1. Lecture 6 Sections 2.1 – 2.2 Objectives: Measure of Center
Measure of Center for Data
Measure of Center for Distributions
Measure of Variability for Data
Measure of Variability for Distributions
The Empirical Rule (Normal Distribution)

2. The Sample Mean
To describe a “typical” or “representative” observation, we will use the sample mean
Most frequently used measure of the center.
Sensitive to extreme observations (outliers).
Useful for the estimation of the center when the distribution is symmetric and is free of outliers.

3. Example
Caustic stress corrosion cracking of iron and steel has been studied because of failures around rivets in steel boilers and failures of steam rotors. Consider the accompanying observations on crack length (μm) as a result of constant load stress corrosion tests on smooth bar tensile samples for a fixed length of time. The data is from the article “On the Role of Phosphorus in the Caustic Stress Corrosion Cracking of Low Alloy Steels”, Corrosion Science, 1989:
a. Find the mean of crack length.
b. Replace 45.0 by and then find the mean of crack length.

4. The Sample Median
Midpoint of the observations in the ordered list. So, 50% of data falls below and 50% falls above.
If n is odd ⇒ the median is the middle value in the ordered list (that is, the (n+1)/2 th observation).
If n is even, there is no unique middle ⇒ the median is the average of the middle pair of values.
Much less sensitive to extreme observations (outliers).
Useful for the estimation of the center when the distribution is skewed.

5. Example Consider the following 5 observations 34 44 56 63 67

6. Trimmed Means
The 100r% trimmed mean is the mean of remaining observations after trimming the largest n*100r% and the smallest n*100r%, where r is a number between 0 and 0.5
Note: The trimmed mean is less sensitive to outliers than the mean but more sensitive than the median.
Example. Consider the following 20 observations, each representing the lifetime (hr) of a certain type of incandescent lamp:
Find the 10% trimmed mean.

7. Population Mean Discrete Distributions
Definition. The mean (or expected value) of a discrete variable x is given by
Example. Plastic parts manufactured using an injection molding process may exhibit one or more defects, including sinks, scratches, black spots, and so on.
Let x represent the number of defects on a single part, and suppose the distribution of x is as follows: x
p(x)
1) If x~B(n,π), then μ=nπ ) If x~Poisson(λ), then μ = λ.

8. Population Mean Continuous Distributions
Definition. The mean (or expected value) of a continuous variable x is given by
Example. The distribution of the amount of gravel (tons) sold by a particular construction supply company in a given week is a continuous variable x with density function
Knowledge of the mean value of x will help the company decide on a price for the gravel. Find the mean of x.

9. Means for Specific Distributions
Continuous distributions
If x ~ N(μ, σ2), then the mean of x is μ.
If x has an exponential distribution with parameter λ, then mean of x is λ.
If x has a lognormal distribution with parameters μ and σ, then the mean of x is
The mean of a Weibull distribution is a somewhat complicated expression involving the parameters α and β.

10. Population Median Median for continuous distribution
The median of a continuous distribution divides the area under the density curve into two equal halves. The defining condition is
Example. Find the median for the distribution of weekly gravel sales.

11. Mean and Median
The mean and the median are the same only if the distribution is symmetrical. The median is a measure of center that is resistant to skew and outliers. The mean is not.
Mean and median for a symmetric distribution
Mean
Median
Mean and median for skewed distributions
Left skew
Right skew
Mean
Median
Mean
Median

12. Measure of Variability for Data
A measure of the center is not enough to describe a distribution well.
Example. Suppose the heights of five starting basketball players on two men’s basketball teams are:
Team I (inches):
Team II (inches):
Range - Simplest measure of variability
Range = the difference between the largest and the smallest sample values.
The range depends on only the two most extreme observations and disregards the positions of the remaining (n-2) values.

13. Sample Variance and Standard Deviation
The sample standard deviation, denoted by s, is the square root of the variance.
The sample variance measures how far, on average, the observations are from the mean.
The more spread out a distribution is around its mean, the larger its standard deviation.
The unit for s is the same as the unit for the data.
Sensitive to outliers.

14. Example
Strength is an important characteristic of materials used in prefabricated housing. Each of 11 prefabricated plate elements was subjected to a severe stress test, and the maximum width (mm) of the resulting cracks was recorded. The data is from the article “Prefabricated Ferrocement Ribbed Elements for Low-Cost Housing” (J. of Ferrocement, 1984: ).
Find the sample variance and sample standard deviation.

15. Measure of Variability for Distributions
Population variance and standard deviation
Discrete distributions
Definition. The variance of a discrete variable x is given by
The standard deviation is σ, the positive square root of the variance.
Example. Revisit the plastic part example. Find the variance and standard deviation.
Variances for specific discrete distributions
1) If x ~ B(n,π), then. σ2 = nπ(1-π)
2) If x ~ Poisson (λ), then σ2 = λ.

16. Measure of Variability for Distributions
Population variance and standard deviation
Continuous distributions
Definition. The variance of a continuous variable x is given by
The standard deviation is σ, the positive square root of the variance.
Example. Revisit the gravel sales example. Find the variance and standard deviation of x.
Variances for specific continuous distributions
If x ~ N(μ, σ2) then the variance of x is σ2
If x has an exponential distribution with parameter λ, then variance of x is λ.
3) If x has a lognormal distribution with parameters μ and σ, then the variance of x is
4) The variance of a Weibull distribution is even more complicated.

17. The Empirical Rule
For any variable x whose distribution is well approximated by a normal curve:
Approximately 68% of the values are within 1 standard deviation of the mean.
Approximately 95% of the values are within 2 standard deviation of the mean.
Approximately 99.7% of the values are within 3 standard deviations of the mean.
N(0,1)

18. Example
Scores on an achievement test taken by all high school seniors in a certain state are known to have, approximately, a bell-shaped distribution with Mean (μ) = 64, Standard Deviation (σ) = 10.
68% of the data will lie in the interval
95% of the scores are between
Almost all of the scores are between

19. Example
The time to complete a standardized exam is approximately bell shaped with a mean of 70 minutes and a standard deviation of 10 minutes.
Using the empirical rule, what percent of students will complete the exam in under an hour?