# Empirical Relationship Between Mean, Median and Mode

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### June 10, 2020

To understand the concept of mean, median, and mode, the relationship between them and the difference, we first need to know that these concepts form the part of measures of central tendency. It refers to a single value that attempts to describe the characteristic of the entire set of data by identifying the central position within that set. It is sometimes also known as measures of central location. Colloquially, measures of central tendency often referred to only averages. But now we have a lot more than averages. The most common measures of central tendency are the mean, the median, and the mode. We have to remember that the measure of central tendency is generally performed on a finite set of values.

The first concept refers to a ‘mean’. By ‘mean’ we refer to an Arithmetic mean. There are other types of ‘mean’ like geometric mean and harmonic mean which is out of the scope of this article. 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 divided by how many numbers there are.

Example: What is the mean of 6, 7, and 11?

Add the numbers: 6 + 7 + 11 = 24

Divide by how many numbers (i.e. 3 numbers): 24 ÷ 3 = 8

So the simple mean is 8.

To calculate the weighted average, we multiply the given numbers with the required weights and divide the result by the sum of the weights.

Example: What is the mean of 6, 7, and 11 whose respective weights are 2, 5, and 8?

Multiply (6*2)=12, (7*5)=35 and (11*8)=88

Add 12, 35, and 88 = 135

Add the sum of weights, 2 + 5 + 8 = 15

Divide, 135/15 = 9

So the weighted mean is 9.

The second concept refers to a ‘median’. By ‘median’ we refer to the middle number of a given data set when it is arranged in either a descending order or in 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. Let us understand the concept of the median by a simple example.

Suppose, we have a data set of {3, 13, 2, 34, 11, 26, 47}. When sorted in the ascending order, it becomes {2, 3, 11, 13, 26, 34, 47}. The amount of numbers in the data set is seven which is odd. Therefore, the median is the number in the middle, which in this instance is 13.

If we had an even amount of numbers, let's say for example, a data set of {3, 13, 2, 34, 11, 17, 27, 47}. On arrangement it becomes, {2, 3, 11, 13, 17, 27, 34, 47}. The median will be the average of the middle two numbers i.e. 13 and 17. This is calculated as (13 + 17) ÷ 2 = 15.

Coming to the third basic concept we have ‘mode’. 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.

For example, if we have a list of numbers, {3, 3, 6, 9, 16, 16, 16, 27, 27, 37, 48}. So, in this case, 16 appears the most and that is our answer. If our data set was {3, 3, 3, 9, 16, 16, 16, 27, 37, 48}, then we would have 2 modes i.e. 3 and 16. Similarly, if each number in the given data set is unique, then we will have no mode.

So, as individually explained, these were some of the basic measures of central tendency. As you can see, there is a huge difference between mean, median, and mode. But, there is also a relationship between mean, median, and mode. We term this relationship as the empirical relationship between mean, median, and mode.

## Relationship Between Mean, Median and Mode

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

1. 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. Thus, the empirical relationship as Mean – Mode = 3 (Mean – Median).

2. In the case of a frequency distribution which has a symmetrical frequency curve, the empirical relation states that mean = median = mode. 1. In the case of a positively skewed frequency distribution curve, mean > median > mode. 1. In the case of negatively skewed frequency distribution mean < median < mode. We have understood the difference between the mean, median and mode and also its relationship. Let us continue understanding the relationship between mean, median and mode formula with the help of an example:-

Question: It is given that in a moderately skewed distribution, median = 10 and mean = 12. Using these values, find the approximate value of mode.

Solution: Let us take mode to be ‘x’. We have been given that the median = 10 and mean = 12. Now, using the relationship between mean mode and median we get, (Mean – Mode) = 3 (Mean – Median)

So, 12 – x = 3 (12 – 10)

12 – x = 3*2

12 - x = 6

-x = -6 or x = 6

So, Mode = 6

Another example could be, find out the mean when you are given that the median = 20.6 and the mode = 26.

Solution: Let us take a mean to be ‘x’. We have been given that the median = 20.6 and mean = 26. Now, using the relationship between mean, mode, and median, we get - (Mean – Mode) = 3 (Mean – Median)

So, x - 26 = 3 (x – 20.6)

x - 26 = 3x - 61.8

-2x = -35.8

x = 17.9

So, Mean = 17.9

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