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Standard Deviation, ZScores, Variance ALGEBRA 1B LESSON 42 INSTRUCTIONAL MATERIAL 2 Standard Deviation Standard Deviation is a number that tells us how a value in the data set differs from the mean. It also tells us how spread out the numbers are. The bigger the standard deviation, the more widespread the data is. The Greek letter sigma, , will represent Standard Deviation. Standard Deviation To find Standard Deviation, there are 2 ways: One way is the “long” way: Calculate Subtract Square the mean the mean from each data value the differences (results from 2nd step) Find the average of the squared differences (This is called “Variance”) Take the square root of the variance Example Given the following Algebra 1 test scores, find the standard deviation: 75, 77, 86, 89, 93, 82 First, find the mean: 75+77+86+89+93+82 = 83.67 6 Cont. Subtract value: the mean from each data 75 – 83.67= -8.67 77 – 83.67 = -6.67 86 – 83.67 = 2.33 89 – 83.67 = 5.33 93 – 83.67 = 9.33 82 – 83.67 = 1.67 Cont. Square the difference and find the average: 75 + 44 + 5 + 28 + 87 + 2 (-8.67)2 = 75 (-6.67)2 = 44 6 (2.33)2 = 5 (5.33)2 = 28 (9.33)2 = 87 (1.67)2 = 2 75+44+5+28+87+2 = 6 40 40 is the Variance. To find the Standard Deviation, take the square root. 40 = 6.3 So, the standard deviation is 6. 3 The 2nd way to find Standard Deviation (the shortcut): Type the data values in your calculator Stat, Edit, L1 Stat, Calc, 1, Calculate – look for the sigma, . Standard Deviation = 6.4 (Between the calculator and paper method, the standard deviation may be slightly different. That is due to rounding. The more places you round to, the more accurate your answer Example 2 Find the Standard Deviation for the following average temperatures: 63, 61, 66, 67, 63, 65, 68 Standard Deviation ( = 2.3) Z-Score The Z-Score tells how many standard deviations a specific number in the data set is away from the mean. If the z-score is positive, the data value will be above the mean. If the z-score is negative, the data value will be below the mean. To find the Z-Score use this formula: 𝑥−µ = , where 𝑥 = the data value, µ = mean, and = standard deviation. 𝑧 Example 3 A student scored an 84 on the Algebra 1 benchmark. The average benchmark score for that class was a 90, while the standard deviation was 2. Find the student’s z-score. 𝑥 = 84 µ = 90 = 2 𝑧 = 𝑥−µ = 84−90 2 = −6 2 = −3 Cont. With a z-score of -3, that means the student scored 3 standard deviations below the mean. Example 4 A student had a z-score of 3 after her math test. The class average was a 82 and the standard deviation was 2. What was the student’s score on the math test? 𝑧 =3 𝑥 =𝑥 µ = 82 =2 𝑧 = 𝑥−µ 3 6 = 𝑥−82 2 = 𝑥 − 82 88 = 𝑥 The student’s scored an 88 on the math test. Example 5 The class average on the box-and-whisker quiz was a 92. The variance was 4. Find the interval for the score of a student that scored within 1.5 standard deviations from the mean. 𝑧 = −1.5 𝑎𝑛𝑑 1.5 𝑥 =𝑥 µ = 92 = 4=2 Cont. −1.5 −3 = 𝑥−92 2 1.5 = 𝑥−92 𝑥 = 𝑥 − 92 3 = 𝑥 − 92 𝑥 = 89 𝑥 = 95 The student must have scored between an 89 and 95. Summary Standard Deviation, Variance, and Z-Score are 3 different ways to analyze data. The Standard Deviation and Variance describe how spread out data values are from each other, as well as from the mean. Given the variance, take the square root to find the standard deviation. Summary Cont. The Z-Score measures how far a data value is away from the mean in terms of Standard Deviation. The Standard Deviation is what you add/subtract to the mean while the zscore is how many times you add/subtract to the mean.