Relation between mean median and mode

General
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relation between mean median and mode

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What is the relation between mean, median, and mode?

Answer: The mean, median, and mode are three common measures of central tendency in statistics. Here’s a detailed look at the relationships and differences between these measures:

Mean

The mean, often referred to as the “average,” is calculated by summing all the values in a dataset and then dividing by the number of values. Mathematically, it’s expressed as:

\text{Mean} (\mu) = \frac{\sum{x_i}}{n}

where x_i represents each value in the dataset, and n is the total number of values.

Median

The median is the middle value in a dataset when the values are arranged in ascending (or descending) order. If the dataset has an odd number of values, the median is the middle one. If the dataset has an even number of values, the median is the average of the two middle values.

Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode, more than one mode, or no mode at all.

Relation in Different Distributions

  1. Symmetrical Distribution:
    • In a perfectly symmetrical distribution (such as a normal distribution), the mean, median, and mode are equal.
\text{Mean} = \text{Median} = \text{Mode}
  1. Positively Skewed Distribution (Right-skewed):
    • In a positively skewed distribution, the tail on the right side is longer or fatter than the left side. Here, the mean is typically greater than the median, which is greater than the mode.
\text{Mode} < \text{Median} < \text{Mean}
  1. Negatively Skewed Distribution (Left-skewed):
    • In a negatively skewed distribution, the tail on the left side is longer or fatter than the right side. Here, the mean is typically less than the median, which is less than the mode.
\text{Mean} < \text{Median} < \text{Mode}

Empirical Relationship (For Moderately Skewed Distributions):

For moderately skewed distributions, there is an empirical relationship between the mean, median, and mode, often referred to as Pearson’s relation. It can be expressed as:

\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})

Rearrangement gives:

\text{Mode} = 3(\text{Median}) - 2(\text{Mean})

This relation is useful for estimating the mode if the median and mean are known.

Summary

Each of these measures of central tendency provides different insights:

  • Mean is sensitive to outliers and skewed data.
  • Median provides a better central value in skewed distributions.
  • Mode provides the most frequent value, useful in categorical data.

Understanding the underlying distribution of your data helps in choosing the appropriate measure of central tendency and interpreting the relationships between the mean, median, and mode.

This comprehensive explanation should help in understanding the relationships and differences between the mean, median, and mode, and how they are affected by different types of distributions.