Measures of Central Tendency

The most frequently used measures of central tendency are the averages. An average can be either weighted or un-weighted. There are three kinds of averages or means, Arithmetic, Geometric and Harmonic. If all numbers in a series are positive, the following relationship between them will always hold;
A >= G >= H,
where equality holds only if all numbers in the series are equal. This relationship holds for weighted as well as un-weighted averages, as long as the weights are the same.

In addition to averages, or means, will median, quintiles, fractiles and mode be introduced.

The un-weighted arithmetic mean is possibly the most frequently used measure of an average, or mean, value. The un-weighted arithmetic mean is often called a “simple” average since all numbers (x_i ) are represented once and given the same weight. The un-weighted arithmetic mean is derived by summing up the numbers and divide by count of numbers (n). 

 

The un-weighted arithmetic average can easily be computed and it is the measure that most people intuitively understand as a measure of average. However, that fact does not make it suitable for all purposes.

A weighted arithmetic mean is obtained by multiplying each number (x_i ) by its weight (w_i ), adding these products, and divide the sum of the products by the sum of the weights.



Weighted arithmetic means are commonly used to construct price and volume indices, e.g., a Laspeyres price index is a weighted arithmetic average of price relatives.

In Bank publications, weighted arithmetic averages are often computed for a group of countries with the countries' population as weights. E.g., the average GNP per capita for a group of countries is computed as a population weighted arithmetic average of the individual countries' GNP per capita in US-dollar terms. This is in practice equal to the sum of the individual countries' GNP divided it by the sum of the countries' population.

An un-weighted geometric mean of a series of numbers is obtained as the n-root of the product of these numbers.




An alternative procedure is to compute the logarithm to each number in the series and derive ln G as the arithmetic mean of the logarithms. G is then derived as the exponential to ln G.




The geometric mean is mostly used for averaging rates of change or ratios. Taking the geometric mean of growth rates, or changes, is equivalent with deriving a compounding growth rate. Using a geometric mean is the most common way of estimating growth in economic aggregates over a period of time. (Note! The Bank uses the least square method for this purpose.)
Furthermore, in the compilation of price indices, geometric averages are sometimes used to average price relatives at the most elementary level where no weights are available. A geometric average is typically used to avoid giving undue importance to a few extreme values.

A weighted geometric mean of a series of numbers is obtained as the product of these numbers raised in the power equal to one over the sum of the weights.

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the numbers in the series. 

 

Harmonic means, or weighted harmonic means are commonly used in averaging indices. A Paasche index is a weighted harmonic mean, and one of the most well-known Paasche indices is the GDP-deflator.

Relationship between the different means

As stated above, the following relationship between the three types of means will always hold if all values of the series are positive; A >= G >= H (-- Equality will hold only if all numbers in the series are equal).

A simple example will show how to compute each of the means, and show that the above given relationship between the arithmetic, the geometric and the harmonic means holds.

The median is also a measure of central tendency. The median of a set of ‘n’ numbers, x₁, x₂, .....,x_n, is the middle value when the numbers are arranged from smallest to largest. If ‘n’ is an odd number, there exists a unique middle value, and it is the median. If ‘n’ is an even number, there are two middle values, and the median is defined as their average. Roughly speaking, the median is the value that divides the data series into equal halves, 50 percent of the observations lie below the median and 50 percent lie above it. Some prefer to call the median the position average.

The difference between the arithmetic average and the median tells something about the distribution of the series. The arithmetic average is a weighted center of a data series in the sense that the series balances even at the point of the arithmetic mean. The arithmetic mean is influenced by any extreme values (small or large) in the series. The median, on the other hand, is not influenced by those extreme values. This point is illustrated in the example below.

The example above illustrates that the median is likely to be a more sensible measure of center than the arithmetic mean when the distribution is (extremely) asymmetrical. This is why reports on income distributions frequently quote the median as a summary measure, rather than the average.

If the number of observations are quite large, it is sometimes useful to extend the notion of the median and to divide the data into quartiles (4), quintiles (5), deciles (10), or percentiles (100) -- fractiles. A fractile is a value at or below which lies a given fraction of a data series. Data are arranged from the smallest number to the largest, so that the value of the third quartile will always be higher than (or at least as high as) the first and the second (the median).

A fractile is found as item (p/q)*n + ½ th, and is measured as the value of this item, where p is the fractile number, q is the number of parts that the series is to be divided, and n is the number of items in the series.

If the numbers of a data series tend to concentrate around one value, this number is called the mode. The mode is in that sense the most typical, or frequently observed, value in a series. The mode is therefor the value that a number selected at random is most likely to take. To obtain the mode, data should be graphed or presented with their frequencies.