Squared errors 375 -count

Figure 8. Root Mean Square Error (RMSE) and Revisit Count for one vs. two step ahead beam and waveform scheduling.

Flow cytometer cell counts are much more precise and more accurate than hemocytometer counts. Hemocytometer cell counts are subject both to distributional (13) and sampling (14—16) errors. The distribution of cells across the surface of a hemocytometer is sensitive to the technique used to charge the hemocytometer, and nonuniform cell distribution causes counting errors. In contrast, flow cytometer counts are free of distributional errors. Statistically, count precision improves as the square root of the number of cells counted increases. Flow cytometer counts usually involve 100 times as many cells per sample as hemocytometer counts. Therefore, flow cytometry sampling imprecision is one-tenth that of hemocytometry. 

Under the simplest conditions, the standard counting error is approximately equal to the square root of the total number of counts or... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

The last column of Table 10-5 shows (1) that the long-term source of variation clearly overshadows the short-term, the ratio of variances exceeding 70 and (2) that the short-term variance is comparable with Arx, which is, of course, the standard counting error squared (Equation 10-8). 

Each abundance was divided by the abundance of that element (except for Rh) in Type / carbonaceous chondrites. Rh abundances were divided by Rh abundances in other types of chondrites as Cl values were not available. Errors in the LBL measurements reflect 1 a values of the counting errors, except for the Au error. The latter is the root-mean-square deviation of six measurements, because the six values were not consistent within counting errors. The Os measurement was on a HNO,-insoluble residue that had been fired to 800°C. Key , this work and O, previous work of Ganapathy. 

The Tb abundance in meteorites is assumed to be 0.5 ppm. Errors for the Danish Cretaceous (3 samples) and Tertiary (3 samples) HNO,-insoluble residues are root-mean-square deviations. Errors for the HNO,-insoluble residues from the Gubbio and Danish boundary layers are 1 a values of the counting errors. Key , Gubbio boundary layer residue O, Danish Cretaceous residues < >, Danish Tertiary residues and , Danish... 

The calibration was represented in the computer program by a fifth-degree polynomial. The conventional method of least-squares was followed to determine the coefficients of the polynomial. The sensitivity of the normal equations made round-off error a significant factor in the calculations. The effect of round-off error was greatly reduced when the calculations were performed with double-precision arithmetic. The molecular weights corresponding to selected count numbers were calculated from the coefficients. The coefficients were input information for the data-reduction program. 

Figure 3. Relationship between leaf area (A), epidermal cell density (B), stomatal density (C) and stomatal index (D) versus altitude for Nothofagus solandri leaves growing on the slope of Mt. Ruapehu, New Zealand (collected in 1999). Black diamonds indicate the mean of ten counting fields on each leaf, white squares are the averages of five to eight leaves per elevation, with error bars of 1 S.E.M. Nested mixed-model ANOVA with a general linear model indicates significant differences for all factors (p = 0.000). Averages per elevation were used for regression analysis A. y = -0.0212 + 73.1 R2 = 0.276 p = 0.147. B. y = 1.70 + 3122 R2 = 0.505 p = 0.048. C. y = 0.164 + 360 R2 = 0.709 p = 0.009. D. linear (dashed) y = 0.004 + 9.33 R2 = 0.540 p = 0.038 non-linear (solid) y = 0.00001 2 - 0.0206 + 21.132 R2 = 0.770.

Counting statistics. The statistical standard deviation of the number of counts in the photopeak is equal to the square root of the integrated photopeak counts. For most useful determinations it is assumed that sufficient activity is produced in order to assure errors due to counting statistics are less than 1%. 

In order to obtain a Larmor resonance line we have to vary the frequency of the microwave field and count the number of spin Hips per unit time. In order to avoid saturation effects the microwave field amplitude was kept low. The resonance curve obtained in the described manner is rather asymmetric. The lineshape can be described using the known spatial configuration of the magnetic field and a thermal distribution of the axial energy. A least squares fit to the data points as shown in Fig. 9 leads to a fractional uncertainty of about 10-6 and the g factor can be quoted with the same error [9]. 

