The normal distributionand standard score

Standardizing a score

Let’s say the SAT has a mean of 500 and a standard deviation of 100 while the ACT has a mean of 24 and a standard deviation of 3.

John Marshall scores a 660 on the SAT and Susan Marshall scores 28 on the ACT.

Who had the better score? It’s hard to tell unless we standardize the

scores…

We call a standard score a Z-score

Here’s how we do it:

x is our observed score (your book uses letter y, but it doesn’t matter)

the Greek letter “mu” is the mean and small-case “sigma” is the population

standard deviation. What are John’s and Susan’s standard

scores?

xz

When Is a z-score BIG?

A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean.

A data value that sits right at the mean, has a z-score equal to 0.

A z-score of 1 means the data value is 1 standard deviation above the mean.

A z-score of –1 means the data value is 1 standard deviation below the mean.

When Is a z-score BIG?

How far from 0 does a z-score have to be to be interesting or unusual?

There is no universal standard, but the larger a z-score is (negative or positive), the more unusual it is.

Often (but not always!), we consider a z-score greater than 2 or less than -2 to be roughly an indication of unusualness. But every data set is different, so be careful!

Being “normal”…

The Normal distribution function is central to the study of Statistics.

Many natural phenomena produce data that fall into a roughly Normal shape.

We even have a mathematical function that models this shape…

This is the standard Normal curve

21

21

2

xy e

The Empirical Rule

The values in the Normal distribution approximately follow this pattern: 68-95-99.7

68% of the values fall within one standard deviation of the mean.

95% of the values fall within two standard deviations of the mean.

99.7% of the values fall within three standard deviations of the mean.

Scoring the decathlon

Three competitors have completed three events. Here are their marks with overall means and st. devs.

Who gets the gold medal?

Whose individual performance was the most extraordinary?

Athlete 100 dash

Shot Put

Long Jump

A 10.1 66’ 26’

B 9.9 60’ 27’

C 10.3 63’ 27’3’’

Mean 10 60 26

St.dev.

0.2 3 6

Sketch a Normal model

The standard Normal distribution N(0, 1) Birthweights of babies are N(7.6, 1.3) ACT scores at College of Bob are N(21.2,

4.4)

The Normal model applied (1)

Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg.

What percent of all cars get less than 12 mpg?

What percent of all cars get between 18 and 30 mpg?

Calculator skills

Look only at 2: and 3: for now.

normalcdf( takes up to four inputs:

lower bound, upper bound, mean, st. dev.

Calculator skills

Look only at 2: and 3: for now.

invNorm( takes only one.

Proportion, mean, st. dev.

Normal models with the calculator

What percent of a standard normal model is found in each area?

a)

b)

c)

In the standard Normal model, what value of z cuts off the region described?

d) The lowest 12%

e) The highest 30%

f) The middle 50%

2.05z

0.33z

0.5z

The Normal model applied (2)

Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg.

What percent of all cars get less than 15 mpg?

What percent of all cars get between 20 and 32?

The Normal model applied (3)

Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg.

Describe the fuel efficiency of the lowest 20% of cars.

What gas mileage represents the third quartile?

The Normal model applied (4)

An environmental group is lobbying for a national goal of no more than 10% of cars having mileage under 20 mpg. If the st. dev. does not change, what should the mean rise to?

Auto makers say that they can only raise the mean to 26 mpg. What standard deviation would allow them to still make the goal of only 10% under 20 mpg?

The normal model applied (5)

At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes.

What percent of employees are late on a typical work day?

The normal model applied (6)

At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes.

A typical worker needs 5 minutes to adjust to their surroundings before beginning duties. What percent of this company’s employees arrive early enough to make this adjustment?

The normal model applied (7)

At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes.

If the mean arrival time does not change, what standard deviation would we need to make sure that virtually all employees are on time to work?