There are several parameters associated with a continuous probability model that will prove useful. These are measures for probability models, so we will classify them into location and scale. To visualize these parameters, consider the probability model in Figure 5.6.
While reading the material below try to locate the parameters on it. Some of these will be "guesses". The answers will be given at the end.
Also consider the following random sample from this model, (I obtained this sample).
24.0720 7.4773 18.4161 10.1440 10.4192 7.0660 3.9749 0.5231 7.9999 28.7453 6.0621 12.3729 5.3767 23.1134 7.9949 16.0966 10.4424 4.1915 8.4586 9.9292As we proceed, calculate the statistics based on this sample which are estimates of the parameters. In fact, the stem leaf plot or the histogram are estimates of the probability curve. These estimates (stem leaf, histogram) are poor because the sample is so small.
The median , ,
is the point located along the horizontal
axis of the probability model which divides the probability mass in two.
That is, half the time the variable is less than
the time the variable is greater than .
Okay! Locate the median
on the probability model above.
Calculate the sample median, Q2. This is the estimate of the median you located on the probability model above.
The mean , , is the center of gravity along the horizontal axis of the probability model. This is the point where the probability mass would balance. Obviously if the probability curve is symmetric, the mean and median would agree . Locate the mean on the probability model above.
The first quartile , q1, is the point located along the horizontal axis of the probability model which divides the probability mass into 1/4 (to the left of q1) and 3/4 (to the right of q1) . Similarly, the third quartile, q3 ,
is the point located along the horizontal axis of the probability model
which divides the probability mass into 3/4 (to the left of q3) and 1/4 (to the right of q3). Their difference,
iqr = q3 - q1, is called the interquartile range
of the probability model. The interquartile range is a parameter. The interquartile
range is a scale or noise parameter. Locate the quartiles on your probability
A second scale parameter is the population standard deviation , . Recall that we gave a formula for it in the discrete case. We would have to use Calculus for the continuous case. But we can discuss it. The sample standard deviation, s , is an estimate of . Calculate your estimate.
|Parameter||Parameter Value||Estimate Based on Sample|
|First quartile||q1 = 5.84||6.31|
|Third quartile||q3 = 14.16||15.17|
|Interquartile range||q3 - q1 = 8.32||8.86|