1. 化工應用數學 授課教師： 郭修伯 Lecture 2 實驗數據的分析

2. 實驗精確程度 The degree of accuracy sought in any investigation should depend upon the projected use of the results, and the accuracy of the required data and calculations should be consistent with the desired accuracy in the results. It is desirable to complete the investigation and obtain the required accuracy with a minimum of time and expense. The accuracy of a number representing the value of a quantity is the degree of concordance between this number and the number that represents the true value of the quantity; it may be expressed in either absolute or relative terms.

3. 誤差來源 Accidental errors of measurement –Such errors are inevitable in all measurements and that they result from small unavoidable errors of observation due to more or less fortuitous variation in the sensitivity of measuring instrument and the keenness of the senses of perception. （例如用了不準的 A 來校正 B ，用 B 的錯誤校正曲 線進行量測） Precision and constant errors –A result may be extremely precise and at the same time highly inaccurate. –Constant errors can be detected only by performing the measurement with a number of different instruments and, if possible, by several independent methods and observers. （例如用了不準確的儀器或樣品取樣在不具代表 性的地方） Errors of Methods –These arises as a result of approximations and assumptions made in the theoretical development of an equation used to calculate the desired result. （例如在計算時，用的錯誤的假設）

4. Variance and distribution of random errors If an experimental measurement is repeated a number of times, the recorded values of the measured quantity almost invariably differ from one another. The data so obtained may be used for two purposes: –to evaluate the precision of the measurement –to obtain an estimate of the probability that the mean of the measurement differs from the true value of the measured quantity by some special amount The “scatter” of the repeated measurements of the quantity is commonly reported in terms of the “variance” or “standard deviation” of the sets of measurements.

5. Sample variance and standard deviation Sample mean Sample variance Sample standard deviation

6. Population variance and standard deviation Population mean Population variance Population standard deviation

7. Population and sample Population Sample 1

8. Normal frequency distribution If an infinite data set the variation in x are random, it was first shown by Gauss that the distribution of values of x about the population mean is given by –f is frequency, or probability of occurrence, of a value of magnitude x.

9. Normal frequency distribution The probability that a single measurement will give a value lying between x - dx/2 and x + dx/2 is The probability is less than 5% that a single measurement of x will differ from by more than twice the standard deviation, i.e. by more than  2 . The range  2  is frequently called the “95 percent confidence belt on x”.

10. More about variance... The sample mean is the best estimate of the population mean. The sample variance is not the best estimate of the population variance. A better estimate is given by sample Population Sample estimate of the population variance

11. Number of measurment By how much can the mean of n measurement be expected to differ from the best value of the population mean? Population, Set 1, k times, s 1 2 Set 2, k times, s 2 2 … n sets Estimate of the set of means, Estimate variance of the set of means,

12. Sample variance of the mean The grand mean The sample estimate of the variance of the set of means The sample estimate of the variance of the set of means may be estimated by a single set sample Population (Sample 猜測的 population mean) (Sample 猜測的 population variance)

13. Confident limits for small samples To associate the magnitude of deviations of from the population mean with the probability of the occurrence of such deviations. It is known that if the sample set contains at least 20 entries, the error introduced by the previous slide is not serious. For smaller samples, however, s 2 /k is not an adequate estimate of  m 2.

14. Student’s t test The solution to small number of entries was first pointed out in Biometrika, Vol. VI, 1908, by W.S. Gossett, who signed his article “STUDENT”. The dimensionless quantity of particular interest in a confidence-limit analysis is called “Student’s t”: It involves estimates obtained from a sample of finite size.

15. Student’s t test t is the difference between the measured sample mean and the true population mean divided by the sample estimate of the standard deviation of the population of means. “Student” derived the frequency distribution for t: –C f is a function of f only –f is the “degree of freedom”, defined as the number of values used to calculate the means on which t is based, less the number of means so calculated.

16. Student’s t test The distribution funcion of t is used to calculate the probability value of the size of the sample (the degrees of freedom f). Probability calculation of this kind have been carried out over a wide range of conditions, and the results are tablulated in the handout given in the course. sample Population t

17. T test 範例 Two methods were used to measure a quantity. It is desired to use these data to obtain the following information:

18. T test 範例（續） The confidence limits to be assigned to the results of procedures 1 and 2 The significance of the difference between the mean values of the results of procedures 1 and 2 The “best value” to be assigned to the sample analysis The confidence limits of the best value

19. T test 範例（續） Confidence limits –The sample mean of procedure 1 is easily found to be 56.6 –The external probability limit on t will be arbitrarily set at 0.05, corresponding to 95 percent confidence limits. –Five measured values are used and one mean is calculated, degree of freedom: f = 5 - 1 = 4 –From the “t table” for f = 4, values of t lying outside  2.776 only are 0.05 probable. –The 95 percent confidence limits on t are: – –The 95 percent confidence limits on x 1 are then

20. T test 範例（續） The significant of the difference –The mean value obtained by procedure 1 is 56.6 with 95 percent limits of –The mean value obtained by procedure 2 is 55.5 with 95 percent limits of –The sample means are different, which might be taken to indicate a systematic difference or bias between the two methods of analysis. –The 95 percent confidence limits analysis shows that the mean of sample 2 is included with the confidence limits of sample 1, and vice versa. –It may be concluded that the difference between the two means has no statitical significance.

21. T test 範例（續） The best value? –Since the difference between the mean value obtained by procedure 1 and that by procedure 2 is not statistically significant, the best value to be assigned to the sample analysis is a weighted combination of the two mean values. –If the difference between the means has been significant, it would have been concluded that one or both of the procedures were affected by non-random factors (errors of method or bias). In this case, the best value cannot be estimated.

22. T test 範例（續） The best value obtainable from a series of sets of measurements exhibiting statistically equivalent means is given by the number of measurements in the i th set Population variance

23. T test 範例（續） The confidence limits of the best value –the variance of the best value is needed: –the degree of freedom: f = 10 - 2 = 8 –the t limits are  2.3 –Consequetly, the 95 percent confidence limits on the best value are

24. Other test ( L test) L test: –A calculation to determine the probability that the samples represent normal populations exhibiting the same population variance  2, but without regard to the population means

25. Other test (F test) F test: –In a single set of data, two types of error are possible - random errors and errors of method or bias. The magnitude of the random errors is estimated by the with-sample or error variance. However, from the single set of data, no estimate of the error due to method is possible. Suppose that other sets of data are available which are likewise subject to both random and method errors. For each data set, the within-sample or error variance may be calculated. –The statistic F is the ratio of the variance which contains both random and method error to the variance which includes random errors only. –The magnitude of F is a measure of the importance of errors of method which differ from one data set to the next.

26. 最小平方法 (least squares) Recall: A best straight line as the one for which the sum of the squares of the error terms is a minimum. The best measure of the precision with which the points fit the line is the variance of estimate: The estimate of the error variance of Y i is The confidence limits of Y i is (  t ) s e 2 (Y i ) degrees of freedom (2 = a, b)