Confidence single measurement

The bias error is a quantity that gives the total systematic error of a measuring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The repeatability error consists of the confidence limits of a single measurement under certain conditions. The mac-curacy or error of indication is the total error of the instrument, including the... 

Standard error of the mean (SEMI Is a statistical assessment of the increased confidence in a result ighen ft isthe meert of a iiQfinM numw or. replicates rather than a single measurement... 

As can be seen in the formula of the previous slide the confidence limit depends very much on the number of measurements. This is demonstrated in this slide showing the confidence limits for a single measurement, for three and for ten measurements... 

The calculation of standard deviation (Equation 5-3) is the same whether you have many measurements or only a few sample size affects only the selection of the t value. For a single measurement taken at random from a small number (n) of measurements, the confidence interval for the desired confidence level is... 

The parameter k is a proportional constant between Ex and the standard deviation. The percent proportional error may be defined within several probability ranges. Standard error refers to a confidence level of 68.3% that is, there is a 68.3% chance that a single measurement will not exceed the °/oEx. For standard error, k = 0.6745. Ninety-five hundredths error means there is a 95% chance that a single measurement will not exceed the °/oEx. The constant k then becomes 1.45. 

This may be described as a 95 percent confidence limit for an individual measurement and is illustrated in Fig. 46. The relationship between a specified uncertainty and the corresponding confidence limit is shown in Table 2. There is a 68.26 percent probability that the walue obtained in a single measurement will lie between ft a and /t + cr. In contrast, the level of confidence that a single. v value will lie between — 1.96cr and /x + 1.96a- is 95 percent. 

Moreover, confidence limits on measured values can be estimated to a good approximation using the mean difference between duplicates. For example, if a single measurement of Y yields a value Kmeasurcd. then there is a 95% probability that the true value of Y falls within the 95% confidence limits (Kmeasured 1-74D,) and (Tmcasured + 1.74Dj). For a measured flow rate of 610 g/min, estimate the 95% confidence limits on the true flow rate. 

Confidence-building measures. To reduce the likelihood that verification data may be misinterpreted, each State party will voluntarily notify the Technical Secretariat of any single chemical explosion using 300 tonnes or more of TNT-equivalent blasting material on its territory. In order to calibrate the stations of IMS, each State party may liaise with the Technical Secretariat in carrying out chemical calibration explosions or providing information on chemical explosions planned for other purposes. 

The Limit of Detection is the smallest concentration/amount of a component of interest that can be measured by a single measurement with a stated level of confidence. This subject is discussed in detail elsewhere (22). 

The data from these experiments were analyzed using the statistical methods described in Chapters 5, 6, and 7. For each of the standard arsenic solutions and the deer samples, the average of the three absorbance measurements was calculated. The average absorbance for the replicates is a more reliable measure of the concentration of arsenic than a single measurement. Least-squares analysis of the standard data (see Section 8C) was used to find the best straight line among the points and to calculate the concentrations of the unknown samples along with their statistical uncertainties and confidence limits. 

If we make a single measurement x from a distribution of known a, we can say that the true mean should lie in the interval x zcr with a probability dependent on z. This probability is 90% forz = 1.64, 95% fore = 1.96, and 99% forz = 2.58, as shown in Figure 7-lc, d, and e. We find a general expression for the confidence interval of the true mean based on measuring a single value x by rearranging Equation 6-2. (Remember that z can take positive or negative values.) Thus,... 

Table VI gives data obtained for RD 1333 lead azide. A statistical analysis using the 11 data points in the table gives a mean of AT of 15°C with a standard deviation of 2°. The 95% confidence interval is 1.3°. The importance of making more than one measurement is made clear because the 95% confidence interval for a single measurement using the same mean and standard deviation is 4°. If we compare data for lead azides (Table VII) for heating rates of 5° and 10°/min, dextrinated material is significantly more sensitive than the other products tested. The explosion temperature is also significantly lower. A higher heating...

Figure 5.2 Calibration line and 95% confidence intervals on concentrations calculated from measurements of unknowns. Data are from calibration of a glucose monitor, (a) Six calibration solutions with a single measurement of the test solution (b) six calibration solutions with three measurements of the test solution (c) three calibration solutions with a single measurement of the test solution. 

In practice, a finite number (n) of measurements are made, all of which would be identical were it not for the random errors involved in each measurement. From this finite (and often small) set of measurements, one estimates the most probable value of the function by averaging the experimental results. The question is if another single measurement were made, how close would this last measurement be to the mean of the previous measurements The answer, of course, is that this depends on the broadness of the distribution function for these measurements, as well as on the degree of confidence to be placed on the answer. For this purpose, one calculates the standard deviation of the individual sample determination, using... 

If the masses of the beans follow a normal distribution, the interval [fj. - 1.96(7, p + 1.96cr] contains 95% of all the possible observations. This means that the single measurement 0.1188 g has a 95% probability of being within this interval. Of course this also implies that there is a 5% chance that it falls outside. If we assume the normal model holds, we can say that we have 95% confidence in the double inequality... 

Due to the directness of the VM method, the exactness of the single measurements should be high. However, to describe the true shape of the population profile a high number of measurements is needed for statistical reliability. To achieve an error of 5% at the 95% confidence level, 740 droplets must be counted (86). 

The overall random error of an analytical process, expressed as the variance v, can be regarded as the sum of two other variances, that due to sampling x), and that due to the remaining measurement components of the process sb. These variances are estimates of the corresponding population variances cr" and its components (tT and <75. The overall standard deviation s is calculated as usual from single measurements on each of h sample increments, and the confidence limits of the true mean value of the population 11 are obtained from ... 

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