Q2Power System Instrumentation

(a) Numerical: Probable Error and Standard Deviation Calculations

Given: standard deviation of 100 readings, sigma=1.380 degrees C; number of readings, n=100.

(i) Probable Error

The probable error of a single reading is defined as the deviation within which there is an equal (50%) chance that any given reading's error lies inside or outside this range, and is related to the standard deviation by the constant factor 0.6745 (derived from the properties of the normal/Gaussian error distribution):

(ii) Standard Deviation of the Mean

The standard deviation of the mean (representing the uncertainty in the estimated average value, as distinct from the spread of individual readings) decreases as the number of readings increases, following the relation:

(iii) Probable Error of the Mean

Applying the same 0.6745 factor to the standard deviation of the mean (equivalently, dividing the single-reading probable error by the square root of n):

(iv) Standard Deviation of the Standard Deviation

The standard deviation of the standard deviation itself (representing the uncertainty in the estimated value of sigma, given only a finite number of readings) is given by:

Summary of results: probable error (single reading) approximately 0.931 degrees C; standard deviation of mean approximately 0.138 degrees C; probable error of mean approximately 0.093 degrees C; standard deviation of standard deviation approximately 0.098 degrees C. These four quantities together allow a complete statistical characterization of the measurement uncertainty for both individual readings and their computed mean, forming the basis for reporting a measured value together with an appropriately quantified confidence interval.

Origin of the 0.6745 Constant

The constant 0.6745 used to convert standard deviation into probable error is not an arbitrary factor but is derived directly from the mathematical properties of the normal (Gaussian) probability distribution. Probable error, r, is defined as the deviation from the mean such that exactly half of all readings (50% probability) are expected to fall within plus-or-minus r of the mean, and the other half fall outside this range. For a normal distribution with standard deviation sigma, the fraction of the total area under the probability curve lying within plus-or-minus z standard deviations of the mean is given by the error function, and setting this cumulative area equal to exactly 0.5 (50%) and solving for z yields z is approximately equal to 0.6745. In other words, 0.6745 standard deviations on either side of the mean bounds exactly the central 50% of the area under the standard normal curve, which is precisely the defining condition for probable error. This is why r=0.6745*sigma, and by the same reasoning, the probable error of the mean is obtained by applying the identical 0.6745 factor to the standard deviation of the mean, since the sampling distribution of the mean is itself assumed to be normally distributed (or very nearly so, by the central limit theorem, for a reasonably large number of readings such as the n=100 used in this problem).

Practical Use of These Statistics in Calibration Certificates

In formal calibration and test reporting practice, a measured quantity is rarely stated as a single bare number; it is instead reported together with a quantified statement of uncertainty, and the statistics computed in part (a) directly supply the basis for this. A calibration certificate for a temperature-measuring instrument, for example, would typically report the mean of the repeated readings as the certified value, together with the standard deviation of the mean (or an expanded uncertainty derived by multiplying it by an appropriate coverage factor) as the associated uncertainty figure, so that a user of the certificate knows both the best estimate of the true value and the confidence that can be placed in it. The probable error is a historically important, closely related alternative expression of the same underlying dispersion, once widely used in physical metrology and instrumentation textbooks specifically because a 50% probable-error band is intuitively interpreted as a 'coin-toss' confidence interval, whereas modern calibration practice has generally shifted toward reporting standard uncertainty (one standard deviation) or expanded uncertainty at a 95% confidence level (approximately two standard deviations), in line with international guidelines such as the Guide to the Uncertainty in Measurement (GUM). Regardless of which specific convention is adopted, the underlying statistical quantities — standard deviation, standard deviation of the mean, and probable error — remain the essential building blocks from which any such calibration uncertainty statement is constructed, and the standard deviation of the standard deviation additionally indicates how reliable the stated sigma value itself is, which becomes particularly relevant when comparing calibration results obtained from a comparatively small number of repeated readings across different laboratories or test occasions.

(b) Accuracy, Precision, Hysteresis, and Probable Error with Examples

Accuracy: the closeness of a measured value to the true (actual) value of the quantity being measured, indicating the degree of freedom from systematic error. For example, a voltmeter that consistently reads 100.0V when the true, independently-verified voltage is exactly 100.0V is highly accurate, even if its repeated readings show some scatter (variation) among themselves; accuracy is typically expressed as a percentage of full-scale reading or as a percentage of the true value.

Precision: the degree of repeatability or consistency among a set of repeated measurements of the same quantity under the same conditions, indicating the closeness of individual readings to one another (regardless of whether they are close to the true value or not). For example, if five repeated voltmeter readings are 98.1, 98.2, 98.0, 98.1, 98.2V, these readings show high precision (very little scatter among themselves) even though they may all be inaccurate (systematically offset from a true value of, say, 100.0V due to a calibration error) — this distinction between accuracy and precision is fundamental to understanding that a highly precise instrument is not necessarily an accurate one, and vice versa.

Hysteresis: a phenomenon in which an instrument's output for a given input value depends on whether that input value was approached from a lower value (increasing) or a higher value (decreasing) direction, causing the instrument's calibration/response curve to trace out a loop rather than a single curve when input is cycled up and then down. For example, a mechanical pressure gauge may read slightly differently for the same actual applied pressure depending on whether the pressure is being gradually increased toward that value or gradually decreased toward it, due to friction and elastic hysteresis in its internal mechanical linkage and Bourdon-tube sensing element; hysteresis error is typically specified as the maximum difference between the increasing and decreasing readings at the same input value, expressed as a percentage of full-scale reading.

Probable error: as calculated numerically in part (a), the probable error defines a deviation range around the mean (or around a single reading) within which there is an equal (50%) probability that the true value lies inside versus outside this range, based on the normal error distribution and the calculated standard deviation — for example, if a temperature measurement's mean is 100.0 degrees C with a calculated probable error of 0.93 degrees C, this means there is a 50% statistical likelihood that the true temperature lies within the range 99.07 to 100.93 degrees C, providing a standard, widely-used way of expressing the statistical confidence associated with a given measured value in precision instrumentation reporting.