Variance, and Standard Deviation

The concepts of mean (average), variance, and standard deviation have been introduced in chapter 1. Here they will be defined. 

The sample mean (arithmetic mean, average) is the result of summing all the results and dividing by the number of data (n)  

The usual symbol for the sample mean is the lower case symbol for the quantity (here x) with a bar across the top. 

Indeed, although it wobbles about to begin with, the means of the data do seem to fall nearer to the population mean (here 10) as n increases. 

This sounds about right, as we know that the data is symmetrically distributed about the mean, so we would expect that the results that were on the high side of the mean to cancel those on the low side, and the more results we had, the better the canceling. This answers our question as to why do repeats. The more repeats that are done, the smaller the uncertainty about the sample mean, and hence the more confident one becomes that the sample mean is a good estimate of the population mean. 

The above precision mean deviation) has however the inconvenience that it cannot be interpreted statistically since both large and small individual deviations, though not being equally probable, have the same weight. The summing of the squares of the differences is preferred which leads to the most largely used definition for the precision (reproducibility) in statistics, known as the variance which is estimated for a set of n measurements by  

The square root of the variance is the standard deviation, represented either, when the number n of measurements is small, by or by cr when a large statistical population of values is available (expression 22.8). The standard deviation is a dispersion index expressed in the same units as x. 

Calculators and related programmable software possess the corresponding functions  

For an infinite number of measurements, s is replaced by a, n is replaced by n — 1 and p is used instead of the arithmetic mean x. 

To compare results or to express the uncertainty of a method, s is often expressed in a relative manner. Calculations are made therefore of the relative standard deviation (or RSD), also called coefficient of variation (CV) and most often expressed as a percentage  

When a measurement is replicated, the set of n values is expressed as a mean, or average, x, i.e., the sum of values divided by their number. 

The spread or dispersion of this set of values is measured by the sample variance, V, which is the sum of the squares of the deviation of each value from the mean, divided by one less than the number, n, of values in the set of samples If, instead of a limited set of samples, the entire set (population) of samples is available, the population variance, V, defined as the sum of the squares of the deviations divided by n, may be used. 

The positive square root of either variance is called the standard deviation, and is widely used. 

In analytical chemistry, generally, we deal with a relatively limited number of replicate determinations and, therefore, use the sample variance and standard deviation. Spreadsheet programs have convenient (functions to calculate the most common statistical functions both Lotus 123 and QPro permit the calculation of V ( VAR) and s ( STD), but only the latter allow direct calculation of V(( VARS) and s( STDS). While s is larger than s, the difference depends on the number of values in the set increases. At what value of n does this difference become 10%, 5%, 1% (Show that the n values are 6, 11, and 51, respectively.) Obviously, as n increases, the differences in s and s (as do V and V) become negligible. 

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This gives some information about the errors (i.e., the variance and standard deviation of each data point), although the goodness of fit, P, cannot be calculated. 

Figure 7. Graphs Relating the Variance and Standard Deviation of Peaks Passing Through an Open Tube against Tube Length...

Draw frequency histograms at intervals of 0.1 for the data in Problems 3.1 and 3.2. Calculate the variance and standard deviation for each of the two sets of data. 

In the same way that one uses the variance and standard deviation to describe the scatter of points around their average, one can also use the variance and standard deviation to describe the scatter, in the vertical (y) direction, of the points about the best-fit line. Statisticians have shown that the variance is given by... 

Mean, Variance, and Standard Deviation Three common statistics can be calculated very easily to give a quick understanding of the quality of a dataset and can also be used for a quick comparison of new data with some... 

An appropriate interpretation of 5 is not completely obvious at this point, but it does turn out to measure the spread of a data set, and to be extremely useful in drawing quantitative inferences from data. (In many, but not all, circumstances met in practice, the range or largest value in a data set minus the smallest value is on the order of four to six times 5.) The variance and standard deviation are time-honored and fundamental quantifications of the variation present in a single group of measurements and, by implication, the data-generating process that produced them. 

The consumption of cyanide is accurately determined by argentometric titration. The method was found to apply both to oxidized starch [439] as well as to cellulose [418], The method is accurate and recently the coefficient of variance and standard deviation for an oxidized cellulose containing 5.60 mmol per 100 g of ketone groups, were found to be 0.0046 and 0.0068, respectively. The corresponding values for the carboxyl groups by the methylene blue method of the same samples were 0.001 and 0.031 [440]. 

Tlie expected value of a random variable X is also called "the mean of X" and is often designated by p. Tlie expected value of (X-p) is called tlie variance of X. The positive square root of the variance is called tlie standard deviation. Tlie terms a" and a (sigma squared and sigma) represent variance and standard deviation, respectively. Variance is a measure of the spread or dispersion of tlie values of the random variable about its mean value. Tlie standard deviation is also a measure of spread or dispersion. The standard deviation is expressed in tlie same units as X, wliile the variance is expressed in the square of these units. 

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

Equation (1) describes the idealized distribution function, obtained from an infinite number of sample measurements, the so-called parent population distribution. In practice we are limited to some finite number, n, of samples taken from the population being examined and the statistics, or estimates, of mean, variance, and standard deviation are denoted then by x, and s respectively. The mathematical definitions for these parameters are given by equations (2H4),... 

Notice that we have deliberately left cells E12 and El 3 blank. We will now use the built-in variance and standard deviation functions of Excel to check our formulas. 

To understand the concept of mean, variance, and standard deviation pertaining both to a large sample (population) and a small sample. 

As we continue with the example we will use the following values for the estimated experimental variance and standard deviation ... 

Chapter 2 presented the basic model requirements for a simple linear regression. The b or slope must be approximately linear over the entire regression data range. The variance and standard deviation of b must be constant (Figure 3.14). 

Dispersion the variance and standard deviation (.square root of the variance) and the ran e... 

The mean (frequently known as the arithmetic average), variance, and standard deviation can be realized in two ways (1) as a true parameter value based on extensive measurement or other know ledge of the entire population, in which case these parameters are designated by the symbols p. o. and o respectively or (2) as estimates of the true values based on samples from the population, in which case they are designated by the symbols, v. S and. S respectively. The equations to calculate the mean, variance, and standard deviation as estimates from a sample are... 

Every statistical test has some assumptions regarding the data and conditions under which they were collected. For /-tests, it is assumed that the sample has been randomly selected from the population (so that the sample is representative of the population), and that the population is normally distributed. The independent measures /-test also assumes that the two populations from which the samples have been selected have the same variance and standard deviation. Thankfully, /-tests are considered to be fairly robust as long as each sample is relatively large (N > 30), which simply means that they still provide valid statistical measures even when there are departures from the assumptions. 

The assumptions made regarding data collection for ANOVAs and ANCOVAs are similar to those for /-tests— the samples have been randomly selected from the population, the population and populations of the samples are normally distributed, and the variances and standard deviations of the subgroups are homogeneous. As with /-tests, ANOVAs and ANCOVAs are relatively robust with respect to non-normality and are relatively robust to heterogeneity of variances when the sample sizes are the same within each subgroup 19). 

If sufficient replicate analytical data are available, the variance and standard deviation for a percent recovery study can be found by pooling the data. For example, consider a thorough percent recovery study using solid-phase extraction (SPE) for the isolation and recovery of the pesticide methoxy-chlor from groundwater. If j replicate injections are made for each of g SPE... 

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