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Prob and Stats, Sep 21 The Empirical Rule and the Z-Score Book Sections: 7.1 & 7.2 Essential Questions: How do I compute and use statistical values? What is the Empirical Rule? What is a Z-Score and how can I compute and use it? How are Z-Scores tied to the Empirical Rule? Standards: PS.SPID.1, .2, .3 The Empirical Rule • A symmetric data set (mean and median about equal) has a consistent distribution that relates the mean, standard deviation, and a percent of data interval for all such distributions. • The Empirical Rule and Z-Scores only apply to symmetric distributions. TheWords Empirical Rule and Graph Together • 68% of the data lies within one standard deviation of the mean • 95% of the data lies between two standard deviations of the mean • 99.7% of the data lies between three standard deviations of the mean Standard Deviation Thresholds • 68% - 1 x Standard deviation • Range ( x - s, x + s) • 95% - 2 x Standard deviations • Range ( x - 2s, x + 2s) • 99.7% - 3 x Standard deviations • Range ( x - 3s, x + 3s) Example • In a symmetrically distributed data set with a mean of 12 and a standard deviation of 3, what is the range of the middle 99.7% of the data? Empirical Rule Problems Backwards • Based on deviations (1, 2, or 3), you are trying to list a range containing a percent. • Given a range, determine a deviation (1, 2, or 3) that corresponds to a percent. Solve this equation for d to determine the deviation: x sd upper _ bound If d = 1, ans = 68%; d = 2, ans = 95%; d = 3, ans = 99.7%. Examples • In a symmetrically distributed data set with a mean of 40 and a standard deviation of 8, what percent of that data lies x sd upper _ bound between 16 and 64? • What is the range of the middle 95% of the data in a symmetrically distributed data set with a mean of 25 and a standard deviation of 2.5? Example • In a symmetrically distributed data set with a mean of 100 and a standard deviation of 15, what percent of that data lies x sd upper _ bound between 70 and 130? Example • In a symmetrically distributed data set with a mean of 77 and a standard deviation of 12, what is the range of the middle 68% of the data? The Standard Score • The standard score, or z-score (z) is a number that is representative of the number of standard deviations a given value of x falls from the mean. __ z= (Value – Mean) Standard Deviation or (x x ) z s In this scheme, you need a mean and standard deviation. The x is the value you want the z-score of. Now its plug and chug. Properties of the Z-Score • A z-score can be positive, negative, or zero • Positive – the x-value was greater than the mean • Negative – the x-value was less than the mean • Zero – the x-value was equal to the mean • A z-score can be used to identify an unusual value of a data set that is symmetrically distributed The Purpose of the Z-Score • A z-score tells you exactly how many standard deviations a value is above or below the mean in a symmetrically distributed data set. • Positive is above the mean, negative is below the mean. The z-score transforms the mean to 0 and the standard deviation to 1. Finding Z-Scores • The weight of polar bears in or near the town of Barry, Canada is symmetrically distributed with a mean weight of 840 pounds with a standard deviation of 67 pounds. Compute the z-score for Slim, a young male bear captured and tagged, who weighed 725 pounds. Example • A symmetrically distributed data set has a mean of 69 and a standard deviation of 7. Compute the z-score for a value of 60. Example • Cars driving along a stretch of I-85 have a mean speed of 71 mph and a standard deviation of 3 mph. What is the zscore of a car clocked here going 81 mph? The Empirical Rule with Z-Scores 99.7% of Data 95% of Data 68% of Data In the world of z-scores, mean is always 0 and standard deviation is 1. Interpreting Z-Scores • It is rare to have a Z-Score smaller than -3 or larger than 3. • It is nearly impossible to have a Z-Score outside the range of [-6, 6]. That would be 6 standard deviations away from the mean. Class work: CW 9/21/16, 1-15 Homework: None