Central Tendency - Measures of Central Tendency

Measures of Central Tendency

The following may be applied to one-dimensional data, after transformation, although some of these involve their own implicit transformation of the data.

• Arithmetic mean (or simply, mean) – the sum of all measurements divided by the number of observations in the data set
• Median – the middle value that separates the higher half from the lower half of the data set
• Mode – the most frequent value in the data set
• Geometric mean – the nth root of the product of the data values
• Harmonic mean – the reciprocal of the arithmetic mean of the reciprocals of the data values
• Weighted mean – an arithmetic mean that incorporates weighting to certain data elements
• Distance-weighted estimator – the measure uses weighting coefficients for x_i that are computed as the inverse mean distance between x_i and the other data points.
• Truncated mean – the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
• Midrange – the arithmetic mean of the maximum and minimum values of a data set.
• Midhinge – the arithmetic mean of the two quartiles.
• Trimean – the weighted arithmetic mean of the median and two quartiles.
• Winsorized mean – an arithmetic mean in which extreme values are replaced by values closer to the median.

Any of the above may be applied to each dimension of multi-dimensional data and, in addition, there is the

• Geometric median - which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently.