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Sigma (Summation) Notation Consider the heights of the Jones family members: Mom 65” Dad 72” Bobby 70” Joey 58” Fluffy 22” Spot 28” Let’s find the mean (m - “mu”). 65 + 72 + 70 + 58 + 22 + 28 = 315 m = 327 ÷ 6 = 52.5” Let’s look at those heights as a “set”. {65, 72, 70, 58, 22, 28} {x1, x2, x3, x4, x5, x6} To find the mean, we added the elements of the set 65 + 72 + 70 + 58 + 22 + 28 x1 + x2 + x3 + x4 + x5 + x6 Summation notation shows it this way: where n = 6 =1 S Is the capital Greek letter “sigma” Mean (m) in Summation Notation m= 1. 2. =1 We are still calculating the mean (average or “mu”) the same as always… Sum the elements of the set Divide by the number of elements. Deviation – how far is a certain piece of data from the mean? So here is our family… Here is the mean of 52.5” that we calculated earlier. Since no single family member is exactly 52.5” tall, all of them deviate from the mean. Some more than others… Mean Absolute Deviation & Standard Deviation These are 2 different statistics to spread (dispersion) of describe the ________ the data. Mean Absolute Deviation (MAD) – is often preferred because it is less affected by __________. outliers Standard Deviation (s) - A more traditional way of indicating dispersion. Calculating Mean Absolute Deviation (MAD) MAD = | −| =1 Step 1: Find the mean m of the data. 52.5” Step 2: Subtract the mean from each data element. 65-52.5” 12.5 72-52.5 19.5 70-52.5 17.5 58-52.5 5.5 22-52.5 -30.5 28-52.5 -24.5 Calculating Mean Absolute Deviation (MAD) | −| =1 MAD = 12.5 19.5 17.5 5.5 -30.5 -24.5 Step 3: Take the absolute value of the values found in Step 2 12.5 19.5 17.5 5.5 30.5 24.5 Step 4: Sum those numbers. 12.5 + 19.5 + 17.5 + 5.5 + 30.5 + 24.5 = 110 Step 5: Divide by “n” (the number of values) 110 ÷ 6 MAD = 18.3 Now that we have it, what is it for? We can use the MAD (18.3), to determine which of our data (family members) are unusually tall or unusually small. The mean was 52.5”, so anyone taller than: 52.5 + 18.3 70.8” or smaller than : 52.5 – 18.3 34.2” is “unusual”. So, Dad (72”), Fluffy (22”) and Spot (28”) have unusual heights for the Jones family.