The Standard Deviation for Discrete MeasurementsIn many cases, physical measurements in a context which involves probability will form a collection of results which can be approximated by a normal distribution. In such cases a large number of experimental results will form a gaussian curve. Consider first an example in which we have a complete set of measurements which is often called a "population". It is useful to calculate the standard deviation from the mean of the population measurements. For a population of N we can determine the mean μ = < x > of the measurements x and the variance σ^{2} of the deviations from the mean. The standard deviation σ is the square root of the variance. The Standard Deviation from a Sample MeanFor an experimental situation in which neither the population N nor it's mean are known, a sample of n measurements can be used to compute an estimate of the population mean. The calculated values of the mean μ_{s} and standard deviation σ_{s} from that sample mean may be taken as estimates of the corresponding values for the population, which are unknown. The question that arises is "why n1 rather than n in the sample variance expression?" Savory points out that since the variance σ_{s}^{2} is measured with deviations from the sample average rather than the population average, the deviations tend to be smaller than the deviations from the population mean. Using n1 rather than n tends to correct for the bias in the estimate of the variance. An empirical example of the effect of the n1 correction is posted by Khan Academy. Another point in this discussion is that the deviation values for the sample are based on the sample average rather than the population average. So once you have determined n1 independent values in the sample measurements based on that average, the last deviation value is predetermined and not independent. As a formal discussion of sample statistics, this approach to a partial correction of sample bias is known as Bessel's correction after Friedrich Bessel.

Index Applied statistics concepts Rohlf Sec 2.1 Kreyzig 22.1, 23.2 Savory Bessel's correction Khan Academy example  

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The Standard Deviation Using a Distribution FunctionThe rootmeansquare deviation of x from its average < x > is called the standard deviation. For a set of discrete measurements, the standard deviation takes the form
and
Determining the average or mean in the above expression involves the distribution function for the variable.

Index Applied statistics concepts Rohlf Sec 2.1 Kreyzig 22.1, 23.2  

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Standard Deviation for Particle PositionA free particle which is constrained to be between x=0 and x=L has a distribution function which is just a constant. The relationship for the standard deviation of the position is the square root of the integral
Normalizing the distribution gives the value for C. The mean value of x is Using this, the standard deviation becomes The average in the square root is The resulting standard deviation for the free particle is

Index Distribution functions Applied statistics concepts  

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