stats_ans.pdf

# Standard deviation variance the standard deviation σ

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Standard Deviation & Variance The Standard Deviation (σ) is a measure of how ______spread out________________ numbers are. The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?" The Variance (which is the square of the standard deviation, ie: σ 2 , is defined as: The average of the squared differences from the Mean. In other words, follow these steps: 1. Calculate the Mean 2. Now, for each number subtract the Mean and then square the result (the squared difference). 3. Find the average of those squared differences. (Why Square?) Example You and your friends have just measured the heights of your dogs (in millimeters): The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. Mean = 394

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Now, we calculate each dog’s difference from the Mean: To calculate the Variance, take each difference, square it, and then average the result: Variance: σ 2 = (42436 + 5776 + 50176 + 1296 + 8836)/5 = 21704 So, the Variance is ___21704________________. And the Standard Deviation is just the square root of Variance, so: Standard Deviation: uni03C3 = uni221A____21704__________ = _____147.32_____________ And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation of the Mean: So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small. Circle which dogs are extra tall and extra short. *Note: Why square?
Squaring each difference makes them all positive numbers (to avoid negatives reducing the Variance). But squaring them makes the final answer really big, and so un-squaring the Variance (by taking the square root) makes the Standard Deviation a much more useful number. Find the mean, median, variance and standard deviation of this set of data: 37, 56, 78, 54, 108, 22, 60, 10 Mean: ___53.13_____________ Median: ______55__________________ Variance, σ 2 ____846.86_______ SD, σ : _____31.11______________
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