# Mean, median, mode, range (formulas, examples)

In mathematics and statistics, such concepts as the arithmetic mean, median and mode are widely used. They allow you to find averages for large amounts of numbers / data, and are an integral part of statistical research. Their second name is measures of central tendency, and with a normal distribution of numbers, the median, mode and arithmetic mean are always equal.

## Measures of descriptive statistics

### Arithmetic mean

The easiest to understand is the arithmetic mean, which is equal to the ratio of the sum of the numbers to their number. So, if we take an array of 500 different elements, put their numerical values in brackets and divide by 500, we get the arithmetic mean. The elements to be averaged are most often research results, statistical data, economic indicators, and so on. Today, this approach is used in most areas of science and natural science, including the humanities, such as history. In general, the formula looks like this:

x = (x1 + x2 + ... + xn) / n,

where x is the arithmetic mean and n is the number of values to be averaged.

Although the statistical mean is used to determine central trends much more often than the median and mode, its accuracy is not high when dealing with heterogeneous (very different) data.

### Median

An equally important measure of the central trend is the median, which is found according to a completely different principle. Array values do not need to be added and divided by their number, but simply arranged in a row: from smallest to largest. The central value of this series will be equal to the median. All values located to the left of it will be less, and to the right - more. The number of numbers in a row doesn't matter, and it can be either 3-5 values or millions/billions. But for the median to be as objective/unambiguous as possible, the number of values must be odd.

With an ideal distribution of numbers, the median and arithmetic mean are equal. But the first one makes it possible to find the central trend much more accurately with a large spread of numbers (in asymmetric distributions). This becomes especially useful when calculating dynamic quantities.

### Fashion

The name of this measure fully conveys its essence. So, "fashionable" is what the majority aspires to. That is, the mode is the value that occurs most frequently in a given row/array. The latter are characterized by the simultaneous existence of several modes at once. For example, if the most common values in the array are a, b, and n, they are added together and divided by the number (3). That is, they find the arithmetic mean.

Most often, the mod is used in non-numerical studies, where certain characteristics/properties are used instead of numbers. For example, colors: blue, green, silver, golden. Or species diversity: terrier, rottweiler, doberman, shepherd dog. To find out which of these colors (or breed of dogs) occurs most often in a series, such a measure as fashion allows. However, with the development of digital technologies, its mathematical affiliation is becoming more and more obvious.

## A bit of history

All three measures were widely used relatively recently - from the 18th to the 20th centuries. The earliest is the concept of fashion, which was invented in the 18th century in Europe, and was originally used only in relation to clothing. Today, fashion is applied to any non-numerical research, including the fields of industry, agriculture, construction.

A little later, in 1843, such a concept as the "median" was introduced - the central value in a series of numbers, ordered in size from smallest to largest. It was introduced by the French mathematician Antoine Augustin Cournot, who conducted psychological and sociological research using this discovery. In addition, the median has found wide application in such a field of science as astronomy.

The most recent invention among those presented is the arithmetic mean. It's hard to believe, but it was only widely used after 1906 - a little more than 100 years ago. The initiator was the famous English scientist Francis Galton, who, during a visit to an agricultural exhibition, calculated the average value from the answers of 787 participants in the competition, dividing the sum of the values by their number. It was about guessing the weight of the bull by eye, and the results of Hamilton's study confirmed that the arithmetic mean of 787 answers turned out to be as accurate as possible, despite the large spread and approximateness of the voiced options.

Summarizing, we can say that today the measures of the central trend are the basis of any statistics. Without them, in principle, accurate planning is impossible: expenses, income, output, etc. To calculate the mode, median or arithmetic mean, today you can use standard formulas or special applications.