Mean, Median, and Mode

Central Tendency

• A central tendency is a central or typical value for a probability distribution

• The most popular measures of central tendency include the following:

  • Mean
  • Median
  • Mode

What is the Mean?

• Roughly speaking, the mean tries to represent the most frequent value from a population distribution while putting a greater weight on the outliers
• The mean can be thought of as an estimate of the mode

• We use the mean as an estimate of the mode because:

  • The formula of the mean is very lightweight
  • We don't have a formula for the mode

• Sometimes the mean represents the most frequent value in our data, but sometimes the mean does not represent the most frequent value in our data

  • In other words, sometimes the mean is a great estimate of the mode, but sometimes the mean is a poor estimate of the mode
  • If our data is normally distributed, the mean is a perfect estimate of the mode
  • If our data is not normally distributed, the mean is usually not a great estimate of the mode

What is the Median?

• Roughly speaking, the median is the middle value in the ordered list of data values
• In other words, the median is the value where there is an equal percentage of data above and below our value
• The formula of the median is heavyweight, so we typically prefer to use the formula for the mean when estimating the mode

• Sometimes the median represents the most frequent value in our data, but sometimes the median does not represent the most frequent value in our data

  • In other words, sometimes the median is a great estimate of the mode, but sometimes the median is a poor estimate of the mode
  • If our data is normally distributed, the median is a perfect estimate of the mode
  • If our data is not normally distributed, the median is usually not a great estimate of the mode

What is the Mode?

• The mode of a set of data values is the value that appears most often
• In other words, the mode is the most probable value
• For a discrete random variable, the mode is the maximum value of its probability mass function
• For a continuous random variable, the mode is the maximum value of its probability density function

• The mean, median, and mode all coincide for symmetric unimodal distributions

  • For example, the mean, median, and mode all coincide with each other for a normal distribution
  • A distribution is considered to be unimodal if it has only one local maxima
  • Also, when a random variable has a probability distribution with multiple local maxima, then it is common to refer to all of the local maxima as modes
• If we draw samples, and the sample and population distributions are both symmetric unimodal distributions, then the sample mean can be used as an estimate of the population mode

References

• Intuition behind Expected Value
• Mode Wiki