1. Standard Deviation

2. 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?

3. 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?

4. 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?)

5. 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.

6. 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.

7. 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?
= - 9.5
Or look at the highest score, 89.
How far away is 89 from the mean of 81.5?
= 7.5
Mean:
81.5

8. 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

9. 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.

10. 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

11. (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

12. 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

13. 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 =
= 15.10

14. 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 

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

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

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

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

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