– 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
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
- 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 ■
You've reached the end of your free preview.
Want to read all 94 pages?
- Fall '12
- Standard Deviation, Mean