Relation Between Mean Median and Mode

Mean has been learned by most of the people in middle school math as the norm. The mean is what you get by combining all the values and dividing the total by the number of values, given a set of values. You have a set of values called the population X1,......,Xn written in math notation.

The mean number is very helpful. It describes the group’s property. It is important to understand that the mean is not an entity-in reality, there can be anything whose value matches the mean, but the mean is a summarized representation of the population.

What is meant by Median?

Often the median is a better representative of a standard group member. If you take all the values in a list and arrange them in increasing order, the median will be the number located at the centre. The median is a quality that belongs to any member of the group. Based on the value distribution, the mean may not be particularly close to the quality of any group member. The mean is also subjected to skewing, as few as one value significantly different from the rest of the group can change the mean dramatically. Without the skew factor introduced by outliers, the median gives you a central group member. If you have a normal distribution, a typical member of the population will be the median value.

What is meant by Mode?

The mode is the group’s most common member. Whether it’s the largest or smallest value in the group, it doesn’t matter whatever the most common value is the mode. Most of these three median measures are the least commonly used, and that’s because they are usually the least meaningful. But it is helpful once in a while. If your data is regular as well as perfect, the median, mean, and mode would all be the same. In real life, this is almost impossible and never happens.

The Relation between Mean, Median and Mode:

The relationship between mean median and mode can be expressed by using Karl Pearson’s Formula as:

(Mean - Median) = ⅓ (Mean - Mode)

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

Mode = Mean - 3(Mean - Median)

Mode = 3 Median - 2 Mean

Thus, the above equation can be used when any of the two values are given and you need to find the third value.

Let’s take a look at a few of the examples:

Example 1:

Suppose we wanted to know something about people’s income. And we’ll imagine a group of 24 individuals. They make income in order as stated below:

$20,000,

$20,000,

$22,000,

$25,000,

$25,000,

$25,000,

$28,000,

$30,000,

$31,000,

$32,000,

$34,000,

$35,000,

$37,000,

$39,000,

$39,000,

$40,000,

$42,000,

$42,000,

$43,000,

$80,000,

$100,000,

$300,000,

$700,000, and

$3,000,000.

Answer:

For calculation of the mean,

The sum of the income is $4789000

As there are total 24 members, 

Mean = 478900024 ÷ 24 

Mean =   199541.66666      

This group’s mean income is about

$199,541.66666.

Now, we set the values for the median, with half of the values on one side and half on the other. We’re going to write the number of thousands without the trailing numbers “000” to make it fit:

20, 20, 22, 25, 25, 28, 30, 31, 32, 34, 35, 37,39,

39, 40, 42, 42, 43, 80, 100, 300, 700, 3000

For the calculation of Median,

Thus, the median is the price on either side of it with the same number of things. 

In this scenario, we have an even number of members in our community, which implies we should choose one of the two.

In this case, as we look at uneven wage distributions, skewing it up and picking the larger of the two will reduce the imbalance, but certainly do not eliminate it. 

Skewing it down will make the difference even bigger than earlier, which will exaggerate the results. 

Thus, picking up the middle values which is higher:

$37000.

For the Mode,

From the definition, the mode in this example is

$25000 .

As, the value

$25000 have fallen in the group most frequently.

This case clearly shows the mean’s skewed effect. The mean value is greater than 80% of the group’s actual members. Even if we select the larger of the median’s two possible values, the mean is more than 5 times larger than the median. The median is a much better measure of the group’s typical member. The mode is not particularly meaningful in this case.

For many techniques people use this tool with statistics, use of the median where the mean is more acceptable, or the mean where the standard is more suitable, is generally followed.

For example, when we discuss income, it’s a very common trick to talk about how a large group of people's income has increased. When, in fact, the group’s typical member has not got any increase-instead one or two outliers have had huge increases, and nobody else has got anything.

Suppose you had 10 employees and you gave them the percentage increase in income as follows:

-2%, -2%, 0%, 0%, 0%, 1%, 3%, 20%

Then the median wage change would be +2%.

Yet, half of the workers saw either no improvement or a, and in addition, virtually all the raise went to just one person.

Take that one user out and the raise goes up by almost a factor of 20% to 0.11% on the median.

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