The dispersion D(x) of the random variable is math1ematical expectation of this random variable calculated according to the formula: [x-M(x)]^2. For the continuous random variable x it looks as follows:
For the random variable with n possible values we get:
The dispersion for the sample selection including n values of the random variable is calculated according to the following formula:
This value is called standard sample variance S.
Dispersion is a very convenient and natural way of statistical analysis, because it considers all deviations of the results from the average and normalizes them accordingly.
The results of the repeated experiment and the corresponding random measuring errors are usually characterized with two statistical criteria:
- The width of the confidence interval [x1, x2], where the results of individual experiments may fall;
- The confidence probability of the fact these results will not fall beyond the interval.
During the statistical analysis and statistical data processing the random variable may have normal or close to normal distribution (as you remember, we mentioned this in the very beginning of our discussion), while the sample selection representing it, appears too small, i.e. is not representative enough. This part of math1ematical statistics devoted to less representative samples (s=<n<20) is also known as micro-statistics.
Micro-statistical estimates of the normally distributed random values are based on Student’s distribution, which links together three major parameters of the sample selection: width of the confidence interval, the corresponding confidence probability and the sample size or the number of sample freedom degrees f=n-1.
This is what the dependence of the probability density on the width of the confidence interval t in Student’s distribution for different number of freedom degrees.
When f is infinite, our curve coincides with the curve for normalized standard distribution. But the fewer degrees of freedom are involved, the flatter is the graph for large |t| values (it gets to the x axis slower). As a result, if we have the same width of the confidence interval, the confidence probability according to Student’s distribution is always lower than the confidence probability of the Gaussian-Laplace normal distribution. Moreover, the less representative is the sample, the higher is the estimates difference.
The confidence estimate of the average result in Student’s distribution looks as follows:
- is the math1ematical expectation of the average result,
- is the confidence probability of the random error of n independent experiments being below