Comparing Means and Deviations

# ENVS 202 On-line Access

In class exercise. Class do the following:

1. Draw a number from the Sacred Box of Sampling in the front of the Lecture room
3. Be prepared to tell the instructor your number if your seat numbers is randomly called
4. That is all

In the Sacred Box of Sampling there are 170 numbers which define this Intrinsic Distribution .

The point of the in class exercises is to demonstarte that a random sampling process done for an intrinsic distribution which is normally distributed (i.e. a bell curve) will provide a robust estimate of the mean and dispersion after just a small number of samples.. While this can be proved with calculus (and in statistics is known as the Central Limit Theorem), the in class example is the probably the best means of demonstrating this.

For this sample as a whole:

• The mean = 175
• The dispersion = 23

The point of the demo in the class is to see how close we can come to recovering this population mean and dispersion from the sample mean and dispersion.

## Comparing Sample Means - What is Significance?

Summary from last time:

• The Sample Mean - Numerical measure of the average or most probable value in some distribution. Can be measured for any distribution knowing the mean value alone for some sample is not very meaningful.

• The Sample Distribution - Plot of the frequency of occurrence of ranges of data values in the sample. The distribution needs to be represented by a reasonable number of data intervals (counting in bins). Refer to the Rainfall Distribution example or for another example of histograms and distributions go here

• The Sample Dispersion - Numerical measure of the range of the data about the mean value. Defined such that +/- 1 dispersion unit contains 68% of the sample, +/- 2 dispersion units contains 95% and +/- 3 dispersion units contains 99.7%. This is schematically shown below:

Refer to document on dispersions for more detail.

In general, we map dispersion units on to probabilities

For instance:

• The Probability that some event will be greater than 0 dispersion units above the mean is 50%
• The Probability that some event will be greater than 1 dispersion units above the mean is 15%
• The Probability that some event will be greater than 2 dispersion units above the mean is 2%
• The Probability that some event will be greater than 3 dispersion units above the mean is 0.1% (1 in 1000)

The calculation of dispersion in a distribution is very important because it represents a uniform way to determine probabilities and therefore to determine if some event in the data is expected (i.e. probable) or is significantly different than expected (i.e. improbable).