Relation Between Mean Median and Mode

October 25, 2022

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In Statistics, “central tendency” is a term used for those statistical measures that identify a single value as representative of an entire distribution. It aims to provide an accurate description of the entire data. It is the single value that is most typical/representative of the collected data.

There are three main measures of central tendency: mean, median & mode. Each of these measures describes a different indication of the typical or central value in the distribution.

Let’s understand the relation between mean median and mode and their properties and uses.

Relation Between Mean Median and Mode(Empirical Formula of Mean Median Mode)

In statistics, for a moderately skewed distribution, there exists a relation between mean, median, and mode. This mean median and mode relation is known as the “empirical relationship” which is defined as Mode is equal to the difference between $3$ times the Median and $2$ times the Mean.

Mathematically, it is written as $\text{Mean } – \text{ Mode } = 3 \left( \text{Mean } – \text{ Median} \right)$.

The above relation between mean median and mode can be rewritten to find any of the measures of central tendencies when the remaining two are known.

Formula to find Mean when Median and Mode are known: $\text{Mean } = \frac{3 \text{ Median } – \text{ Mode}}{2}$
Formula to find Median when Mean and Mode are known: $\text{Median } = \frac{2 \text{ Mean } + \text{ Mode}}{3}$
Formula to find Mode when Mean and Median are known: $\text{Mode } = 3 \text{ Median } – 2 \text{ Mean}$

Examples

Ex 1: Find the mode of the data, using an empirical formula when it is given that median = $41$ and mean = $34$.

According to the empirical formula, $\text{Mode } = 3 \text{ Median } – 2 \text{ Mean}$

$=>\text{Mode } = 3 \times 41 – 2 \times 34 = 123 – 68 = 55$.

Ex 2: Find the median of the data, using an empirical formula when it is given that mean = $20$ and mode = $22$.

According to the empirical formula, $\text{Median } = \frac{2 \text{ Mean } + \text{ Mode}}{3}$

$=>\text{Median } = \frac{2 \times 20 + 22}{3} = \frac{40 + 22}{3} = \frac{62}{3} = 20.67$.

Ex 3: Find the mean of the data, using an empirical formula when it is given that median = $55$ and mode = $60$.

According to the empirical formula, $\text{Mean } = \frac{3 \text{ Median } – \text{ Mode}}{2}$

$=>\text{Mean } = \frac{3 \times 55 – 60}{2} = \frac{165 – 60}{2} = \frac{105}{2} = 52.5$

Mean Median Mode in Statistics

Mean, median, and mode are the measures of central tendency, used to study the various characteristics of a given set of data. A measure of central tendency describes a set of data by identifying the central position in the data set as a single value. We can think of it as a tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendency are Mean, Median, and Mode. Choosing the best measure of central tendency depends on the type of data we have.

When Do We Use Mean Median Mode?

You know there are three measures of central tendencies.

Mean: Also known as arithmetic mean or average is the ratio of the sum of all observations and the count of observations. Read more about Mean.
Median: It is the middlemost value when data items are arranged either in ascending or descending order. Read more about Median.
Mode: It is the most occurring value in the dataset. Read more about Mode.

Following are the uses of Mean Median Mode.

When Do We Use the Mean?

The mean is used when both of the following conditions are met:

Data is Scaled: Data with equal intervals like speed, weight, height, temperature, etc.
Distribution is Normal: The mean is sensitive to outliers that are found in skewed distributions, you should only use the mean when the distribution is more or less normal.

When Do We Use the Median?

The median is used when either one of two conditions is met.

Data is Ordinal: Data with socioeconomic status (“low income”, ”middle income”, ”high income”), an education level (“high school”, ”B.Sc.”, ”M.Sc.”, ” Ph.D.”), income level (“less than 50K”, “50K-100K”, “over 100K”), satisfaction rating (“extremely dislike”, “dislike”, “neutral”, “like”, “extremely like”)
Distribution is Skewed or Non-Normal: The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.

When Do We Use the Mode?

The mode is used when you want to know the most frequent response, number, or observation in a distribution.

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Can Mean Median and Mode be Equal?

When a frequency distribution is perfectly symmetrical, then the mean, median, and mode are equal. Such a symmetrical distribution is commonly known as Normal Distribution.

When $\text{Mean } = \text{ Median } = \text{ Mode}$, the distribution is called a normal distribution.
When $\text{Mode } \lt \text{ Median } \lt \text{ Mean}$, the distribution is called positive skewed where there is a longer or fatter tail on the right side of the distribution.
When $\text{Mean } \lt \text{ Median } \lt \text{ Mode}$, the distribution is called negative skewed where there a longer or fatter tail on the left side of the distribution.

Practice Problems

What is the empirical formula defining the relation between mean, median, and mode?
Find the mean of the data, using an empirical formula when it is given that median = $28$ and mode = $30$.
Find the median of the data, using an empirical formula when it is given that mean = $42$ and mode = $38$.
Find the mode of the data, using an empirical formula when it is given that mean = $10$ and median = $11$.
Check whether the distribution is normal, left-skewed, or right-skewed.
- Mean = $12$, Median = $12$, Mode = $12$
- Mean = $14$, Median = $13$, Mode = $12$
- Mean = $12$, Median = $13$, Mode = $14$

FAQs

What is mean mode & median?

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set

What is the empirical formula for mean median & mode?

The empirical formula showing the relation between mean, median and mode in case of slightly skewed distribution is $\text{Mean } – \text{ Mode } = 3 \left( \text{Mean } – \text{ Median} \right)$.

Conclusion

Mean, median, and mode are the measures of central tendency, used to study the various characteristics of a given set of data. There is a relation between mean median and mode in case of a slightly skewed distribution which is called the empirical formula.

Relation Between Mean Median & Mode – With Examples