Measures of Central Tendency: Mean, Median, Mode, and Weighted Mean

Measures of Central Tendency

Measures of central tendency are statistical values that represent the typical or average value of a dataset. In statistics, we commonly use three measures of central tendency: mean, median, and mode. This article provides a detailed explanation of these measures, their advantages, and when to use them.

Mean

The mean, also known as the arithmetic mean, is calculated by summing all the values in a dataset and then dividing by the total number of observations. Mathematically, it is expressed as the sum of individual observations (xi) divided by the number of observations (N):

The mean is the most commonly used measure of central tendency. It is simple to calculate, and it provides a good representation of the dataset when all observations are considered equally. However, the mean is sensitive to outliers and extreme values, especially when the sample size is small.

Median

The median is the value that divides the dataset into two equal halves when arranged in ascending or descending order. If there are an odd number of observations, the median is the value at the middle position. If there are an even number of observations, the median is the average of the two middle values.

The median is not sensitive to outliers and is more robust to extreme values than the mean. It is especially useful when there are extreme values or when the distribution is skewed.

Mode

The mode is the value that occurs most frequently in a dataset. If all values occur only once, the dataset does not have a mode. In a bimodal distribution, there are two values that occur most frequently.

The mode is the only measure of central tendency that can be used for data measured in a nominal scale. It is not sensitive to outliers and is easy to calculate.

Weighted Mean

The weighted mean is a measure of central tendency that takes into account the relative importance of each observation. It is calculated by multiplying each observation by its corresponding weight, then summing the products and dividing by the total weight.

Weighted mean is used when certain values in a dataset are more important than others, and it provides a more accurate representation of the central tendency when the values are not equally important.

In conclusion, the choice of the appropriate measure of central tendency depends on the nature of the data and the specific context. The mean is generally considered the best measure of central tendency and is the most frequently used one. However, the median and mode may be preferred in certain situations, such as when there are few extreme scores, some scores have undetermined values, or data are measured in a nominal scale. Weighted mean is used when some data are given higher or lower weights depending on the objectives.