Relation Between Mean Median and Mode

The relation between mean, median, and mode is very important to understand in statistics and is useful while dealing with similar problems. In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median. i.e., (Mean - Mode) = 3 (Mean - Median). This can be rearranged as 3 Median = 2 Mean + Mode and this is easy to remember.

Before learning about the relation between mean, median, and mode formula, let us revise the concepts of mean, median, and mode.

• The arithmetic mean refers to the average of a data set of numbers. It can either be a simple average or a weighted average. To calculate a simple average, we add up all the numbers given in the data set and then divide it by the total frequency.
• The median is the middle number of a given data set when it is arranged in either a descending order or ascending order. If there is an odd amount of numbers, the median value is the number that is in the middle whereas if there is an even amount of numbers, the median is the simple average of the middle pair in the dataset. Median is much more effective than a mean because it eliminates the outliers.
• The mode refers to the number that appears the most in a dataset. A set of numbers may have one mode, or more than one mode, or no mode at all.

The formula to define the relation between mean, median, and mode in a moderately skewed distribution is 3 (median) = mode + 2 mean. The proof of the mean, median, mode formula can be understood using Karl Pearson’s formula, which states:

(Mean - Median) = 1/3 (Mean - Mode)

3 (Mean - Median) = (Mean - Mode)

3 Mean - 3 Median = Mean - Mode

3 Median = 3 Mean - Mean + Mode

3 Median = 2 Mean + Mode

We will understand the empirical relation between mean, median, and mode by means of a frequency distribution graph. We can divide the relationship into four different cases:

• In the case of a moderately skewed distribution, Mean – Mode = 3 (Mean – Median).
• In the case of a frequency distribution that has a symmetrical frequency curve, the empirical relation states that mean = median = mode.
• In the case of a positively skewed frequency distribution curve, mean > median > mode.
• In the case of negatively skewed frequency distribution, mean < median < mode.

Observe the graph for each of the above-mentioned cases and understand the relation and positioning of mean, median, and mode in the frequency distribution.

☛Related Topics:

• Mean, Median and Mode
• Mean Median Mode Formula
• Difference Between Mean and Median
• Measures of Central Tendency

What is the Relation Between Mean Median and Mode Class 10?

The most general relationship between mean, median, and mode is defined as 3 median = mode + 2 mean. This relation works for a moderately skewed frequency distribution.

What is the Relationship Between Mean, Median, and Mode for Positive Skewness?

The relationship between mean, median, and mode for positive skewness is mean > median > mode. Here, mean is greater than median and mode, and mode is the smallest value among mean, median, and mode.

What is the Empirical Relation between Mean Median and Mode?

The empirical mean, median, and mode relation is Mean – Mode = 3 (Mean – Median) for moderately skewed frequency distribution.

How are the Mean, Median, and Mode Related?

In general, mean, median, and mode are related with a formula 3 median = 2 mean + mode. If any two values are given, we can find the third value by using this formula.

What is the Relationship Between Mean, Median, and Mode in a Symmetrical Distribution?

In a symmetrical frequency distribution, mode = median = mean. In short, all three values are equal.