1. Show that x(i)
is the sample quantile of order i
/ (n + 1). . Virtual Laboratories > Random Samples > 1 2 3 4 5 6 7 [8] 9
Suppose that we observe real-valued data
x1, x2, ..., xn
from a random sample of size n. We are interested in the question of whether the data could reasonably have come from a continuous distribution (taking values in an interval) with distribution function F.
First, we order that data from smallest to largest (these are the observed values of the order statistics)
x(1) < x(2) < ··· < x(n).
1. Show that x(i)
is the sample quantile of order i
/ (n + 1). .
2.
Show that the distribution
quantile of order i
/ (n + 1) is
yi = F-1[i / (n + 1)]
If the data really do come from the distribution, then we would expect the points
(x(i), yi); i = 1, 2, ..., n
to be close to the diagonal line y = x; conversely, strong deviation from this line is evidence that the distribution did not produce the data. The plot of these points is referred to as a probability plot.
In the following exercises, we will explore probability plots for the normal, exponential, and uniform distributions.
3. In
the probability plot
experiment, set the sampling distribution to the standard normal
distribution and the sample size to n = 20. For each test distribution
given below, run the experiment 50 times and note the geometry of the
probability plot.
4. In
the probability plot
experiment, set the sampling distribution to the uniform
(0, 1) distribution and the sample size to n = 20. For each test distribution
given below, run the experiment 50 times and note the geometry of the
probability plot.
5. In
the probability plot
experiment, set the sampling distribution to the
exponential (1) distribution and the sample size to n = 20. For each test distribution
given below, run the experiment 50 times and note the geometry of the
probability plot.
Usually, we are not trying to fit the data to a particular distribution, but rather to a parametric family of distributions (such as the normal, uniform, or exponential families). We are usually forced into this situation because we don't know the parameters; indeed the next step, after the goodness of fit, may be to approximate the parameters. Fortunately, the probability plot method has a simple extension for any location-scale family of distributions.
Suppose that G is a given distribution function. Recall that the location-scale family associated with G has distribution function
F(x) = G[(x - a) / b],
where a is the location parameter and b > 0 is the scale parameter.
6. For p in
(0, 1), let zp denote the quantile of order p for G
and yp the quantile of order p for F.
Show that
yp = a + b zp.
From Exercise 6, it follows that if the probability plot constructed with distribution function F is nearly linear (an in particular, if it is close to the diagonal line), then the probability plot constructed with distribution function G will be nearly linear. Thus, we can use the distribution function G without having to know the location and scale parameters.
7. In the probability plot
experiment, set the
sampling distribution to normal distribution with mean 5 and standard deviation 2. Set the
sample size to n = 20. For each of the following test distributions, run the experiment 50
times and note the geometry of the probability plot:
8. In the probability plot
experiment, set
the sampling distribution to the uniform distribution on (4, 10) Set the sample size to
n = 20. For each of the following test distributions, run the experiment 50 times and note the
geometry of the probability plot:
9. In the probability plot
experiment, Set
the sampling distribution to the exponential distribution with parameter 3. Set the sample
size to n = 20. For each of the following test distributions, run the experiment 50 times and
note the geometry of the probability plot:
10. Draw the
normal probability plot with Michelson's velocity of light
data. Interpret the results.
11. Draw the
normal probability plot with Cavendish's density of the earth
data. Interpret the results.
12. Draw the
normal probability plot with Short's parallax of the sun
data. Interpret the results.
13. Draw the
normal probability plot for the petal length variable in Fisher's iris
data, using the
following cases. Compare the results.
From your experiments, we hope that you have reached a few general conclusions. First, the probability plot method is of very little use with small samples. With just five points, for example, it is essentially impossible to judge the linearity of the probability plot. Even with large samples, the results can be rather subtle. For example, a sample from a normal distribution frequently seems to fit the uniform distribution rather well. Experience with a variety of distributions helps in making the fine judgments.