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Algebra III Lesson 61 Single-Variable Analysis – The Normal Distribution – Box-and-Whisker Plots Single-Variable Analysis Basic Statistics Mean - the average of the items in question Range - the difference between the highest and the lowest items in question New Statistics Standard Deviation - it is a way of describing how tightly the items are ‘packed’, and a way of showing how representative any given number is from the whole Deviation - the first step in finding standard deviation. Take each number and subtract the mean from it. This will show how far an item is from the mean and if it is higher or lower. Variance - the second step. Consider it the ‘squared’ average of the deviations. Standard Deviation - take the square root of the variance. Example 61.1 The measured distances are 12, 7, 15, and 10. Find the mean and the standard deviation of the distances. 1st – find mean mean = 2nd – find deviations 12 + 7 + 15 + 10 4 12 – 11 = 15 – 11 = = 11 1 7 – 11 = 4 10 – 11 = 2 2 2 2 ( ) ( ) ( ) ( ) 1 4 4 1 + − + + − 3rd – find variance var iance = 4 4th – find std dev σ = 8.5 = 2.92 -4 -1 = 34/4 = 8.5 The Normal Distribution Run enough experiment and the results will become a normal distribution. The graph looks like a symmetric mountain, with the peak right over the mean of the items. And the sides drop off the same on both sides of the peak. Median Mode - list the numbers in order and find the middle number ( or the average of the middle 2, if an even amount) -most commonly occurring result For normal distributions the mean, median, and mode all occur at the same spot. mean, mode, median A normal distribution takes on a ‘bell’ shape, and is commonly also known as a bell curve. An interesting fact about bell curves is that 1 std dev either side of the mean will include about 68% of the results. This is the average zone. 2 std devs includes about 95% of the results. 3 std devs includes 99% of the results. How sharp or flat the curve is depends upon the size of the std dev. σ=8 σ=2 Example 61.2 The mean of a distribution that is approximately normal is 100 and the standard deviation is 5. Use the standard deviation to tell how the data are distributed with respect to the mean. 1 std dev: 68% of the data 95 Æ 105 100 - 1σ = 95 100 + 1σ = 105 2 std dev: 95% of the data 90 Æ 110 100 - 2σ = 90 100 + 2σ = 110 3 std dev: 99% of the data 100 - 3σ = 85 100 + 3σ = 115 85 Æ 115 Box-and-Whisker Plots A way of graphically showing the results from an experiment, survey, or test, etc. To do a box and whisker plot: 1st – put the numbers in order 2nd – find the median, and put a line, and label it, in the list to mark the median. Once the median is known half the numbers are above the median and half below. Not necessarily evenly spread out, just split in half. 3rd – find the median for the lower half, this marks the upper edge of the first quarter of the list. Called the first quartile, Q1. 4th – find the median for the upper half, this marks the upper edge of the third quarter of the list. Called the third quartile, Q3. 5th – make a picture like the one below showing this data. min Q1 med Q3 max Stem and leaf plot 1st - Arrange the data in order 2nd – Make a vertical list of the leading digits (make sure they all stop at the same place value) 3rd – list the remaining digits next to its appropriate leader digit(s) in order. Sample: 16,22,28,33,35,40,50,120 12 5 4 3 2 1 0 0 0 3,5 2,8 6 Example 61.3 The measurements are 16, 22, 28, 33, 35, 40, 50, and 120. find the mean, the standard deviation, and make a box-and-whisker plot. 1st – find mean = 2nd – find deviations 3rd 16 + 22 + 28 + 33 + 35 + 40 + 50 + 120 8 16 – 43 = 28 – 43 = 35 – 43 = 50 – 43 = = 43 -27 22 – 43 = -21 -15 33 – 43 = -10 -8 40 – 43 = -3 7 120 – 43 = 77 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 27 21 15 10 8 3 7 77 − + − + − + − + − + − + + – find variance = 8 = 7546/8 = 943.25 4th – find std dev σ = 943.25 = 30.71 1st – put the numbers in order 16,22,28,33,35,40,50,120 2nd – find the median, and put a line, and label it, in the list to mark the median. 16,22,28,33,35,40,50,120 median = 34 3rd – find the median for the lower half, this marks the upper edge of the first quarter of the list. Called the first quartile, Q1. 16,22,28,33,35,40,50,120 Q1 = 25 Q3 = 45 4th – find the median for the upper half, this marks the upper edge of the third quarter of the list. Called the third quartile, Q3. 5th – make a picture like the one below showing this data. min 16 Q1 25 med 34 Q3 45 max 120 Practice 9,11,12