# The Standard Deviation as a Ruler + The Normal Model

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The Standard Deviation as a Ruler + The Normal Model
(Chapter 6) The Standard Deviation as a Ruler + The Normal Model

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 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 Results
Z-Scores (Standardized Values): No units because we’re measuring distance from the mean in standard deviations

What would these z-scores mean?
2 -1.6 -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. Which one will be dropped Does this seem fair?

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 Data When 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-Scores What 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?

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 is interesting?
3 (+ or -) or more is rare 6,7 call for attention

Homework: page 123 1 – 11 (odd)

Normal Models Appropriate 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

If we standardize with a Normal model…
Standard Normal model/standard Normal distribution

Normality Assumption When we apply the Normal model, we assume a distribution is normal There is no way to check And most likely, it’s not true Nearly Normal Condition: the shape of the data’s distribution is unimodal an dsymmetric

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. How often will you arrive at school in less than 22 minutes? How often will it take you more than 24 minutes? Do you think the distribution of your driving times is unimodal and symmetric? What does this say about the accuracy of your predictions? Explain.

Normal Models: Make a picture!

Homework: Page 124 15-27 (odd)

The SAT has 3 parts: Writing, Math, and Critical Reading (verbal)
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. Suppose you score 600 on one part; where do you stand among all students? What if you scored 200? 800?

What about this data? The 2007 freshman class at Uconn had an average score of 1192 The 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 1,2,3 etc. standard deviations away
What if we’re not exactly 1,2,3 etc. standard deviations away? How can we find our percentile? Find the z-score Use Table Z to find the percentage of individuals in a standard Normal distribution falling below that score These are called Normal Percentiles

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 scores fall between 450 and 600?

What is the z-score cut point for the 25th percentile?
Make a picture Look in the table With your calculator—invnorm What z-score cuts off the highest 10% of the data?

Suppose a college only admits those with verbal SAT scores in the top 10 percent. What do you need?

Normal Probability Plot
If 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! 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!)

Page 126 29 – 37 (odd) 41,43,45,47 Homework:

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