The Practice of Statistics Chapter 2: Normal Curves Section 2: Density Curves & Z-Scores Copyright © 2008 by W. H. Freeman & Company

Recall from last chapter… Data Analysis Toolbox To answer a statistical question of interest involving one or more data sets, proceed as follows. DATA Organize and examine the data. Answer the key questions. GRAPHS Construct appropriate graphical displays. NUMERICAL SUMMARIES Calculate relevant summary statistics We add one more step we can often use to help draw conclusions If possible, describe the pattern of the data using a smooth curve.

Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. Doing so can help us describe the location of individual observations within a distribution. The curve is a mathematical model– an idealized description (not exact).

Density curves An idealized model The blue area in the histogram represents 0.303 of the scores– the 30th percentile. The blue area in the density curve has an area of 0.293– the 29th percentile. Only 0.010 away from the histogram results, the density curve gives a good approximation of areas given by the histogram.

Measures of relative standing Two main measures Z-Scores (use with mean & standard deviation) Percentiles (use with median & five-number summary)

Measures of relative standing and density curves Was your last test score a good score? Lets pretend you scored an 86. What other factors besides the raw score might determine whether you felt good about the score?

Test scores Let’s assume here’s a list of all the class test scores. How did your 86 fall relative to the center of the distribution?

A z-score tells us how many standard deviations away from the mean the original observation falls, and in which direction.

Do you think observations larger than the mean are positive or negative when standardized? Observations smaller than the mean?

Comparing Tests The next day after getting your statistics test score, you get a 82 on your chemistry test. Are you disappointed? For the chemistry test, the mean was 76 and standard deviation was 4. Compare the z-scores for the two tests. Which did you do better on relative to the class?

Percentiles The percentile of a distribution is the value with p percent of observations less than or equal to it.

Test scores revisited What percentile is your 86? You got the 22nd highest score out of 25 total tests, so you scored at the 88th percentile. Your friend got a 73. What percentile did they score in?

Percentiles Note that some people define the pth percentile of a distribution as the value with p percent of observations below it. (as opposed to at or below it). Calculating percentiles is not an exact science. Be sure and include your work and explanations.

Relating z-scores to percentiles What percent of observations fall within 2 standard deviations of the mean? 5 standard deviations of the mean? Chebyshev’s inequality relates percents and standard deviations but does not give exact values!