3.1.1 Measures of Central Tendency

Central tendency refers to the middle point of a distribution. Its aim is to characterize the statistical data by a single number, representing the order of magnitude of the whole set of observations:

Arithmetic mean

The most commonly used measure of the central tendency is the arithmetic mean or simple average. The arithmetic mean of a variable X with N observations Xi is the ratio of the sum of all observations divided by the number of observations, N.

This measure is commonly used in statistical inference procedures. A major disadvantage of this measure is its sensitivity to departure from symmetry between the left and right extremes of the distribution. In such cases, the preferable measure of centrality is the Median.

Median

The value of the variable X that splits the sample into two halves is called the Median. Fifty percent of cases fall below the median and fifty percent fall above the median. If there is an even number of cases, the median is the average of the two middle cases when they are ordered in ascending or descending order.

If the median value is very different from the mean, then the distribution of the data is skewed.

The median has certain disadvantages. Certain statistical procedures that use the median are more complex than those, which use the mean.

Mode

The mode is the value that occurs in the data set with the highest frequency. If several values share the highest frequency, each of them is a mode. Like the median, the mode is not unduly affected by extreme values.

If the data come from a normal distribution, the mean, median and mode are all equal. If mean and median are very different, most likely there are outliers or the distribution is skewed. If this is the case, then the median is probably a better measure of location. The mean is very sensitive to extreme values and can be seriously contaminate even by one observation!

Quantiles

Consider a variable, whose values Xi are set in ascending order. The quantiles, denoted by q1, q2, qp–1 are the values of X that divide the objects into p equal parts.

The deciles, denoted by q1, q2,……,q9 are the values of X that divide the objects into ten equal parts. Similarly, the quartiles denoted by q1, q2, q3 are the values of X that divide the objects into four equal parts.