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statistics

Branch of mathematics concerned with the collection and interpretation of data. For example, to determine the mean age of the children in a school, a statistically acceptable answer might be obtained by calculating an average based on the ages of a representative sample, consisting, for example, of a random tenth of the pupils from each class. Probability is the branch of statistics dealing with predictions of events.

Mean, median, and mode

The mean, median, and mode are different ways of finding a ‘typical’ or ‘central’ value of a set of data. The mean is obtained by adding up all the observed values and dividing by the number of values; it is the number that is commonly used as an average value. The median is the middle value, that is, the value that is exceeded by half the items in the sample. The mode is the value that occurs with greatest frequency, the most common value. The mean is the most useful measure for the purposes of statistical theory. The idea of the median may be extended and a distribution can be divided into four quartiles. The first quartile is the value that is exceeded by three-quarters of the items; the second quartile is the same as the median; the third quartile is the value that is exceeded by one-quarter of the items.

Standard deviation and other measures of dispersion

The mean is a very incomplete summary of a group of observations; it is useful to know also how closely the individual members of a group approach the mean, and this is indicated by various measures of dispersion. The range is the difference between the maximum and minimum values of the group; it is not very satisfactory as a measure of dispersion. The mean deviation is the arithmetic mean of the differences between the mean and the individual values, the differences all being taken as positive. However, the mean deviation also does not convey much useful information about a group of observations. The most useful measure of dispersion is the variance, which is the arithmetic mean of the squares of the deviations from the mean. The positive square root of the variance is called the standard deviation, a measure (symbol ςor s) of the spread of data. The deviation (difference) of each of the data items from the mean is found, and their values squared. The mean value of these squares is then calculated. The standard deviation is the square root of this mean.

It is usual to standardize the measurements by working in units of the standard deviation measured from the mean of the distributions, enabling statistical theories to be generalized. A standardized distribution has a mean of zero and a standard deviation of unity. Another useful measure of dispersion is the semi-interquartile range, which is one-half of the distance between the first and third quartiles, and can be considered as the average distance of the quartiles from the median. In many typical distributions the semi-interquartile range is about two-thirds of the standard deviation and the mean deviation is about four-fifths of the standard deviation.

Applications

One of the most important uses of statistical theory is in testing whether experimental data support hypotheses or not. For example, an agricultural researcher arranges for different groups of cows to be fed different diets and records the milk yields. The milk-yield data are analysed and the means and standard deviations of yields for different groups vary. The researcher can use statistical tests to assess whether the variation is of an amount that should be expected because of the natural variation in cows or whether it is larger than normal and therefore likely to be influenced by the difference in diet.

Correlation

Correlation measures the degree to which two quantities are associated, in the sense that a variation in one quantity is accompanied by a predictable variation in the other. For example, if the pressure on a quantity of gas is increased then its volume decreases. If observations of pressure and volume are taken then statistical correlation analysis can be used to determine whether the volume of a gas can be completely predicted from a knowledge of the pressure on it.