Every sample has some intrinsic dispersion which determines the overall possible range of values for some attribute. The bigger this range, the more difficult it is to accurately characterize the sample with just a few data points.

For instance, suppose I go to my son's second grade class and I want to determine the mean age of the class. Well, a priori I know that kids of age 6 to 8 are in the second grade and so, to determine the mean age, I really only need to ask 2 or 3 kids, essentially to verify that this is a normal second grade class and not filled with small 40 year olds.

However, to determine the mean age of the students in this class would require considerably more samples, because the dispersion or range of ages is larger.

The measure of a dispersion is a somewhat mathematically complex procedure which you don't need to know. This will be done automatically for you using the provided tools. What you need to do is to understand how to interpret it.

The measure of dispersion assumes that the sample can be adequately represented by a normal or guassian distribution. We will discuss this in detail later but for now we can assume that most samples are adequately represented in this way. Let's return to the rainfall example and show the results of fitting the data to a mean value plus a dispersion. This is shown here:

The mean value for this data is 51.5 inches and the dispersion is 8 inches. So how do you interpret this?

A measure of dispersion is also a measure of probability. It is DEFINED in such a way that +/- 1 dispersion unit (usually called 1 sigma) contains 68% (about 2/3) of the sample. In the rainfall example given above, the fit to the data means that 68% of the time in eugene the average annual rainfall is between:

43.5 and 59.5 inches.

Furthermore, there is no significant difference for quantities that are separated by less than one dispersion unit. That is, it would be inappropriate to say that a 55 inch annual rainfall is significantly above normal.

A dispersion is then a measure of the statistical fluctuation around some mean quantity. You must have knowledge of this if you are to identify an event as being significant. Here is a simple guide:

• Differences of +/- one sigma (one dispersion unit) are not significant
• 95% of a sample is contained within +/- 2 dispersion units. Differences of +/- two sigma are marginally significant (and because of this, always difficult to interpret)
• 99.9% of a sample is contained within +/- 3 dispersion units. Differences of +/- three sigma are statistically significant.

Let's return again for rainfall. We have determined that the 25 year average was 51.5 inches. Last year, 1995, Eugene set a new annual rainfall record of 65.56 inches. Is this significant? Well

65.56 inches is 65.56 - 51.5 inches above normal or about 14 inches above normal. For this data set, one dispersion unit is 8 inches and so 14 inches represents 14/8 = 1.75 sigma and is therefore not all that significant. In otherwords, if 51.5 +/- 8 inches represents the normal rain fall, then its rather probable that some year will have 65 inches. But, it would be very improbable for some year to have 51.5 + (3*8) = 75 inches since that value is 3 dispersion units away from the mean and would happen less than 99.9% of the time (meaning one time in one thousand years).

Example Let's return to the example of the 6 people in the store with ages distributed as follows:

23	33	43	25	37		51

Use the Tool

I find that the dispersion in this population is 23 years.

Now a sample size of 6 does not represent a sufficiently large sample to accurately determine a dispersion (as we will see later, minimum sample sizes are N=25--30). If I thought that 6 did form a representative sample then in the future I would expect to see people in my store covering the age range 35 +/- (2*23) since that represents 95% of the population, according to my statistical sampling. This implies a range of ages from -6 to 81 years. The last time I looked I didn't see any pre-born people in the store.

Hence, there are some physical limits that you can apply to some one's estimation of the dispersion in a population to check that value.

Following are plots showing normal distributions with differeing dispersions to give you a graphical feeling for how dispersion is equated with the range of the data.





In many cases, there are physical limits which slightly skew the distribution. For instance, if I measure the size of bacteria in a test tube, I know there will be no bacterize of size zero. If I measure flood levels, there will be no such thing as a zero size flood. Those kinds of data tend to produce distributions like this



Although its not symmetrical, the same general principles of dispersion measurement can be applied with high accuracy.

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