1. December 12, 2011 Lesson #21: Describing Numbers with the Mean & Standard Deviation

2. Some Terminology 1.Mean – the average of the data. Will be shown by the symbol x (pronounced x- bar). 2.Deviation – the distance a number is from the mean.

3. Some Terminology 3.Standard Deviation – the average of all the deviations for the data set. Will be shown by the symbol “s”. 4.∑ = the symbol which represents summation. When you see this, you add up all of the numbers which are part of its problem.

4. How to find the Standard Deviation The following ten numbers represent grade point averages: 2.0 3.2 1.8 2.9 3.6 0.9 4.0 3.3 2.9 0.8 Step 1: Find the mean of the data. X – Bar = (2.0 + 3.2 + 1.8 + 2.9 + 3.6 + 0.9 + 4.0 + 3.3 + 2.9 + 0.8) / 10 X – Bar = 2.54

5. Step 2 – Find each deviation X minus X-Bar X minus X-Bar (Deviation) -0.54 0.66 -0.74 0.36 -1.64 1.46 0.76 0.36 1.06 -1.74 X (the data)X – Bar 2.02.54 3.22.54 1.82.54 2.92.54 0.92.54 4.02.54 3.32.54 2.92.54 3.62.54 0.82.54

6. Step 3 – Square the Deviations and Find their Sum X minus X-Bar (Deviation) -0.54 0.66 -0.74 0.36 -1.64 1.46 0.76 0.36 1.06 -1.74 X (the data) X – Bar 2.02.54 3.22.54 1.82.54 2.92.54 0.92.54 4.02.54 3.32.54 2.92.54 3.62.54 0.82.54 (X minus X-Bar) Squared 0.2916 0.4356 0.5476 0.1296 2.6896 2.1316 0.5776 0.1296 1.1236 3.0276 ∑(x-xbar)² = 11.084

7. Step 4 – Find the Variance Step 5 – Find the Standard Deviation Step 4 – Find the Variance Divide the value from step 3 by the number in the data set minus 1 (n-1) 11.084 / 9 ≈ 1.2316 Step 5 – Find the Standard Deviation Take the square root of the Variance √1.2316 ≈ 1.11

8. What this means The average distance all the numbers are from the mean is 1.11 units (grade points in this instance). When we get to Chapter 5 and are introduced to the Empirical Rule, we will discuss how Standard Deviation tells us how much of the data will lie within 68% of the mean, 95% of the mean, and 99.7% of the mean. This information will describe to you how “good” your data is (in these terms, how much “better” was your GPA in relation to your peers?)

9. Now, More about Standard Deviation The standard deviation describes the spread of the data. If s = 0, there is no spread (the data is all the same). The larger the value of s, the more wide spread the data. Standard deviation works best with a symmetric graph. When you describe a symmetric graph, use the mean as the center and standard deviation as the spread.

10. What if it’s not a Symmetric Graph? When you have a skewed graph, there are likely to be outliers. The standard deviation value will also be skewed as a result (much bigger than it should be). With a skewed graph, quartiles and the median are less affected, so it is best to use a 5 number summary to describe the data.

11. Homework #20 Worksheet to be handed out Don’t forget…You have a quiz tomorrow on the 5 number summary, outliers, and the box plot.