In statistics, the term central tendency relates to the way in which quantitative data is clustered around some value.[1] A measure of central tendency is a way of specifying - central value. In practical statistical analysis, the terms are often used before one has chosen even a preliminary form of analysis: thus an initial objective might be to “choose an appropriate measure of central tendency”.
In the simplest cases, the measure of central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. However, the term is applied to multidimensional data as well as to univariate data and in situations where a transformation of the data values for some or all dimensions would usually be considered necessary: in the latter cases, the notion of a “central location” is retained in converting an “average” computed for the transformed data back to the original units. In addition, there are several different kinds of calculations for central tendency, where the kind of calculation depends on the type of data (level of measurement).
Both “central tendency” and “measure of central tendency” apply to either statistical populations or to samples from a population.
Basic measures of central tendency
The following may be applied to individual dimensions of multidimensional data, after transformation, although some of these involve their own implicit transformation of the data.
· Arithmetic mean - the sum of all measurements divided by the number of observations in the data set
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term “arithmetic mean” is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation’s population.
While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers. Notably, for skewed distributions, the arithmetic mean may not accord with one’s notion of “middle”, and robust statistics such as the median may be a better description of central tendency.
Suppose we have sample space 
. Then the arithmetic mean A is defined via the equation
.
If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean.
· Median - the middle value that separates the higher half from the lower half of the data set
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values.[1][2]
In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size), and, if there is such a member, there may be more than one so that the median may not uniquely identify a sample member. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.
At most, half the population have values less than the median, and, at most, half have values greater than the median. If both groups contain less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {a, b, c} is b, and, if a < b < c < d, then the median of the list {a, b, c, d} is the mean of b and c; i.e., it is (b + c)/2.
The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors. A disadvantage of the median is the difficulty of handling it theoretically.
· Mode - the most frequent value in the data set
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution.[1] In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score.[2]
Like the statistical mean and the median, the mode is a way of capturing important information about a random variable or a population in a single quantity. The mode is in general different from the mean and median, and may be very different for strongly skewed distributions.
The mode is not necessarily unique, since the same maximum frequency may be attained at different values. The most ambiguous case occurs in uniform distributions, wherein all values are equally likely.
· Geometric mean - the nth root of the product of the data values
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken.
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is 2√2 × 8 = 4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is 3√4 × 1 × 1/32 = ½ .
More generally, if the numbers are , the geometric mean G satisfies
and hence
The latter expression states that the log of the geometric mean is the arithmetic mean of the logs of the numbers.
The geometric mean can also be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a right cuboid with sides whose lengths are equal to the three given numbers.
The geometric mean only applies to positive numbers.[1] It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.
The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between.
· Harmonic mean - the reciprocal of the arithmetic mean of the reciprocals of the data values
In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.
The harmonic mean H of the positive real numbers x1, x2, …, xn > 0 is defined to be
From the third formula in the above equation it is more apparent that the harmonic mean is related to the arithmetic and geometric means.
Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is
· Weighted mean - an arithmetic mean that incorporates weighting to certain data elements
The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.
If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counter-intuitive properties, as captured for instance in Simpson’s paradox.
The term weighted average usually refers to a weighted arithmetic mean, but weighted versions of other means can also be calculated, such as the weighted geometric mean and the weighted harmonic mean.
Example
Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:
Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98
Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99
The straight average for the morning class is 80 and the straight average of the afternoon class is 90. The straight average of 80 and 90 is 85, the mean of the two class means. However, this does not account for the difference in number of students in each class, and the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes:
Or, this can be accomplished by weighting the class means by the number of students in each class (using a weighted mean of the class means):
Thus, the weighted mean makes it possible to find the average student grade in the case where only the class means and the number of students in each class are available.
· Truncated mean - the arithmetic mean of data values after a certain number or proportion of the highest and lowers data values have been discarded.
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both.
For most statistical applications, 5 to 25 percent of the ends are discarded. In some regions of Central Europe it is also known as a Windsor mean, but this name should not be confused with the Winsorized mean: in the latter, the observations that the trimmed mean would discard are instead replaced by the largest/smallest of the remaining values.
· Midrange - the arithmetic mean of the maximum and minimum values of a data set.
In statistics, the mid-range or mid-extreme of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set,[1] or:
As such, it is a measure of central tendency.
The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-robust statistic (having a breakdown point of 0, meaning that a single observation can change it arbitrarily), and it is rarely used in statistical analysis.
The midhinge is the 25% trimmed mid-range, and is more robust, having a breakdown point of 25%.
