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5/3/2016 Students will be able to: find the z-scores of a data set z-scores When a set of data is normally distributed, we can standardize each score by converting it into a z-score. z-scores can compare data values measured on different scales. A z-score reflects how many standard deviations above or below the mean. The z-score is positive if the data value lies above the mean and negative if it’s below the mean. z-score formula z x x : each value of the data set, :the mean :the standard deviation Example 1: Suppose SAT scores are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score? Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score? 700 500 z 2 100 Her z-score would be 2 which means her score is two standard deviations above the mean. Example 2: • A set of math test scores has a mean of 70 and a standard deviation of 8. • A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? To solve: Find the z-score for each test. 78-70 math z -score = 1 8 78-74 English z -score= .25 16 The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean. Try it out! What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2? What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2? Using the formula for z-scores: z x 23 1.5 2 x 3 x 23 x 26 The Toyota Camry would be expected to use 26 mpg of gasoline. Try it out! A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative? A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative? The z-score must be positive since the element of the data set is above the mean. How to Use the “Z- table” • • • Values found in this table tell us the area under the curve to the left of that z-score. If your calculated z-score is -1.46 , the value you find on the z-table tells you the area under the curve to the left of -1.46. If your calculated z-score is 0.81, the value you find on the z-table tells you the area under the curve to the left of 0.81. Looking up -1.46 on a z-table Looking up +0.81 on a z-table Now try it: Suppose the heights (in inches) of adult females (ages 20−29) in the United States are normally distributed with a mean of 64.1 inches and a standard deviation of 2.75 inches. 1. Find the percent of women who are no more than 63 inches tall. 2. At least 66 inches tall