Normal error curve properties

Equation 2.54 has the form of the normal error curve and from the geometrical properties of the curve we can show that... 

A normal error curve has several general properties (a) The mean occurs at the central point of maximum frequency, (b) there is a symmetrical distribution of positive and negative deviations about the maximum, and (c) there is an exponential decrease in frequency as the magnitude of the deviations increases. Thus, small uncertainties are observed much more often than very large ones. 

The properties of the normal error curve are useful because they permit statements to be made about the probable magnitude of the net random error in a given measurement or set of measurementsprovidedthe Stan-ilard deviation is known. Thus, one can say that it is 68,3% probable that the random error associated with any single measurement is within t lo, that it is 0.5,4% probable that the error is within 2o-, and. so forth. The standard deviation is clearly a useful quantity for estimating and reporting the probable net random error of an analytical method. 

A useful property of the normal error curve is that, regardless of the magnitude of n and a, the area under the curve within defined limits on either side of [I (usually expressed in multiples of a) is a constant proportion of the total area. Expressed as a percentage of the total area, this indicates that a particular percentage of the population will be foimd between those limits. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

Fig. 9. Viscoelastic properties of ultrasound-treated ex vivo porcine muscle specimen. Muscle samples were coagulated with focused ultrasounds in selected regions. MRE using the method of Ref. 23 was carried out and shear moduli were calculated in normal and heated regions at different shear wave frequencies. Upper curve FUS-treated tissue. Lower curve normal tissue. Error bars are for standard deviations. (From Ref 47, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.)...

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