Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 6 The Standard Deviation as a Ruler and the Normal Model

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 3 The Standard Deviation as a Ruler The trick in comparing very different-looking values is to use standard deviations as our rulers. The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group. As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 The Standard Deviation as a Ruler Consider Math 1107/01, with a test 1 average of 10 and standard deviation of 2. Also, consider Math 1107/02 with a test 1 average of 150 and standard deviation of 15. Which score is better, a 14 from Math 1107/01 or a 160 from Math 1107/02?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Standardizing with z-scores We compare individual data values to their mean, relative to their standard deviation using the following formula: We call the resulting values standardized values, denoted as z. They can also be called z-scores.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Standardizing with z-scores (cont.) Standardized values have no units. z-scores measure the distance of each data value from the mean in standard deviations. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. What is the original test score for a z-score of -1.5 in Math 1107/01?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 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. The larger a z-score is (negative or positive), the more unusual it is.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 When Is a z-score Big? An observation that is unusually large or small, relative to the other values in a data set, is called an outlier. An observation is considered to be unusually large or small if the absolute value of its z-score is at least 2. Is a test score of 135 unusual in Math 1107/02? Is a test score of 15 unusual in Math 1107/01?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 When Is a z-score Big? (cont.) There is no universal standard for z-scores, but there is a model that shows up over and over in Statistics. This model is called the Normal model (You may have heard of “bell-shaped curves.”). Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric. These distributions provide a measure of how extreme a z-score is.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide When Is a z-score Big? (cont.) There is a Normal model for every possible combination of mean and standard deviation. We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ. We use Greek letters because this mean and standard deviation do not come from data—they are numbers (called parameters) that specify the model.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Rule (cont.) The following shows what the Rule tells us:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Rule (cont.) Heights of men follow a normal distribution with an average of 69" and a standard deviation of 2.8". This indicates that 68% of men are from to tall. What percentage of men are between 60.6" and 77.4" tall? What height is considered unusually tall for a man? What height is considered unusually short for a man?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Rule (cont.) Consider a class whose test results have a distribution of N( 75,6). What grades comprise the central 68% of the students? What is an unusually low grade on this test? What percent of grades are below 57? If 500 students took this test, how many students earned a grade less than 57?