The Standard The Standard Deviation as a Ruler Deviation as a Ruler

+ + The Normal ModelThe Normal Model

(Chapter 6)

Reading Quiz (10 points):Reading Quiz (10 points):

When we rescale data, how are measures of center and spread affected?

Why do we use z-scores?What does a z-score measure?To use a Normal model what shape

must our data be?If a distribution is roughly Normal, a

Normal probability plot shows what kind of line?

What do we use standard What do we use standard deviations for?deviations for?

To compare different values (like cm and seconds in a heptathlon)

Compares an individual value to the group

How far is a value from the mean?

Standardizing ResultsStandardizing ResultsZ-Scores (Standardized Values):

No units because we’re measuring distance from the mean in standard deviations

s

xxz

What would these z-scores What would these z-scores mean?mean?

2-1.60-3

Which of these values is the most statistically surprising?

Your Statistics teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 80 on test 2. You’re all set to drop the 80 until she announces that she grades “on a curve.” She standardized the scores in order to decide which is the lower one. If the mean on the first test is 88 with a standard deviation of 4 and the mean on the second was a 75 with a standard deviation of 5.

a.Which one will be droppedb.Does this seem fair?

Shifting Data: Remember?Shifting Data: Remember?Adding (or subtracting) a

constant to each value, all measures of position (center, percentiles, min, max) will increase (or decrease) by the same constant, but does not change any measures of spread

Rescaling DataRescaling DataWhen we multiply (or divide) by a

constant, our measures of position get multiplied (or divided) by the same constant, as do our measures of spread

Z-ScoresZ-ScoresWhat are really doing in terms of

shifting and rescaling?

What will the new value of the original mean be?

What happens to the standard deviation when we divide by s?

s

xxz

Standardizing:Standardizing:Does not change the shape of

the distribution of a variable

The center (mean) becomes:_____

The spread (standard deviation) becomes:______

How do we know if a z-score How do we know if a z-score is interesting?is interesting?

3 (+ or -) or more is rare6,7 call for attention

Homework:Homework:page 123◦1 – 11 (odd)

Normal ModelsNormal ModelsAppropriate for unimodal, roughly

symmetric distributions

Why do we have new notation for mean, standard deviation?◦These are the parameters for our model

rather than numerical summaries of the data

),( N

If we standardize with a If we standardize with a Normal model…Normal model…

Standard Normal model/standard Normal distribution

Normality AssumptionNormality AssumptionWhen we apply the Normal

model, we assume a distribution is normal

There is no way to checkAnd most likely, it’s not true

Nearly Normal Condition: the shape of the data’s distribution is unimodal an dsymmetric

68-95-99.7 Rule68-95-99.7 Rule

Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Suppose a Normal model is appropriate for the distributions of driving times.

a.How often will you arrive at school in less than 22 minutes?

b.How often will it take you more than 24 minutes?

c.Do you think the distribution of your driving times is unimodal and symmetric?

d.What does this say about the accuracy of your predictions? Explain.

Normal Models:Normal Models:Make a picture!

Homework:Homework:

Page 124◦15-27 (odd)

The SAT has 3 parts: Writing, Math, and Critical Reading (verbal). Each part has a distribution that is roughly unimodal and symmetric and designed to have an overall mean of about 500 and a standard deviation of 100 for all test takers. In any one year the mean and standard deviation may differ from the target by a small amount, but they’re a good overall approximation.

a.Suppose you score 600 on one part; where do you stand among all students?

b.What if you scored 200? 800?

What about this data?What about this data?The 2007 freshman class at Uconn

had an average score of 1192The 2007 freshman class at Umass

had an average math score of 559 and an average verbal score of 561

The 2008 class at URI has an average SAT score of 1659.

At NYU, to take Calculus you must score at least a 750 on math

What if we’re not exactly What if we’re not exactly 1,2,3 etc. standard 1,2,3 etc. standard deviations away? How can deviations away? How can we find our percentile?we find our percentile?Find the z-scoreUse Table Z to find the

percentage of individuals in a standard Normal distribution falling below that score

These are called Normal Percentiles

With technology…With technology…Go to the distribution menu

◦normalpdf—used for graphing

◦normalcdf—finds the area between two z-score cut points

What proportion of SAT What proportion of SAT scores fall between 450 and scores fall between 450 and

600?600?

What is the z-score cut point What is the z-score cut point for the 25for the 25thth percentile? percentile?

Make a pictureLook in the tableWith your calculator—invnorm

What z-score cuts off the highest 10% of the data?

Suppose a college only admits Suppose a college only admits those with verbal SAT scores those with verbal SAT scores

in the top 10 percent. What do in the top 10 percent. What do you need?you need?

Normal Probability PlotNormal Probability PlotIf the distribution of data is

roughly Normal, this plot is roughly a diagonal straight line

Use this data:◦22,17,18,29,22,23,24,23,17,21◦Statplot—the last one!

What can go wrong!What can go wrong!Only use a Normal model when the

distribution is symmetric and unimodal!

Don’t use the mean and standard deviation when outliers are present!

Don’t round too soon—be as precise as possible

Don’t round any results in the middle of a calculation

Don’t worry about minor differences in results (just like with quartiles and median!)

HomeworHomework: k:

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