Normally distributed population

Earlier we noted that 68.26% of a normally distributed population is found within the range of p, lo. Stating this another way, there is a 68.26% probability that a member selected at random from a normally distributed population will have a value in the interval of p, lo. In general, we can write... 

According to Table 4.11, the 95% confidence interval for a single member of a normally distributed population is... 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

A second example is also informative. When samples are obtained from a normally distributed population, their values must be random. If results for several samples show a regular pattern or trend, then the samples cannot be normally distributed. This may reflect the fact that the underlying population is not normally distributed, or it may indicate the presence of a time-dependent determinate error. For example, if we randomly select 20 pennies and find that the mass of each penny exceeds that of the preceding penny, we might suspect that the balance on which the pennies are being weighed is drifting out of calibration. 

A normally distributed population of data can be characterized by two parameters. The centre, or location, of the population is described by the parameter i... 

Univariate case data from normally distributed populations generally have a higher information value associated with them but the traditional hypothesis testing techniques (which include all the methods described in this section) are generally neither resistant nor robust. All the data analyzed by these methods are also, effectively, continuous that is, at least for practical purposes, the data may be represented by any number and each such data number has a measurable relationship to other data numbers. 

A normally distributed population is assumed, and the results are sensitive to outliers. 

The extreme data point X is not an outlier and comes from a normally distributed population with sample mean X and standard deviation s... 

A sample of 36 observations was drawn from a normally distributed population having a mean of 20 and variance of 9. What portion of the population can be expected to have values greater than 26 ... 

Assume underlying normally distributed populations for each fertilizer group. All groups have constant population variance a1 — o2 = o3 ... 

Use Student s t-test for small, normally distributed populations (number of samples is less than 30). 

The requirements are similar to those for the two-sample f-test. Each set of data should be drawn from a normally distributed population and they should all have the... 

This rather complicated equation can be interpreted as follows. The function f (x) is proportional to the probability that a measurement has a value v for a normally distributed population of mean /< and standard deviation a. The function is scaled so that the area under the normal distribution curve is 1. 

Figure 5.5 Some properties of a normal distribution (population (o) or sample extracted (s)).

On many occasions, sample statistics are used to provide an estimate of the population parameters. It is extremely useful to indicate the reliability of such estimates. This can be done by putting a confidence limit on the sample statistic. The most common application is to place confidence limits on the mean of a sample from a normally distributed population. This is done by working out the limits as F— ( />[ i] x SE) and F-I- (rr>[ - ij x SE) where //>[ ij is the tabulated critical value of Student s t statistic for a two-tailed test with n — 1 degrees of freedom and SE is the standard error of the mean (p. 268). A 95% confidence limit (i.e. P = 0.05) tells you that on average, 95 times out of 100, this limit will contain the population... 

The F-distribution has great utility in a statistical test referred to as analysis of variance (ANOVA). ANOVA is a powerful tool for testing the equivalence of means from samples obtained from normally distributed, or approximately normally distributed, populations. As an example, suppose that the following are the content uniformity values on 20 tablets from each of four different lots lot A mean = 99.5%, standard deviation = 2.6% lot B mean = 100.2%, standard deviation = 2.8% lot C mean = 90.5%, standard deviation = 2.1% and lot D mean = 100.3%, standard deviation = 2.7%. 

This ratio will have a known probability distribution for variances estimated from samples drawn from a normally distributed population. This probability distribution is called the F distribution. Fig. 3.9 shows a typical F distribution. The actual shape of the curve depends on the numbers of degrees of freedom of the sample variance estimates. Tabulated critical F ratios are given in Appendix Statistical Tables at the end of this book. 

Figure 2.2 Dose-response relationships drawn on three different models for four populations, (a) Doses and responses in linear scale, (b) Doses in log scale and responses in linear scale, (c) Doses in log scale and responses in probits. (1) Sensitive population with normally distributed sensitivity and LD50 = 2.5 units. (2) A mixed population with 75% of (1) and 25% resistant individuals. (3) Intermediate sensitive population with normally distributed sensitivity, but more scattered than (1), and LD50 = 5 units. (4) Less sensitive, but normally distributed population, similar to (1), but with LD50 = 6.5 units.

Figure 2.1 Means of n data randomly drawn from a normally distributed population with fx= 10 and a= 1 as a function of n.

In reality, both of these cases are unlikely but not impossible. Our group could have been two classes of fifth- and eleventh-grade students that would have been a bimodal population very similar to that described above. In most cases, the magnitude of the variability is intermediate between these extremes for most normally distributed populations. Just as we can use weight as a descriptor, we can use other measurements to describe disease or toxicity induced by drugs or pesticides. The statistical principles are the same for all. 

We have discussed, in previous sections, ways of estimating, for a normally distributed population, the central value (mean, x), the spread of results (standard deviation, s), and the confidence limits (t test). These statistical values hold strictly for a large population. In analytical chemistry, we typically deal with fewer th 10 results, and for a given analysis, perhaps 2 or 3. For such small sets of data, other estimates may be more appropriate. 

It is sometimes said that by log-transforming AUCs we are actually considering the ratios of medians rather than the ratio of means. The argument goes like this. If the log-AUCs are a sample from a Normally distributed population, then the mean and median of this distribution are identical. The AUCs themselves, however, will follow a log-Normal distribution and for this distribution the mean is higher than the median. 

In population biology deterministic or stochastic corpuscular models are used, describing the behavior of a population of single cells or organisms. If cell division is considered a random process and the distribution of cycle times is assumed to be described by a normal distribution, population size models of the form... 

This test is also based on the assumption of a normally distributed population. It can be applied to series of measurements (3-150 measurements). The null hypothesis that x is not an outlier within the measurement series of n values is accepted at level a, if the test quantity T is... 

The normal distribution is valid if there are many (theoretically infinitely many) measurements. In practice, there is always a finite number N of measured values. They represent a random selection of the (infinite) number of possible values of the entire population. In statistics, we refer to this as a random sample. The arithmetic mean of a random sample taken from a normally distributed population (discussed at the beginning of the section) is a suitable estimate of the expected value. 

In theory, any random sample taken from a normally distributed population should give a normally distributed sample. In practice, we cannot take a random sample from the exploration of the SSp of a PN system. However, by using a quota sampling strategy, we can direct our exploration toward getting the result of a random sampling, a pseudo-normally distributed sample. 

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