This statistical calculator finds the arithmetic, geometric, and harmonic averages, as well as the median value for a given data set. Enter the elements of the original data set in the corresponding field of this Average Calculator and find the above parameters and also the number of input data set items (count), minimum and maximum values of the original dataset, the range and sum of its elements. You can paste the data copied from a spreadsheet or csv-file or enter manually using comma, space or enter as separators.
Averages in Statistics
In statistics, a central tendency is a central or typical value for a probability distribution. Usually measures of central tendency are called averages. They are actually different ways to tell us what value in a data set is typical or best represents this data set.
In the case of a dataset of \(n\) values \(\{ { x }_{ 1 },{ x }_{ 2 },…,{ x }_{ n }\}\), we have the following definitions for the most common averages.
Arithmetic Average
Arithmetic average (or arithmetic mean) is just the sum of all the elements divided by the number of elements in the data set:
$$\bar { x } =\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { x }_{ i } } .$$
The arithmetic mean is a data set’s simplest, most commonly used, and readily understood measure of central tendency.
While the arithmetic mean is often used to report central tendency, in many cases it is not a reliable statistic. The point is that it is heavily affected by outliers (values much larger or smaller than most others).
For skewed distributions, such as the income distribution, in which the incomes of a few people are significantly higher than the incomes of most people, the arithmetic mean may not coincide with the intuitive notion of “average”. In this case, a robust statistic, such as the median, gives a better description of central tendency.
Apart from mathematics and statistics, the arithmetic mean is often used in fields such as economics, anthropology, history, and others where large deviations from the mean are rare.
Geometric Average
In mathematics, the geometric average, often referred to as geometric mean, is defined as the \(n\)-th root of the product of \(n\) numbers, i.e., for a set of numbers \({ x }_{ 1 },{ x }_{ 2 },…,{ x }_{ n }\), the geometric mean is defined as
$${\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}} = {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}$$
It is important to note that the geometric mean applies only to positive numbers.
Interestingly, the concept of geometric average has a clear geometric meaning. Indeed, the geometric mean of two numbers \(a\) and \(b\) is the length of the side of a square whose area is equal to the area of a rectangle with sides \(a\) and \(b\). Similarly, the geometric mean of three numbers, \(a\), \(b\) and \(c\), is the length of the edge of a cube whose volume is equal to the volume of a cuboid with sides \(a\), \(b\) and \(c\).
The geometric average is used in a number of problems where a growth rate is of interest, for example in calculating compound interest rates, financial returns or risk and loses, area and volume averages, in computing indexes, and others. It is useful also in surveys and socials studies. The geometric mean is often used in geometry.
Harmonic Average
In mathematics, the harmonic average, or harmonic mean, sometimes referred to as subcontrary mean, of the \(n\) positive real numbers \({ x }_{ 1 },{ x }_{ 2 },…,{ x }_{ n }\) is defined as:
$${\displaystyle H={\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}} ={\frac {n}{\sum \limits _{i=1}^{n}{\frac {1}{x_{i}}}}}.}$$
In certain cases, the harmonic mean provides a better representation of “average”. For example, if for half the distance of a trip a car moves at 30 miles per hour and for the other half of the distance it moves at 60 miles per hour, then the average speed for the trip is given by the harmonic mean of 30 and 60, which is 40 miles per hour. This is exactly the distance divided by the total amount of time spent for the trip. The simple (arithmetic) mean would give us 45 miles per hour.
Various applications of the harmonic mean can be found in electricity, finance, geometry and other sciences.
Median
This is the middle value that separates the higher half from the lower half of the data set. When we have a data set ordered from lowest to highest value \(\{ { x }_{ 1 }\le { x }_{ 2 }\le …\le { x }_{ n }\}\), the median \(\overset { \sim }{ x }\) is the data point separating the upper half of the data values from the lower half.
In case \(n\) is odd the median is the data set value at position \(k\):
$$\overset { \sim }{ x }={ x }_{ k }, \,\,\,\, k=\frac { n+1 }{ 2 } .$$
In case \(n\) is even the median is the average of the values at positions \(k\) and \(k+1\):
$$\overset { \sim }{ x } ={ \frac { { x }_{ k }+{ x }_{ k+1 } }{ 2 } } , \,\,\,\, k=\frac { n }{ 2 } .$$
The main feature of the median in describing data as compared to the mean is that it is not skewed by a small fraction of extremely large or small values and therefore provides a better representation of the center.
The median income, for example, may be the best way to describe the center of the income distribution, because an increase in the highest income, for example, does not in itself affect the median.
Range
Range in statistics is the difference between the largest and smallest values in a data set. Range is one common measure of variability that describes how far away data points are from each other and from the mean.
Along with various “averages”, this parameter is also calculated by our Average Calculator.
Related calculators
Check out our other statistics calculators such as Mean, Median, Mode Calculator or Midrange Calculator.