Find the Mean Find the Mean There are other statistical measures of central tendency that should not be confused with means - including the 'median' and 'mode'. Statistical analyses also commonly use measures of dispersion, such as the range, interquartile range, or standard deviation. Note that not every probability distribution has a defined mean; see the Cauchy distribution for an example. For a data set, the arithmetic mean is equal to the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the "sample mean" () to distinguish it from the "population mean" ( or x). For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals.
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The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean. For a probability distribution, the mean is equal to the sum or integral over every possible value weighted by the probability of that value. In the case of a discrete probability distribution, the mean of a discrete random variable x is computed by taking the product of each possible value of x and its probability P(x), and then adding all these products together, giving. As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. Examples of means are listed below. The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data. Nevertheless, many skewed distributions are best described by their mean – such as the exponential and Poisson distributions. Truncated mean :- Sometimes a set of numbers might contain outliers, i.e. a datum which is much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. In this case one can use a truncated mean.
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It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values. FrĂŠchet mean :- The FrĂŠchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold. Unlike many other means, the FrĂŠchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher).
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