Measures of central tendency

There is a great number of different average or mean sizes which can be defined for a given particle size distribution. The purpose of such measures 

Fovir particle size distributions of a given particle population. 

The mode, median, and mean of a particle size distribution. 

Evaluation of the various means required for a given particle size distribution is based on Equation 2.21, which may also be written as 

The mode is the most commonly occurring size, i.e. the size corresponding to the peak on the size distribution frequency curve. Some distributions may have more than one peak and are commonly referred to as multimodal distributions. The median or the 50% size is the size such that half the particles (by the given quantity ) are larger and half are smaller, i.e. the size which divides the area under the distribution frequency curve into two halves. The median is most easily determined from the cumulative percentage curves (see section 2.5) where it corresponds to 50%. 

Comparing equation 2.12 for the arithmetic mean (g(x)=x), quadratic mean (g(x) = x ) and cubic mean (g(x) = x ) with equations 2.1, 2.2 and 2.3 respectively, it may be shown that, for example, the number arithmetic mean relates the total length (if all particles are placed next to each other in a line) to the total number in the population and is therefore known as the number length mean xnl Table 2.5 summarizes these relationships. 

If either/(x) or F(x) are available as analytical functions (see section 2.5), the desired mean diameters are evaluated by integration following equation 2.12. 

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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. 

A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, 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]

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). 

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. 

Sensitivity, specificity, odds ratio, and relative risk Types of data and scales of measurement Measures of central tendency and dispersion Inferential statistics Students s t-distribution Comparing means Comparing more than two means Regression and correlation Nonparametric tests The x2-test Clinical trials INTRODUCTION... 

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. 

Less commonly used is the mode, the most frequently occurring value in the dataset. The mode is the appropriate measure of central tendency for nominal data. Other measures of location (but not of central tendency) are percentiles or quartiles. The percentile of xis the percentage of the total cases that falls at or below xin value. Commonly used percentiles are the 25th and 75th percentiles (or 1st and 3rd quartiles). The median is the 50th percentile. The distance between the first and third quartile is the interquartile range (Figure 21.1). 

The median of the results for each property is reported. There is evidence that tensile properties at break (strength and elongation) follow a double exponential distribution when the mode would be the best measure of central tendency. This has never been widely adopted, largely because it is... 

Descriptive Statistics involves the presentation of summary statistics, which are concise yet meaningful summaries of large amounts of data. One category of descriptive statistics is the measurement of central tendency. 

One of the most commonly used measures of central tendency is the mean, more correctly (but rarely) called the arithmetic mean, a term that unambiguously distinguishes it from the geometric mean. While very informative in some circumstances, the geometric mean is less commonly used, and, in the absence of the prefix arithmetic or geometric, the default interpretation of the term mean is the arithmetic mean. This is the convention followed in the rest of this book. The mean of a set of data points is therefore defined as their sum divided by the total number of data points. 

Measures of central tendency provide an indication of the location of the data. For data measured on a scale of 1-100, a mean of 89 would suggest that the data are, in general, located closer to the top end of the scale than to the bottom end. 

If chaotic dynamics are present, the experimental errors do not originate exclusively from classical randomness. Thus, the measures of central tendency used to describe or treat experimental data are questionable, since averaging is inappropriate and masks important information in chaotic systems [234]. 

The analytical plan of epidemiological studies should use descrip tive and analytical techniques in describing the sample and results. Descriptive statistics, such as frequency distributions, cross-tabulations, measures of central tendency, and variation, can help explain underlying distributions of variables and direct the assessment of appropriateness of more advanced statistical techniques. Careful weighing of study findings with respect to the design and methods helps to ensure the validity of results. 

Means or other measures of central tendency, expressed as atom ratios relative to Ca. n.d. Data for 6 pastes from 3 cements. 

Table 10.5 Analysis of impact of measure of central tendency selection on overall occupational risk assessment outcome... 

For short-term exposures, the median is the appropriate measure of central tendency. This guidance is based on the recognition that, for the two most prevalent distribution types (log-normal, and normal), the median approximates the mean (i.e. geometric mean for log-normal distribution and arithmetic mean for normal distribution). 

Non-parametric statistical techniques (i.e. those that make minimal assumptions about the error distribution) can be used to handle the raw data. Such methods are generally resistant towards the effects of extreme values, often because they use the median (see Section 6.2) as a measure of central tendency or measure of location. Such methods have the further advantage of extreme simplicity of calculation in many cases, but while popular in the behavioural sciences they are less frequently used in the analytical sciences. 

The purpose of an average is to represent a group of individual values in a simple and concise manner in order to obtain an understanding of the group. It is important therefore that the average should be representative of the group. All averages are a measure of central tendency, which are only modestly affected by the relatively few extreme values in the tails of the distribution. 

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