1. Standard Deviation and the Normally Distributed Data Set
The Empirical Rule
Standard Deviation and the Normally Distributed Data Set

2. Normal Distribution 5 6 7 8 9 10 11 Data Values S -3 -2 -1 0 1 2 3 x
The Mean is always graphed in the middle of normally distributed data and is represented with the number 0 on the number line as the mean is zero distance from itself, the mean.
If we take a look at this data set,
5, 6, 6, 7, 8, 8, 9, 10, 10,11
The mean of those values is 8
The standard deviation of this group is s= 2
So 1 standard deviation is the mean of 8 +/- 2
So 2 standard deviations is the mean of 8 +/- 4 (1 additional +/- 2)
So 3 standard deviations is the mean of 8 +/- 6 (1 additional +/- 2) S x
Data values

3. The Empirical Rule When Then The standard deviation is known
The mean is known
& the data is normally distributed
(like a bell-shaped mound)
Then
The Empirical Rule can be used to generalize some properties about the distribution (data)
Also known as the sigma rule; rule

4. Normal Distribution
Percent of values under portions of the normal curve
34.13%
34.13%
13.59%
13.59%
2.14%
2.14%
0.13%
0.13% S x

5. The Empirical Rule When data is normally distributed:
Approx. 68% of the values will fall within +/- 1 S from the x
68% S

6. The Empirical Rule When data is normally distributed:
Approx. 95% of the values will fall within +/- 2 S from the x
95% S

7. The Empirical Rule When data is normally distributed:
Approx. 99% of the values will fall within +/- 3 S from the x
99.7% S

8. Approximate Empirical Calculations
Approximate Calculations
68% = x +/- 1(s)
[ x – 1(s), x + 1(s)]
95% = x +/- 2(S)
[ x – 2(s), x + 2(s)]
99% = x +/- 3(S)
[ x – 3(s), x + 3(s)]

9. SAT Distribution m = 490 and s = 100
What percentage of seniors scored between 390 and 590 on this SAT test?
One student scored 795 on this test. How did this student do compared to the rest of the scores?
A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university?