Statistical normal curve

Mean aud standard deviation The statistical normal curve shows a definite relationship among the mean, the standard deviation, and normal curve. The normal curve is fully defined by the mean, that locates the normal curve, and the standard deviation that describes the shape of the normal curve. A relationship exists between the standard deviation and the area under the curve. 

NOTE If Z(N) is greater than the absolute value of the z-statistic (Normal Curve one-tailed) we reject the null hypothesis and state that there is no significant difference in rl and r2 at the selected significance level. 

There are two main families of statistical tests parametric tests, which are based on the hypothesis that data are distributed according to a normal curve (on which the values in Student s table are based), and non-parametric tests, for more liberally distributed data (robust statistics). In analytical chemistry, large sets of data are often not available. Therefore, statistical tests must be applied with judgement and must not be abused. In chemistry, acceptable margins of precision are 10, 5 or 1%. Greater values than this can only be endorsed depending on the problem concerned. 

To determine the critical region, we must know the distribution of the test statistic. In this case, Z is distributed as the standard normal distribution. With H0 [t<48 and a=0.05, we determine that the critical region will include 5% of the area on the high end of the standard normal curve Fig. 1.6. The Z-value that cuts off 5% of the curve is found to be 1.645, from a table of... 

The area under the Normal curve is of considerable interest in Statistics. That is, it is of considerable interest to define and quantify the area bounded by the Normal curve at the top and the x-axis at the bottom. This area will be defined as 1.0, or as 100%. Given this interest, the final point in Section 6.6 raised an issue that appears problematic. That is, it appears that, if the two lower slopes of the Normal curve never quite reach the x-axis, the area under the curve is never actually fully defined and can therefore never be calculated precisely. Fortunately, this apparent paradox can be solved mathematically. In the Preface of this book I noted that, in several cases, I had resisted the temptation to provide an explanation of subtle points. This case, I believe, is a worthwhile exception. An understanding of the qualities of the Normal distribution and the Normal curve is extremely helpful in setting the scene for topics covered in Chapters 7 and 8, namely statistical significance and clinical significance. 

Of particular interest in Statistics is that the means of many large samples taken from a particular population are approximately distributed in this Normal fashion, i.e., they are said to be Normally distributed. This is true even when the population data themselves are not Normally distributed. The mathematical properties of a true Normal distribution allow quantitative statements of the area under the curve between any two points on the x-axis. In Section 6.6.1 it was shown that the total area under the Normal curve is 1, or 100%. It is also of interest to know the proportion of the total area under the curve that lies between two points that are equidistant from the mean. These points are typically represented by multiples of the SD. From the properties of the mathematical equation that governs the shape of the Normal curve, it can be shown that ... 

Statistical formulas are based on various mathematical distribution functions representing these frequency distributions. The most widely used of all continuous frequency distributions is the normal distribution, the common bellshaped curve. It has been found that the normal curve is the model of experimental errors for repeated measurements of the same thing. Assumption of a normal distribution is frequently and often indiscriminately made in experimental work because it is a convenient distribution on which many statistical procedures are based. However, some experimental situations subject to random error can yield data that are not adequately described by the normal distribution curve. 

There is considerable individual variation in nutrient requirements. It is generally assumed that requirements follow a more or less statistically normal (Gaussian) distribution, as shown in the upper curve in Figure 1.1. This means that 95% of the population has a requirement for a given nutrient within the range of 2 SD about the observed mean requirement. Therefore, an intake at the level of the observed (or estimated) mean requirement plus 2 x SD will be more than enough to meet the requirements of 97.5% of the population. This is the level that is generally called the RDI, RDA, RNI, or PRI. 

Use a suitable statistical computer program to generate predicted normal curves from the Y and s values of your sample(s). These can be compared visually with the actual distribution of data and can be used to give expected values for a x -test or a G-test. 

Having introduced the normal distribution and discussed its basic properties, we can move on to the common statistical tests for comparing sets of data. These methods and the calculations performed are referred to as significance tests. An important feature and use of the normal distribution function is that it enables areas under the curve, within any specified range, to be accurately calculated. The function in Equation (1) is integrated numerically and the results presented in statistical tables as areas under the normal curve. From these tables, approximately 68% of observations can be expected to lie in the region bounded by one standard deviation from the mean, 95% within jjl 2o, and more than 99% within x 3a. 

The extreme regions of the normal curve containing 5% of the area are illustrated in Figure 3 and the values can be obtained from statistical tables. The selected portion of the curve, dictated by our limit of significance, is referred to as the critical region. If the value of the test statistic falls within this area then the hypothesis is rejected and there is no evidence to suggest that the samples come from the parent source. From statistic tables, 2.5% of the area is below - 1.96o- and 97.5% is above 1.96o. The calculated value for z of 1.85 does not exceed the tabulated z-value of 1.96 and the conclusion is that the mean sodium concentrations of the analysed samples and the known parent sample are not significantly different. 

A more accurate sample size can be established using statistical methods. It is known that averages of samples x drawn from a normal distribution of observations are distributed normally about the population mean /a. The variance about the population mean /i equals cr /n, where n equals the sample size and cr equals the population variance. Normal curve theory leads to the following confidence interval ... 

Such a procedure would require a very large analytical effort and the assumption is usually made that the distribution curve at this second stage can be described either as a normal curve or as a curve. The statistical validity of this second stage estimate of precision is based on the form of the first Stage distribution curve, i.e. Figure 2.1. An estimate of the precision of mixture quality is only possible if the sample distribution curve described in Figure 2.1 is that of a normal distribution. The test for normality can be carried out by an adaptation on the test and is illustrated in section 2.3. 

In Equation (19.9), z represents the number of standard deviations from the mean. The mathematical fimction that describes a normal-distribution curve or a standard normal curve is rather complicated and may be beyond the level of your current understanding. Most of you will learn about k later in your statistics or engineering classes. For now, using Excel, we have generated a table that shows the areas under portions of the standard normal-distribution curve, shown in Table 19.11. At this stage of your education, it is important for you to know how to use the table and solve some problems. A more detailed explanation will be provided in your future classes. We will next demonstrate how to use Table 19.11, usii a number of example problems. 

The distribution of random errors should follow the Gaussian or normal curve if the number of measurements is large enough. The shape of Gaussian distribution was given in Chapter 3 (Fig. 3.4). It can be characterized by two variables—the central tendency and the symmetrical variation about tjie central tendency. Two measures of the central tendency are the mean, X, and the median. One of these values is usually taken as the correct value for an analysis, although statistically there is no correct value but rather the most probable value. The ability of an analyst to determine this most probable value is referred to as his accuracy. 

There are a number of different statistical treatments for the data obtained. The most commonly used method is to calculate the range into which 95% of the values fall and call this the normal range. If the values have a symmetrical (normal) distribution, 95% of the individuals are found in the mean 2SD range. If the distribution of the values is skewed, the standard deviation cannot be used. However, if the logarithm of the concentration is plotted against the numbers, an approximately normal curve is obtained from which the standard deviation can be calculated. 

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