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STUDENTS WILL DEMONSTRATE UNDERSTANDING OF THE CALCULATION OF STANDARD DEVIATION AND CONSTRUCTION OF A BELL CURVE Standard Deviation & The Bell Curve

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Standard Deviation 1st find the variance for a set of data Variance is the average squared deviation from the mean of a set of data

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Computing the Variance Grades from recent quiz in AP Biology: 96, 96, 93, 90, 88, 86, 86, 84, 80, 70 1 st Step: find the mean: 96 + 96 + 92 + 90 + 88 + 86 + 86 + 84 + 80 + 70 ÷ 10 868 ÷ 10 = 86.8 87

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Computing the Variance 2 nd Step: determine the deviation from the mean for each grade then square it: (96 – 87)² = (9)² = 81 (96 -87)² = (9)² = 81 (92 – 87)² = (5)² = 25 (90 – 87)² = (3)² = 9 (88 – 87)² = (1)² = 1 (86 – 87)² = (-1)² = 1 (84 – 87)² = (-3)² = 9 (80 – 87)² = (-7)² = 49 (70 – 87)² = (-17)² = 289

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Computing the Variance now add these 10 #’s and ÷ 10 81 + 81 + 25 + 9 + 1 + 1 + 1 + 9 + 49+ 289 ÷ 10 546 ÷ 10 = 54.6 = variance

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Formula for Variance

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Standard Deviation shows the variation in data if data pts are close together the standard deviation will be small if data pts are spread out the standard deviation will be larger σ or S = symbol used for standard deviation

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The Bell Curve represents a normal distribution of data & is used to show what standard deviation represents

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The Bell Curve 1 standard deviation from the mean (μ is symbol for 1 standard deviation) in either direction on horizontal axis represents 68% of the data

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The Bell Curve 2 μ = 2 standard deviations from the mean and will include ~95% of your data 3μ = 3 standard deviations form the mean and will include ~99% of your data

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Standard Deviation Calculation Standard Deviation is the square root of the variance

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Back to the Example Quiz Grades Mean = 87 Variance = 55 σ = √55 = 7.42 = 7.4 Mean = 87 so (87 -7.4) & (87 + 7.4) is the range of 1 standard deviation

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Example Continued My range for 1 standard deviation is : 79.6 to 94.4 80 to 94 in my class scores 93, 90, 88, 86, 86, 84, 80 are w/in 1 standard deviation of the mean check: 1 standard deviation is where 68% of data should fall: my class scores: 70%

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If data are a sample of larger group: Use the formula:

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for my class data: S = √ variance ÷ n – 1 S = √ 55 ÷ 9 = √ 6.1 =2.5 which means the measurements vary by +/- 2.5 from the mean

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so for my class data: Mean: 87 S = 2.5 1 µ would be (87 – 2.5) thru (87 + 2.5) or 84.5 to 89.5 2 µ would be (87 – 5) thru (87 + 5) 3 µ would be (87 – 7.5) thru (87 – 7.5)

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How Can I Use S? Suppose instead of grades my data described the average height of a certain plant (in cm). We can use S to predict the probability of finding a plant at a particular height: What is the probability of finding a plant that is 100 cm tall? 100 cm falls in the 3 rd standard deviation range from a bell curve we know that represents ~2% of a population so… There is a 2% chance a plant will grow to 100cm.

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Make Friends with the Bell Curve

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