Comparing Means and Deviations

ENVS 202 On-line Access

Comparing Sample Means - What is Significance?

Summary from last time:

In general, we map dispersion units on to probabilities

For instance:

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).

Now that we have an understanding of means and dispersions we have a simple way for determining if two distributions are fundamentally different. Again let's use the example of rain.

Seattle Eugene

mean = 51.5 inches

mean = 39.5 inches

dispersion = 8.5

dispersion = 7.0

On average, does it rain significantly more in Eugene than Seattle?

Here is the wrong way to do this problem:

A proper comparison makes use of a tenant of statistical theory which states that

    The error in the mean is calculated by dividing the dispersion by the square root of the number of data points.

Seattle Eugene

mean = 51.5 inches

mean = 39.5 inches

dispersion = 8.1

dispersion = 7.0

N = 25

N = 25

error in mean = 8.1/5

error in mean = 7.0/5

error in mean = 1.6

error in mean = 1.4

The difference in mean rainfall between Seattle and Eugene is (51.5 - 39.5) = 12 inches which is 12/1.6 = 7.5 dispersion units difference in the mean value.

Thus there is a highly significant difference in the mean annual rainfall between Eugene and Seattle.

Note this method is only an approximation. A more exact and proper way to compare two sample means will be given later.

Another way to look at this rainfall comparison is as follows:

We have already determined that 65 inches is not a significant amount of rainfall in Eugene compared to the normal value of 51.5 inches. Would 65 inches be a significant amount of rain in Seattle?

For the case of Seattle, 65 inches is 65-39.5 = 26.5 inches above normal. The dispersion in the Seattle data is 7 inches and so 26.5 inches is 26.5/7 = 3.8 dispersion units above the mean. This is highly significant which again reinforces the notion that there is a significant difference in mean rainfall between

Eugene and Seattle (note also this difference in community web pages).

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