CHAPTER 2 DESCRIBING LOCATION IN A DISTRIBUTION Section

CHAPTER 2: DESCRIBING LOCATION IN A DISTRIBUTION Section 1 - Measures of Relative Standing and Density Curves

WHAT IF…… Suppose Jane earned an 82 (out of 100) on her next Stats test. Should she be disappointed or satisfied with her performance? What factors should she consider?

NOW…. . WHAT IF…… What if her classmates made the following…… � 79, 77, 91, 90, 62, 21, 89, 84, 90, 99, 43, 83 What factors should she consider?

NOW…. . WHAT IF…… What if her classmates made the following…… � 79, 81 , 80 , 77, 73, 83, 74, 93, 78, 80, 75, 67, 73, 77, 83, 86, 90, 79, 85, 83, 89, 84, 82, 77, 72 For this data: � The mean is: � Standard deviation is: Now how does Jane compare to her classmates? What does this data look like on a graph (shape)?

STANDARDIZING… We standardize values to determine how many standard deviations an observation is from the mean. IF x is an observation from a distribution that has a known mean and standard deviation, the standardized value is: z= This value is called a z-score.

BACK TO JANE…… So… should Jane be satisfied or disappointed with her score? ? ?

SOME THINGS TO TAKE NOTE Observations LARGER than the mean yield in a positive z-score. Observations SMALLER than the mean yield in a negative z-score.

REMINDER…P-TH PERCENTILE Remember, the p-th percentile represents the p percent of observations below it. � 100 th percentile= � 95 th percentile= � 30 th percentile=

CHEBYSHEV’S INEQUALITY How can we convert z-scores to percentiles? � CHEBYSHEV’S INEQUALITY In any distribution, the percent of observations that fall within k standard deviations of the mean is at least

CHEBYSHEV’S INEQUALITY … So what happens when…. � K=1 � K=2 � K=3 � K=4 � K=5

MORE…CHEBYSHEV’S INEQUALITY … It is VERY unusual for an observation to fall more than 5 standard deviations away from the mean in any distribution Note: Chebyshev’s Inequality does not help us determine the percentile corresponding to a given z-score. It only helps us gain insight about how observations are distributed within distributions

DENSITY CURVES A density curve is a curve that � Is always on or above the horizontal axis, and � Has area exactly 1 underneath it They describe overall patterns of a distribution

NORMAL CURVE Our Favorite!!!!!

MEDIAN AND MEAN OF DENSITY CURVES Median of a Density Curve: � is the “equal-area point” which divides the area under the curve in half Mean of a Density Curve: � is the “balance point” of the data

LET’S SPEAK GREEK…… Mean of a density curve, µ Standard Deviation of a density curve, σ

CAUTION ABOUT THOSE GREEKS…… µ can be located by “eye” on a density curve (remember…balance point) σ cannot be located by eye on most density curves