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Central tendency

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Central tendency is a term in descriptive statistics which gives an indication of the typical score in that data set. The three most common measures typically used for this: the mean, the median (not to be confused with Medium) and the mode.[1][2] However, there are other measures of central tendency.

Average is an often-used term that may refer to any measure of central tendency, though in casual conversation it is generally assumed to refer to the mean.

Averages[edit]

Arithmetic Mean[edit]

The arithmetic mean is easily calculated by summing up all scores in the sample and then dividing by the number of scores in that sample. The mean for the sample 5, 6, 9, 2 would therefore be calculated as follows:

\frac{5+6+9+2}{4}{=5.5}

The mathematical formula can be expressed as:

\frac1n\sum_{i=1}^na_i

where n is the total number of samples.

The above is the arithmetic mean; there are also other means, such as the geometric mean and the harmonic mean, but usually the arithmetic mean is meant if the type of mean is not specified.

Median[edit]

The median is officially defined as the value that lies in the middle of the sample, i.e. it has the same number of scores above as below it.[1] It is calculated by ranking all the scores in order and taking the one in the middle. When there is an even number simply take the mean of the two in the middle. For example, the median for the sequence 2, 12, 12, 19, 19, 20, 20, 20, 25 would be 19, as it is the fifth rank and there are nine scores. However, if we were to take the sequence 2, 12, 12, 19, 19, 20, 20, 20, 25, 25, the median would be 19.5 as 19.5 is the mean of 19 and 20, which are the two scores in the middle.

Mode[edit]

The mode is simply the most frequently occurring score. For example, the mode of the sample 3, 3, 2, 1, 4, 3 would be 3. If there is a second number that occurs just as frequently within the sample, then it can be described as having two modes or being 'bimodal'. However, some books will identify such a sample as having no mode.[3]

Comparison[edit]

The three methods can be useful in different ways, and how they relate can give information about your statistical sample. The mean is the most intuitive measure of the concept of "average", while the median is most useful for breaking samples into groups (e.g., quartiles). For evenly distributed samples, or symmetrically distributed samples, the median and mean (and usually the mode) should match. The difference between them is an indicator of how skewed the data is. For instance, in economics, income is not evenly distributed (not by a long shot), nor even symmetrically distributed, and the mean value is easily shifted by those high earners at the top — for this reason the median is most often used.[4][5]

Other measures of central tendency[edit]

  • Geometric Mean: all the values are multiplied and then the nth root is taken (where n is the total number of scores). Useful in some geometric (heh) contexts; for example, the area of an ellipse is equal to that of a circle whose radius is the geometric mean of the ellipse's semi-major and semi-minor axes. Has the neat property of being equal to e raised to the arithmetic mean of the natural logarithms of the values being averaged. (The same principle holds for other bases, so for example it could be defined as ten raised to the arithmetic mean of the common logarithms.) The mathematical formula can be expressed as:
\left(\prod_{i=1}^n x_i\right)^\frac1n
  • Harmonic Mean: the mean obtained by taking the reciprocal of the arithmetic mean of the reciprocals of a set of (nonzero) numbers. One of the most memorable uses of the harmonic mean is in physics, where the equivalent resistance of a set of resistors in parallel is the harmonic mean of their resistances divided by the number of resistors. The same principle applies to capacitors placed in series. The mathematical formula can be expressed as:
\frac{n}{\displaystyle\sum\limits_{i=1}^n\frac1{x_i}}
  • Weighted Mean
  • Truncated Mean
  • Midrange

References[edit]

  1. 1.0 1.1 C. P. Dancey & J. Reidy (2002) Statistics without maths for psychology, 37
  2. About.com - Median, mean, and mode
  3. http://www.purplemath.com/modules/meanmode.htm
  4. See the Wikipedia article on median household income.
  5. See the Wikipedia article on mean household income.