10_R

A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset [2]. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution. In statistics, the three most common measures of central tendency are the mean, median, and mode. Each of these measures calculates the location of the central point using a different method.

Mean

The arithmetic mean is the most common measure of central tendency [3]. It is simply the sum of the numbers divided by the number of numbers. The symbol "μ" is used for the mean of a population. The symbol "M" is used for the mean of a sample. The formula for μ is shown below:

μ = ΣX/N

where ΣX is the sum of all the numbers in the population and N is the number of numbers in the population.

The formula for M is essentially identical:

M = ΣX/N

where ΣX is the sum of all the numbers in the sample and N is the number of numbers in the sample.

There are a few variants of the mean [1]:

Weighted mean In ordinary mean, all data points are treated equally and equal weights are allocated (implicitly) to all data points. In weighted mean, some data are given higher (or lower) weights depending on the objectives.
Geometric mean Unlike ordinary mean, geometric mean multiplies N values and take N-th root of the product. So, for two values 2 & 8, the geometric mean would be 4.
Harmonic mean It is another kind of mean that is calculated by taking reciprocals of data points, then taking their overage and finally taking reciprocal of the result.

Median

The median is the middle value. It is the value that splits the dataset in half [4]. To find the median, data is sorted from smallest to largest, and then find the data point that has an equal amount of values above it and below it. The method for locating the median varies slightly depending on whether a dataset has an even or odd number of values.

[2, 5, 6, 7, 9, 10, 12, 13, 15, 16, 18, 21, 23]

In the dataset with the odd number of observations, notice how the number 12 has six values above it and six below it. Therefore, 12 is the median of this dataset.

[17, 19, 22, 23, 24, 26, 27, 19, 30, 32, 33, 35, 38, 40]

When there is an even number of values, count in to the two innermost values and then take the average. The average of 27 and 29 is 28. Consequently, 28 is the median of this dataset.

Mode

The mode is the value that occurs the most frequently in the data set [4]. On a bar chart, the mode is the highest bar. If the data have multiple values that are tied for occurring the most frequently, it has a multimodal distribution. If no value repeats, the data do not have a mode.

In the dataset below, the value 5 occurs most frequently, which makes it the mode.

[5, 5, 5, 4, 4, 3, 2, ,2, 1]