Central tendency measures

Table 1.1 Effect of various factors on pesticide exposure. All data are unit exposure values ( xg/kg of active ingredient (a.i.) handled), taken from PHED (1992). Values are central tendency measures based on high confidence data sets...

Data Representation and Interpretation Erequency Distributions Measures of Central Tendency Measures of Dispersion Probability... 

The mean is the most common estimator of central tendency. It is not considered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean s value. For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g ... 

As shown by Examples 4.1 and 4.2, the mean and median provide similar estimates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensitive to measurements with extreme values. For example, introducing the transcription error discussed earlier for the mean only changes the median s value from 3.107eto3.112e. 

If the mean or median provides an estimate of a penny s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all... 

Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. Eirst, does our measure of central tendency agree with the true, or expected value Second, why are our data scattered around the central value Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 

Accuracy is a measure of how close a measure of central tendency is to the true, or expected value, Accuracy is usually expressed as either an absolute error... 

Although the mean is used as the measure of central tendency in equations 4.2 and 4.3, the median could also be used. 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, is given 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 median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

One further point might be made here. Although the example illustrates the difference between the two types of molecular weight average, the weight average molecular weight in this example cannot be said to be truly representative, an essential requirement of any measure of central tendency. In such circumstances where there is a bimodal, i.e. two-peaked, distribution additional data should be provided such as the modal values (100 and 100000 in this case) of the two peaks. 

Distributions are characterized by measures of central tendency The median is the value of X (e.g., crap scores) that divides the distribution into equal areas. The value of x at the peak... 

Mean The measure of central tendency of a distribution, often referred to as its arithmetic average. 

A location parameter is the abscissa of a location point and may be a measure of central tendency, such as a mean. 

Mean, arithmetic More simply called the mean, it is the sum of the values in a distribution divided by the number of values. It is the most common measure of central tendency. The three different techniques commonly used are the raw material or ungrouped, grouped data with a calculator, and grouped data with pencil and paper. 

In particle size analysis it is important to define three terms. The three important measures of central tendency or averages, the mean, the median, and the mode are depicted in Figure 2.4. The mode, it may be pointed out, is the most common value of the frequency distribution, i.e., it corresponds to the highest point of the frequency curve. The distribution shown in Figure 2.4 (A) is a normal or Gaussian distribution. In this case, the mean, the median and the mode are found to fie in exactly the same position. The distribution shown in Figure 2.4 (B) is bimodal. In this case, the mean diameter is almost exactly halfway between the two distributions as shown. It may be noted that there are no particles which are of this mean size The median diameter lies 1% into the higher of the two distri-... 

Limitations on our ability to measure constrain the extent to which the real-world situation approaches the theoretical, but many of the variables studied in toxicology are in fact continuous. Examples of these are lengths, weights, concentrations, temperatures, periods of time, and percentages. For these continuous variables, we may describe the character of a sample with measures of central tendency and dispersion that we are most familiar with the mean, denoted by the symbol x and also called the arithmetic average, and the standard deviation SD, denoted by the symbol [Pg.870]

It is evident that the mean of n results is 4n times more reliable than any one of the individual results. Therefore, there exists a diminishing return from accumulating more and more replicate meaurements. In other words, the mean of 9 results is 3 times as reliable as 1 result in measuring central tendency (i.e., the value about which the individual results tend to cluster) the mean of 16 results is 4 times as reliable etc. 

When dealing with sets of numbers, there are measures used to describe the set as a whole. These are called measures of central tendency and they include mean, median, and mode. 

Once data have been collected, the values will be distributed around a central point or points. Various terms are used to describe both the measure of central tendency and the spread of data points around it. 

The measure of central tendency used throughout this book is the mean, sometimes called the average [Arkin and Colton (1970)]. It is defined as the sum (E) of all the response values divided by the number of response values. In this book, we will use y as the symbol for the mean (see Section 1,3). 

The idea behind measures of location and central tendency is contained within the notion of the average. There are predominantly three summary statistics that are commonly used for describing this aspect of a set of data the arithmetic mean - normally shortened to the mean, the mode and the median. 

In general, bias refers to a tendency for parameter estimates to deviate systematically from the true parameter value, based on some measure of the central tendency of the sampling distribution. In other words, bias is imperfect accuracy. In statistics, what is most often meant is mean-unbiasedness. In this sense, an estimator is unbiased (UB) if the average value of estimates (averaging over the sampling distribution) is equal to the true value of the parameter. For example, the mean value of the sample mean (over the sampling distribution of the sample mean) equals the mean for the population. This chapter adheres to the statistical convention of using the term bias (without qualification) to mean mean-unbiasedness. 

Arithmetic mean A measure of central tendency. It is calculated as the sum of all the values of a set of measurements divided by the number of values in the set. 

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