Engineering statistics variance

Introduction to Statistics for Engineers Table 1.30 Analysis of variance... 

The characteristic bell shape of many RTDs can be fit to well-known statistical distributions. Hahn and Shapiro (Statistical Models in Engineering, Wiley, 1967) discuss many of the standard distributions and conditions for their use. The most useful distributions are the gamma (or Erlang) and the gaussian together with its Gram-Charlier extension. These distributions are represented by only a few parameters that can be used to determine, for instance, the mean and the variance. 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

I 5 Statistical Models in Chemical Engineering Table 5.44 Synthesis of the analysis of variances for two factors - Example 5.6.2. 

A probability density function that is an approximation to the biomodal distribution and is characterized by its mean being equal to its variance. See Mezei, L.M., Practical Spreadsheet Statistics and Curve Fitting for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, NJ, 1990 Dowdy, S.M. and Wearden, S., Statistics for Research, Wiley, New York, 1991 Balakrishnan, N. and Nevzorov, V.B., A Primer on Statistical Distributions, Wiley, Hoboken, NJ, 2003. 

The critical value for this test is given by Z, j = Zo 25 = 1-96. Hence, there is no statistical evidence to support the rejection of the nuU hypothesis, and the proper decision would be to purchase both weUs under the management interpretation of these results. However, the same clever engineer who questioned the first test results again questioned the validity of using o-j and al in the calculations. The question was then posed, Can we perform a similar test without knowing the population variances The statistician responded yes as the same example used to calculate Xj and X2 could be used to estimate erf and erf with Sj and SI, respectively. 

Limited space precludes a full discussion of these (and other) risk measures. Therefore, in the remainder of this section we will limit our remarks to the variance, a statistic that has proven most popular in use as well as in the literature of engineering economy. 

FIGURE 4.2.1 Various probability distributions important in biology. The normal distribution is used for most applications. The t-distribution is used for small sample sizes from a normal distribution. The log normal distribution fits some data better than a normal distribution. The F distribution is used to check equality of variances, and the yj- (chi square) distribution is used to check expected values of data. The curves shown here are for various values of distribution parameters. (From Barnes, J.W., Statistical Analysis for Engineers and Scientists A Computer-Based Approach, McGraw-Hill, New York, 1994.)... 

In engineering calculations, it can be important to determine how uncertainties in independent variables (or inputs) lead to even larger uncertainties in dependent variables (or outputs). This analysis is referred to as error analysis. Due to the uncertainties associated with input variables, they are considered to be random variables. The uncertainties can be attributed to imperfect measurements or uncertainties in unmeasured input variables. Error analysis is based on the statistical concepts of means and variances, considered in the previous section. 

---