identifies the number of standard deviations a particular value is from the

Identifies the number of standard deviations a

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– identifies the number of standard deviations a particular value is from the mean of its distribution Z = (x - µ) ¸ σ x = the data value of interest µ = the population mean σ = the population standard deviation Example: Calories in Hamburgers – Z-score for the WHOPPER Hamburger Type Restauran t Cal ori es Cheeseburger McDonald' s 300 Single with Everything Wendy's 430 Big Mac McDonald' s 540 Whopper Burger King 670 Bacon Cheeseburger Sonic 780 Baconator Wendy's 840 Triple Whopper with Cheese Burger King 123 2/3 lb. Monster Thickburger Hardee's 142 Sample Mean 776 .25 Standard Deviation 385 X = 640 µ = 776.25
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σ = 385.1 Z = (640 – 776.25) ¸ (385.1) = -0.35 Z-Score Attributes Positive for values above the mean Negative for values below the mean Zero for values equal to the mean Has no units, even though the original values will normally be expressed in other measurements - Analogous to expressing original data in other units Outliers – extreme values in the data set that require special consideration Can be identified using the z-score Within (+ / -) 3 = not considered outliers A z-score that is 3.1 or -3.6 is considered an outlier Calculating Z-Scores in Excel =STANDARDIZE (x, mean, standard_deviation) Example from above X = 640 µ = 776.25 σ = 385.1 =STANDARDIZE (640, 776.3, 385.1) The Empirical Rule Empirical Rule – if a distribution follows a bell-shaped, symmetrical curve centered around a mean, we would expect approximately 68%, 95%, and 99.7% of the values to fall within one, two, and three standard deviations above and below the mean, respectively Almost all of the values will fall w/in 3 SD of the mean AKA 68-95-99.7 Rule Example: - Average exam score is 80 points & SD is 5 points - Average PLUS and MINUS standard deviation (5) - à 68% of exam scores will fall b/w 75 and 85 points Formula for Expressing the z-Score in Terms of X X = µ + zσ - X = average - µ = mean - z = standard deviation 68% fall within one standard deviation of the mean Find data value that are one standard deviation above the mean - Set z = 1 Find data value that are one standard deviation below the mean
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- Set z = -1 95% fall within two standard deviations of the mean Find data value that are two standard deviations above the mean - Set z = 2 Find data value that are two standard deviations below the mean - Set z = -2 99.7% fall within three standard deviations of the mean Find data value that are three standard deviations above the mean - Set z = 3 Find data value that are three standard deviations below the mean - Set z = -3 Chebyshev’s Theorem Chebyshev’s Theorem – a mathematical rule that states that for any number z > 1, the percent of the values that fall within z standard deviations from the mean will be AT LEAST = (1-1/z²) • (100) Like empirical rule but applies to any distribution - Not just bell-shaped, symmetrical distributions **can only be stated for z-scores that are greater than 1 Ex: when z = 2 - [1 – (1/4)] = 0.75 (100) = 75% - à 75% of the data values will fall w/in 2 standard deviations of the mean (at least) Conclusions
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