In this chapter, we will look at using the standard deviation as a measuring stick and some properties of data sets that are normally distributed.

Chapter 5Standard Deviation as a Ruler and Normal Models

z-ScoresDefinition & Formula

The number of standard deviations a particular observation in a data set is away from the mean of that set is called it z-score, and is denoted zx.

It is calculated as follows:

z-ScoresConceptual Ideas

If zx > 0, then the observation is above average

If zx < 0, then the observation is below average

z-scores have no units; when observations are changed into their z-scores it is called standardizing the data

Once standardized, observations from different data sets having different centers and spreads can be compared.

Example 1

In the year 2009, cars sold in the United States had an average of 135 horsepower with a standard deviation of 40 horsepower.

(a) Find the z-score for a car with 120 horsepower.

(b) Find the z-score for a car with 200 horsepower.

(c) A certain car has z-score of 1.2. How much horsepower does it have?

(d) A certain car has z-score of 0.6. How much horsepower does it have?

Example 2

Adult Dalmatians have an average weight of 52 pounds with a standard deviation of 6 pounds. Adult German Shepherds have an average weight of 77 pounds with a standard deviation of 3.6 pounds. Which is more unusual, an adult Dalmatian weighing 63 pounds or an adult German Shepherd weighing only 68 pounds?

Normal ModelsDefinition and Ideas

If the histogram of a data set is unimodal and fairly symmetric, then its distribution is called normal.

If such a distribution has mean m and standard deviation , it can be modeled with a normal model, the notation of which is , and the shape is below:

+ + 2 + 3 - - 2 - 3

Normal ModelsDefinition and Ideas

The standard normal curve is .

This is commonly called the z–curve.

It has shape below:

0 1 2 3-1-2-3

Normal Models68 – 95 – 99.7 Rule

For a normal distribution:

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

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

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

Example 3

IQ scores are modeled by .

(a) Draw the 68-95-99.7 diagram for IQ scores. This should be used for the remaining parts of this problem.

(b) About what percent of people have IQ scores 68 and 84?

(c) About what percent of people have IQ scores above 116?

(d) About what percent of people have IQ scores below 68?

(e) What is the 84th percentile of IQ scores? That is, about what score can we say about 84% of people get that score or lower?

Normal ModelsMore General Use

For a normal distribution, the 68-95-99.7 diagram is somewhat limited in terms of what questions can be answered.

For example, in the previous example, if we wanted to know what percentage of people have IQ scores between 90 and 120, the 68-95-99.7 diagram would not be very helpful.

For such questions we can use z-scores and tables published in statistics books or we can use technology.

We will focus on technology.

Normal ModelsTechnology

gives the proportion of data that falls between the values a and b from a normal model with mean and standard deviation .

• if there is no lower bound, the a can be replaced with -∞

• if there is no upper bound, the b can be replaces with ∞

To find this command press on the TI-83/84.

Example 4

Birth weights of babies are well modeled by

. (a) About what percent of babies are born between 3200g and

3600g?

(b) About what percent of babies are born weighing more than 4000g?

(c) About what percent of babies are born weighing less than 2000g?

Normal ModelsMore Technology

We can also find percentiles using the TI 83/84.

gives the value on the normal curve such that

(100a)% of the data is less than or equal to this value.

This command is found by pressing

Example 5

Birth weights of babies are well modeled by

.

(a) Find the 15th percentile of birth weights.

(b) Find the lowest 5% of all birth weights.

(c) Find the highest 2% of all birth weights.