1. Section 4.4: Interpreting Center and Variability: Chebyshev’s Rule, The Empirical Rule, and z-scores

2. To compare information such as the mean and standard deviation it is useful to be able to describe how far away a particular observation is from the mean in terms of standard deviation.

3. Suppose we have a data set of scores on a standardized test with mean of 100 and standard deviation of 15. We can make the following statements:
Because 100 – 15 = 85, we say that a score of 85 is “ 1 standard deviation below the mean” similarly = 115 is “1 standard deviation above the mean”
Because 2 standard deviations is 2(15) = 30 and = 130 and 100 – 30 = 70 scores between 70 and 130 are within 2 standard deviations of the mean.
Because (3)(15) = 145, scores above 145 exceed the mean by more than 3 standard deviations

4. Chebyshev’s Rule

5. For specific values of k Chebyshev’s Rule reads
At least 75% of the observations are within 2 standard deviations of the mean.
At least 89% of the observations are within 3 standard deviations of the mean.
At least 90% of the observations are within 3.16 standard deviations of the mean.
At least 94% of the observations are within 4 standard deviations of the mean.
At least 96% of the observations are within 5 standard deviations of the mean.
At least 99% of the observations are with 10 standard deviations of the mean.

6. Example – Chebyshev’s Rule
Consider the student age data
Color Code: within 1 standard deviation of the mean
within 2 standard deviations of the mean
within 3 standard deviations of the mean
within 4 standard deviations of the mean
within 5 standard deviations of the mean

7. Example continued Interval Chebyshev’s Actual
within 1 standard deviation of the mean
 0%
72/79 = 91.1%
within 2 standard deviations of the mean
 75%
75/79 = 94.9%
within 3 standard deviations of the mean
 88.8%
76/79 = 96.2%
within 4 standard deviations of the mean
 93.8%
77/79 = 97.5%
within 5 standard deviations of the mean
 96.0%
79/79 = 100%

8. Notice that Chebyshev gives very conservative lower bounds and the values aren’t very close to the actual percentages.

9. Empirical Rule
If the histogram of values in a data set is reasonably symmetric and unimodal (specifically, is reasonably approximated by a normal curve), then
Approximately 68% of the observations are within 1 standard deviation of the mean
Approximately 95% of the observations are within 2 standard deviations of the mean
Approximately 99.7% of the observations are within 3 standard deviations of the mean

11. Z-Score
The z-score is how many standard deviations the observation is from the mean.
A positive z-score indicates the observation is above the mean and a negative z-score indicates the observation is below the mean

12. Computing the z score is often referred to as standardization and the z score is called a standardized score.

13. Example
A sample of GPAs of 38 statistics students appear below (sorted in increasing order)
Mean = and s =

14. The following stem and leaf indicates that the GPA data is reasonably symmetric and uimodal
2 0
2 233
2 55
3 0001
3 7
3 889
Stem: Units digit
Leaf: Tenths digit

16.  68%  95% 99.7% Interval Empirical Rule Actual
within 1 standard deviation of the mean
 68%
27/38 = 71%
within 2 standard deviations of the mean
 95%
37/38 = 97%
within 3 standard deviations of the mean
99.7%
38/38 = 100%

17. Notice that the empirical rule gives reasonably good estimates for this example.

18. Comparison of Chebyshev’s Rule and the Empirical Rule
The following refers to the weights in the sample of 79 students. Notice that the stem and leaf diagram suggest the data distribution is unimodal but is positively skewed because of the outliers on the high side. Nevertheless, the results for the Empirical Rule are good.
10 3
11 37
19 5
20 00
21 0
22 55
23 79
Stem: Hundreds & tens digits
Leaf: Units digit

19. Interval
Chebyshev’s Rule
Empirical Rule
Actual
within 1 standard deviation of the mean
 0%
 68%
56/79 = 70.9%
within 2 standard deviations of the mean
 75%
 95%
75/79 = 94.9%
within 3 standard deviations of the mean
 88.8%
99.7%
79/79 = 100%
Notice that even with moderate positive skewing of the data, the Empirical Rule gave a much more usable and meaningful result.