Measures of central tendency are statistical measures that provide a single, central value representing a set of data.

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These measures give an idea of where the center of the data distribution lies. The three main measures of central tendency are the mean, median, and mode.

Mean:

The mean, also known as the average, is calculated by summing up all the values in a dataset and dividing the sum by the number of observations. The formula for the mean ((\bar{X})) of a set of (n) values is:
[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} ]
The mean is sensitive to extreme values, and it may not accurately represent the center of the distribution if there are outliers.

Median:

The median is the middle value of a dataset when it is arranged in ascending or descending order. If there is an even number of observations, the median is the average of the two middle values. The median is less sensitive to extreme values and is a good measure of central tendency for skewed distributions.

Mode:

The mode is the value that occurs most frequently in a dataset. A dataset may have one mode (unimodal), more than one mode (multimodal), or no mode at all. The mode is especially useful for categorical data, and it is less affected by extreme values in numerical datasets.

These measures have different properties and are applicable in various situations:

For Symmetrical Distributions:
For symmetrically distributed data with no outliers, the mean, median, and mode are approximately equal.
For Skewed Distributions:
In positively (right) skewed distributions, the mean is typically greater than the median, and the median is greater than the mode. In negatively (left) skewed distributions, the relationships are reversed.
For Normal Distributions:
In a perfectly normal distribution, the mean, median, and mode coincide at the center of the distribution.
For Categorical Data:
The mode is particularly useful for categorical data, where it represents the most frequently occurring category.
For Outliers:
The mean is sensitive to outliers, meaning that extreme values can disproportionately influence it. In the presence of outliers, the median is often a more robust measure of central tendency.

It’s important to choose the appropriate measure of central tendency based on the characteristics of the data and the specific goals of the analysis. While the mean is commonly used due to its mathematical simplicity, the median and mode offer alternatives that may be more suitable in certain situations, especially when dealing with non-normally distributed or skewed data.