Central Tendency and Spread

The ability of the frequency curve to accurately represent the underlying distribution increases with the number of observations. With a small number of results only an approximation is possible, and the divergence may be relatively large. 

Sometimes it is preferable to express the frequencies in terms of proportional frequencies rather than actual frequencies. For each given observation, the probability of occurrence is defined as the proportional frequency with which it occurs in a large number of observations. The resulting curve is referred to as the probability distribution. 

The frequency curve and the histogram described previously have very explicit attributes which can be used as the basis of criteria to characterize the distribution  

There is a certain value (for example the arithmetic mean) that represents the center of the distribution and serves to locate it. 

The values are spread around this central value, extending over a range. 

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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... 

Indices of distribution central tendency and spread are not reviewed here (see Vose 2000, Section 3.2.1). The concept of skewness of a distribution relates to deviations from symmetry of the pdf. The normal distribution has a skewness of zero (the distribution is symmetric, with the familiar bell-shaped pdf). For a distribution with positive skewness the right tail of the distribution is more extended than the left tail a distribution with the left tail more extended has negative skewness. In many cases, it seems that skewness is associated with a constraint on the permissible values of a variable (Vose 2000, Section 6.7). The idea is that the distribution tail can be more extended in the direction opposite to a bound than in the direction of the bound. 

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. 

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. 

Commonly used descriptive statistics include measures that describe where the middle of the data is. These measures are sometimes called measures of central tendency and include the mean, median, and mode. Another category of measures describes how spread out the data is. These measures are sometimes called measures of variability and include the range, variance, and standard deviation. Additional descriptive measures can include percentages, percentiles, and frequencies. In safety performance measurement, the safety professional must determine the format of the data (i.e., ratio, interval, ordinal, or categorical) that will be collected and match the data format to the appropriate statistic. As will be discussed in the following sections, certain descriptive statistics are appropriate for certain data formats. 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

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... 

A characteristic of biological systems is variability, with most values of a variable clustered around the middle of the range of observed values, and fewer at the extremes of the range. The measure of location or central tendency gives an indication where the distribution is centred, while a measure of dispersion indicates the degree of scatter or spread in the distribution. The most widely used measure of central tendency is the arithmetic mean or average of the observed values, i.e, the sum of all variable values divided by the number of observations. Another measure of central tendency is the median, the middle measurement in the data (if n is odd) or the average of the two middle values (if n is even). The median is the appropriate measure of central tendency for ordinal data. 

Figure 4.4 shows a histogram of C-C bond lengths in C(ar)-CN fragments [5]. The histogram conveys three types of information. Firstly, it tells us something about the central tendency of the distribution (the bond lengths are centred around a value of about 1.445 A). Secondly, it conveys information about spread, or dispersion (individual observations vary between about 1.42 and 1.46 A). Thirdly, it shows the shape of the distribution (approximately bell-shaped, but perceptibly skewed). In this section, we discuss how these three different types of information can be described numerically. 

Dispersion n The variation in the value of a variable within a set of observations. A small dispersion indicates closely spaced values whereas a large dispersion indicates widely spaced values. There are several measures of dispersion which define quantitative values related to the spread of the values in a set of observations. These include range, interquartile range, standard deviation, and variance. A measure of dispersion along with a measure of central tendency give a simple quantitative description of shape of the distribution of values within the set of observations. 

There are many different descriptive statistics that can be chosen as a measurement of the central tendency of a data set. These include arithmetic mean, the median, and the mode. Other statistical measures such as the standard deviation and the range are called measures of spread and describe how spread out the data are. 

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