Mean of a data set

State the following in terms of null and alternative hypotheses a) the mean of a data set is equal to zero b) the mean of a data set is equal to six c) the mean of a data set is equal to or less than six d) the variance of one data set is not different from the variance of a second data set e) the variance of one data set is equal to or less than the variance of a second data set. 

Possible means of a data set are more closely clustered than the data... 

A second type of error, called systematic (or determinate) error, causes the mean of a data set to differ from the accepted value. For example, the mean of the results in Figure 5-1 has a systematic error of about —0.2 ppm Fe. The results of analysts 1 and 2 in Figure 5-3 have little systematic error, but the data of analysts 3 and 4 show systematic errors of about —0.7% and —1.2% nitrogen. In general, a systematic error in a series of replicate measurements causes all the results to be too high or too low. An example of a systematic error is the unsuspected loss of a volatile analyte while heating a sample. 

In this spreadsheet exercise, we learn to calculate the mean of a data set. First, we define formulas to calculate the mean, and then we use the built-in functions of Excel to accomplish the task. 

There are many cases in which a scientist or an engineer needs to compare the mean of a data set with a known value. In some cases, the known value is the true or accepted value based on prior knowledge or experience. In other situations, the known value might be a value predicted from theory or it might be a threshold value that we use in making decisions about the presence or absence of a constituent. In all these cases, we use a statistical hypothesis test to draw conclusions about the population mean y. and its nearness to the known value, which we call p.Q. 

Standard deviation of the mean the standard deviation associated with the mean of a data set consisting of N measurements,... 

There are many ways of computing the mean of a data set depending on how it is believed to be distributed. One of the most common methods is to use the arithmetic mean, which is defined as ... 

Variance was introduced in Chapter 4 as one measure of a data set s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the variance, s, of a data set consisting of n measurements is given as... 

The purpose of translation is to change the position of the data with respect to the coordinate axes. Usually, the data are translated such that the origin coincides with the mean of the data set. Thus, to mean-center the data, let be the datum associated with the kth measurement on the /th sample. The mean-centered value is computed as = x.f — X/ where xl is the mean for variable k. This procedure is performed on all of the data to produce a new data matrix the variables of which are now referred to as features. 

In general, air quality data are classified as a function of time, location, and magnitude. Several statistical parameters may be used to characterize a group of air pollution concentrations, including the arithmetic mean, the median, and the geometric mean. These parameters may be determined over averaging times of up to 1 year. In addition to these three parameters, a measure of the variability of a data set, such as the standard deviation... 

The goal of factor analysis (FA) and their essential variant principal component analysis (PCA) is to describe the structure of a data set by means of new uncorrelated variables, so-called common factors or principal components. These factors characterize frequently underlying real effects which can be interpreted in a meaningful way. 

A box contains only five slips of paper on which are found the numbers 3, 4, 3, 2, and 5. What is the mean of this data set, and what symbol should be used to represent it ... 

From the mean of the data set (y,) to the value predicted by the model (y,). This distance is a measure of the effectiveness of the model in explaining the variation in the data set. It is allowed by the degrees of freedom associated with the coefficients of the factor effects (the other P s). 

From the mean of the data set (y,) to the response itself (y ). This is a distance that has been corrected or adjusted for the mean. 

Regression equation The equation obtained by the correlation of a data set with a correlation equation by means of simple or multiple linear regression analysis. 

When you find the standard deviation of a data set, you are finding the average distance from the mean for the n measurements. It cannot be negative, and when two sets of measurements are compared, the larger the standard deviation, the larger the dispersion. 

The F test is a very simple but powerful statistical test, as many other tests require the variances of the data or populations to be similar (i.e., not significantly different). This is quite logical it would be rather inappropriate to test the means of two data sets if the precisions of the data were significantly different. As mentioned previously, the precision is related to the reproducibility of the data collected. If we have poor reproducibility, then the power and the significance of further testing are somewhat limited. 

The one thing at which Excel does not excel is statistical analysis. The (very basic) level of material covered in this chapter is about at (if not beyond) the limit of its capabilities. It can be used to generate means, medians, SDs and quartiles, but while the first three are OK, the quartile values generated are somewhat unconventional and will not be pursued further. The mean, median and SD of a data set can be generated by using either worksheet functions or the Data Analysis tool. 

Example 10 Comparison of two variances. A simplified analytical procedure is proposed for a routine laboratory test. It is necessary to determine not only whether the new procedure gives the same results as the old, i.e., whether the means of a duplicate set are the same, but also whether the precision of the new test is as good as the current test. The data for the two tests are as follows ... 

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