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Used to determine if a random sample comes from a specific distribution (e.g., Normal or Exponential).
Basis:We
compare the (100*p)th percentile of the sample data set
with the (100*p)th percentile of the distribution we
think it came from, and see how close they are to each other.
We will take the ithsmallest
observation as the [100*(i-0.5)/n]th percentile
of the sample data set
Example: Consider the following sample data set of
n=5 points (arranged in ascending order): Y1= 3.01, Y2=3.35,
Y3=4.79, Y4=5.96, Y5=7.89. Could
this have come from a Normal distribution with m=5
and s=2, i.e., N(5,2) distribution?
Question: Why (i-0.5)/n?
Answer: Exact percentiles are impossible to compute for a data set of arbitrary sample size n! This is a compromise: for each i there are (i-1) values that are smaller and i values that are equal or larger, so we take the average of i and (i-1)...
So we get
the table below (n=5):
| i |
(i-0.5)/n
|
Yi | Zi | Xi=5+2*Zi |
| 1 | 0.1 (10th sample percentile)= | 3.01 | -1.28 |
|
| 2 | 0.3 (30th sample percentile)= | 3.35 | -0.525 | 3.95 = (30th N(5,2) percentile) |
| 3 | 0.5 (50th sample percentile)= | 4.79 | 0 | 5.00 = (50th N(5,2) percentile) |
| 4 | 0.7 (70th sample percentile)= | 5.96 | 0.525 | 6.05 = (70th N(5,2) percentile) |
| 5 | 0.9 (90th sample percentile)= | 7.89 | 1.28 | 7.56 = (90th N(5,2) percentile) |
A plot of Xi vs. Yi should thus yield a straight line with slope=1 (45°) if the values are close to each other.
Normal Probability Plots
Used to determine if a random sample
comes from a Normal distribution (with some unknown m
and s). Based on the fact that x
= m + zas.
So, if the sample did indeed come
from a Normal distribution, and every sample percentile was exactly
equal to its theoretical Normal percentile (xa), then
a plot of the data against its corresponding z percentile (za)would
yield a straight line with slope s and intercept
m!
Our example yields:
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Next: Joint Distributions
Previous: Other Continuous Distributions
Jayant Rajgopal