lecture 4

The mean, mode and median do a nice job in telling where the center of the data set is, but often we are interested in more. For example, a pharmaceutical engineer develops a new drug that regulates iron in the blood. Suppose he finds out that the average sugar content after taking the medication is the optimal level. This does not mean that the drug is effective. There is a possibility that half of the patients have dangerously low sugar content while the other half has dangerously high content. Instead of the drug being an effective regulator, it is a deadly poison. What the pharmacist needs is a measure of how far the data spread apart. This is what the variance and standard deviation do.
The most common measures of dispersion or variability are (Range, Variance, Standard deviation and Coefficient of variation).

Range (R):
The range is the difference between the largest (XL) and the smallest (XS) values in a set of observations.

R = XL - XS

Note: The range is poor measure of dispersion? Because it only takes into account two of the values.

Variance:
The variance is the most commonly used to measure of spread in biological statistics. For a population is defined as the sum of squares of the deviation from the mean (SS), dividing by the total number of the deviations, and by one less than the total number of the deviation (degree of freedom, df) for a sample.


or ……..……….. (Population)
or ……..……..….. (Sample)
Notes:
1. The variance is a measure that uses the mean as a point of reference.
2. The variance is small when all values are close to the mean.
3. The variance is large when all values are spread away from the mean.



Why the separate formula for sample?
The formula for sample divided by n-1 to:
1. Correct for probability that most extreme cases will be excluded from a smaller sample.
2. Makes the sample more representative of the population for every small sample.
3. Reduces the denominator to a larger extent (If n=5 they we have a 20% reduction in the denominator). But in large samples the n-1 correction does not have as large effect.

Example:
We want to compute the sample variance of the following sample values:
10, 21, 33, 53, 54.
Solution: n=5

* First method:






* Second method:









10
21
33
53
54 -24.2
-13.2
-1.2
18.8
19.8 585.64
174.24
1.44
353.44
392.04






* Third method:


10 21 33 53 54


100 441 1089 2809 2916






Standard Deviation (s) or (sd):
Is defined as a positive square root of variance.

……………. (Population) ……………… (Sample)

The mean squared difference from the sample mean will, on average, underestimate the population variance. In some samples, it will overestimate it, but most of the time it will underestimate it, if the formula is modified so that the sum of squared deviations is divided by n-1 rather than N, then the tendency to underestimate the population variance is eliminated.