There are two important statistical tests which can be used to determine whether the differences between two sets of data are real and significant or are just due to chance errors. Both assume that the experimental results are independently and normally distributed. One is the t-test and the other is the chi-squared test. The t-test applies only to continuous data-usually a measurement. The chi-squared test in many cases can approximate frequency or count type of data. 

Expression and Interpretation of Results. Archaeological interpretation of a radiocarbon age may depend critically on the error associated with that age. Errors are commonly expressed as a variance range attached to the central number (e.g., 2250 80 years). The 80 years in this example may correspond to the random error for a single analytical step. Both decay and direct-atom counting are statistical in nature, and lead to errors that vary as the square root of the number of counts. The error may also be expressed as the overall random experimental error (the sum of individual errors.) Overall random error can be determined only by analyzing replicate samples. 

The final measurement can contribute considerably to imprecision and for optical immunoassays is instrument-related. For radioimmunoassays, considerable improvements in both time and precision may be made by employing y-labelled drugs and counting in a 16-head y-counter. Scintillation counters have an accuracy of measurement which is approximately the square root of the total number of counts, e.g. about 1% counting error when measuring 10000 counts. 

The analytical errors in [ Th/ U] and 5 U(m) generally follow counting statistics, such that the fractional error is inversely proportional to the square root of the number of counts per analysis. At the 2a level ... 

There are two sources of error that have to be taken into account when assessing the experimental error in a measured intensity. There are errors that arise from the random fluctuations in the detection system these errors follow a Poisson distribution and are proportional to the square root of the measured value (the count, hence the term counting statistics ). The total error in an intensity, represented by the estimated standard deviation (e.s.d.) cr(I), is approximated by counting statistics, and the second source of error, the instrumental uncer-... 

The sample area however is difficult to measure due to the air inclusions (holes) and the rough boundaries of the sample. Therefore we scanned the samples with a HP deskscan (ScanJet 3c) at highest resolution (600 dpi) to a pcx-file. Every pixel corresponds to a square of 42.3 pm X 42.3 pm. A determination of the surface area is possible by a summation of the number of black pixels (i.e. the pixels that are covered by the sample area). The error is due to the fact, that the CCD-elements of the scanner counts a pixel as black if more than 50 % (but not necessarily all) of the pixel area is covered by the sample. This happens only at the border of the sample. Therefore the error is as low as 0.5 %. 

The scattered intensity is usually represented as the total number of the accumulated counts, counting rate (counts per second - cps) or in arbitrary units. Regardless of which units are chosen to plot the intensity, the patterns are visually identical because the intensity scale remains linear and because the intensity measurements are normally relative, not absolute. In rare instances, the intensity is plotted as a common or a natural logarithm, or a square root of the total number of the accumulated counts in order to better visualize both strong and weak Bragg peaks on the same plot. The use of these two non-liner intensity scales, however, always increases the visibility of the noise (i.e. highlights the presence of statistical counting errors). A few examples of the non-conventional representation of powder diffraction patterns are found in the next section. 

Obviously, the quality of intensity measurements in powder diffraction is inversely proportional to the statistical measurement errors and, therefore, it is directly proportional to the square root of the total number of registered photon counts. Assuming constant brightness of the x-ray source, the most certain way to improve the quality of the diffraction data is to use a lower scanning rate or longer counting time in continuous and step scan experiments, respectively. 

The fixed monitor count used in the data collection is chosen to optimise its statistical errors. Spectra are accumulated over several hours or overnight (or longer) for weak samples. On FANS a good signal to noise spectrum from co 8 g of a typical organic compound can be recorded in about five hours (excluding the time to cool the sample ( 3.5.2)). The errors in INS spectroscopy are, as in infi ared, Raman and NMR, governed by Poisson statistics, so the error on a data point of n counts, is fn This tyranny of the square root requires the measurement time to double before the statistics improve by only V2. ... 

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