Standard Deviation

Two Trig Honors classes took a recent probability quiz Two Trig Honors classes took a recent probability quiz. There were 10 students in each class, and each class had an average score of 81.5 Since the averages are the same, can we assume that the students in both classes all did pretty much the same on the exam? Why or why not?

Mean The average (mean) does not tell us anything about the distribution or variation in the grades! Observe the dot plots! Which has the Greater spread?

We need a way to measure the distribution of the data (not just the average) Why not just give the range of the data (different between the highest and lowest values?)

Observe a set of data with an average of 17.95 and the range is 23. Here is the average And here is the range Most of the numbers are in this area, and are not evenly distributed throughout the range.

The Standard Deviation is a number that measures how far away each number in a set of data is from their mean If the Standard Deviation is large, it means the numbers are spread out from their mean. If the Standard Deviation is small, it means the numbers are close to their mean.

72 76 80 81 83 84 85 89 scores on the Quiz for Team A: The Standard Deviation measures how far away each number in a set of data is from their mean. 72 76 80 81 83 84 85 89 Let’s start with the lowest score, 72. How far away is 72 from the mean of 81.5? 72 - 81.5 = - 9.5 Or look at the highest score, 89. How far away is 89 from the mean of 81.5? 89 - 81.5 = 7.5 Mean: 81.5

Distance from Mean Distance from Mean 72 76 80 81 83 84 85 89 -9.5 7.5 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 Step The first step in finding the Standard Deviation is to find all the distances from the mean

Distance from Mean Distances Squared 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 Step Next you need to square each of the distances to turn them into positive numbers.

Distance from Mean Distances Squared 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 Step Then sum up all the distances squared Sum: 214.5

(where n represents the number of data points) Distance from Mean Distances Squared 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 Step Sum: 214.5 10 Divide by n (where n represents the number of data points) = 21.45

Finally, take the square root of the average distances Distance from Mean Distances Squared 72 76 80 81 83 84 85 89 - 9.5 - 5.5 - 1.5 - 0.5 1.5 2.5 3.5 7.5 90.25 30.25 2.25 0.25 6.25 12.25 56.25 Step Sum: 214.5 10 Finally, take the square root of the average distances AND YOU HAVE THE STANDARD DEVIATION!!! = 21.45 = 4.63

Distance from Mean Distances Squared 57 65 83 94 95 96 98 93 71 63 - 24.5 - 16.5 1.5 12.5 13.5 14.5 16.5 11.5 - 10.5 -18.5 600.25 272.25 2.25 156.25 182.25 210.25 132.25 110.25 342.25 Sum: 2280.5 10 If we followed the same process, we’d get a STANDARD DEVIATION OF 15.91 = 228.05 = 15.10

Team A Team B Average on the Quiz 81.5 Standard Deviation 4.63 15.10 Mean We can see the standard deviation gives us a much better understanding of the distribution/spread 

Let’s recap that formula (include on your notesheet ) = mean x1, x2, … xn,…  Data Values n = # of data values in set

Use the formula to find the standard Deviation of the data set: 43, 48, 41, 51, 42

Use the formula to find the standard Deviation of the data set (on notesheet): 43, 48, 41, 51, 42

Thinking Questions: 1. What kind of data set would have a Standard deviation of zero? 2. Can a standard deviation be negative? Explain.

Thinking Questions: 1. What kind of data set would have a Standard deviation of zero? 2. Can a standard deviation be negative? Explain.