1. Interpreting
Center &
Variability

2. Density Curves Can be created by smoothing histograms
ALWAYS on or above the horizontal axis
Has an area of exactly one underneath it
Describes the proportion of observations that fall within a range of values
Is often a description of the overall distribution
Uses m & s to represent the mean & standard deviation

3. z score Standardized score Creates the standard normal density curve
Has m = 0 & s = 1

4. What do these z scores mean?
-2.3
1.8
6.1
-4.3
2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean

5. Jonathan wants to work at Utopia Landfill
Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with m = 45 and s = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations below the mean. Jonathan scores 35 on this test. Does he get the job?
No, he scored 2.78 SD below the mean

6. At least what percent of observations is within 2 standard deviations of the mean for any shape distribution?
Chebyshev’s Rule
The percentage of observations that are within k standard deviations of the mean is at least
where k > 1
can be used with any distribution
75%

7. Chebyshev’s Rule- what to know
Can be used with any shape distribution
Gives an “At least ” estimate
For 2 standard deviations – at least 75%

8. Normal Curve Bell-shaped, symmetrical curve
Transition points between cupping upward & downward occur at m + s and m – s
As the standard deviation increases, the curve flattens & spreads
As the standard deviation decreases, the curve gets taller & thinner

9. Can ONLY be used with normal curves!
Empirical Rule
Approximately 68% of the observations are within 1s of m
Approximately 95% of the observations are within 2s of m
Approximately 99.7% of the observations are within 3s of m
See p. 181
Can ONLY be used with normal curves!

10. The height of male students at PWSH is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches.
a) What percent of the male students are shorter than 66 inches?
b) Taller than 73.5 inches?
c) Between 66 & 73.5 inches?
About 2.5%
About 16%
About 81.5%

11. First, find the mean & standard deviation for the total setup time.
Remember the bicycle problem? Assume that the phases are independent and are normal distributions. What percent of the total setup times will be more than minutes?
First, find the mean & standard deviation for the total setup time.
Phase
Mean SD
Unpacking
3.5
0.7
Assembly
21.8
2.4
Tuning
12.3
2.7
2.5%