20 is 2. If the variance

is 16, what’s the mean?

348 Chapter 8

It’s time to standardize. We’ll give you a distribution and value,

and you have to tell us what the standard score is.

1. N(10, 4), value 6 2. N(6.3, 9), value 0.3

3. N(2, 4). If the standard score is 0.5, what’s the value?

4. The standard score of value 20 is 2. If the

variance is 16, what’s the mean?

z = x - μ

= 6 - 10

= -2

z = x - μ

= 0.3 - 6.3

= -2

z = x - μ

0.5 = x - 2

0.5 x 2 = x - 2

x = 1 + 2

= 3

This is the reverse of previous problems. We’re

given the standard score, and we have to find the

original value. We can do this by substituting in the

values we know, and finding x.

z = x - μ

2 = 20 - μ

2 x 4 = 20 - μ

μ = 20 - 8

= 12

This is a similar problem to question 3. We have to

substitute in the values we know to find μ.

sharpen solution

So we’ve found our

distribution, standardized it,

and found Z. Now can we find

the probability of my blind date

being taller than me?

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using the normal distribution

Now that we have a standard score, we can use probability tables to find

our probability. Standard normal probability tables allow you to look up

any value z, and then read off the corresponding probability P(Z < z).

P(Z < z) is this area here.

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

–3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002

–3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003

–3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005

–3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007

–3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010

–2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014

–2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019

–2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026

–2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036

–2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048

–2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064

Z ~ N(0, 1)

So how do you use probability tables?

Start off by calculating z to 2 decimal places. This is the value that you

will need to look up in the table.

To look up the probability, you need to use the first column and the top

row to find your value of z. The first column gives the value of z to

1 decimal place (without rounding), and the top row gives the second

decimal place. The probability is where the two intersect.

As an example, if you wanted to find P(Z < –3.27), you’d find –3.2

in the first column, .07 in the top row, and read off a probability of

0.0005.

Here’s the

row for

z = -3.2x,

where x is

some number.

Here’s the column for

.07, the second decimal

place for z.

This is where

-3.2 and .07

meet. It gives

the value of

P(Z < z).

Step 3: Look up the probability in your handy table

We’ve put all

the probability

tables you need in

Appendix ii of the

book.

Just flip to pages 658-659 for the normal

distribution tables you need to find

probabilities in this chapter.

350 Chapter 8

Julie’s probability is in the table

Let’s go back to our problem with Julie. We want to find P(Z > -1.56), so let’s

look up -1.56 in the probability table and see what this gives us.

So, looking up the value of –1.56 in the probability table gives us a probability of

0.0594. In other words, P(Z < –1.56) = 0.0594. This means that

P(Z > –1.56) = 1 - P(Z < –1.56)

= 1 – 0.0594

= 0.9406

In other words, the probability that Julie’s date is taller than her is 0.9406.

You can find normal probability

tables in the appendix at the

back of the book.