Relationship of mean, median and mode | Derivation
The relationship between mean median and mode is the difference between mean and mode is almost equal to the three times of the difference between mean and median. It is also known as the empirical relation between mean median and mode. it is given by,
Mean − Mode = 3 (Mean − Median)
Empirical relation between mean median and mode
The difference between Mean and mode is equal to the three times the difference between mean and median. that is given by, Mean − Mode = 3 (Mean − Median).
As we understand that the empirical relation between mean median and mode is Mean − Mode = 3 (Mean − Median). but how to get relation to the reference of mean, median and mode. so here below are the relation between mean median and mode taken mean, median, mode as single term.
general relation between mean, median and mode is, Mean − Mode = 3 (Mean − Median).
take Mean as single term:-
Mean − Mode = 3 (Mean − Median)
Soln:-
= Mean − Mode = 3 Mean − 3 Median
= 3 Median − Mode = 3 Mean − Mean
= 3 Median − Mode = 2 Mean
= (3 Median − Mode)/2 = Mean
∴ Mean = (3/2 Median − 1/2 Mode
take Median as single term:-
Mean − Mode = 3 (Mean − Median)
soln:-
= Mean − Mode = 3 Mean − 3 Median
= 3 Median − Mode = 3 Mean − Mean
= 3 Median − Mode = 2 Mean
∴ Median = 2/3 Mean + 1/3 Mode
take mode as single term:-
Mean − Mode = 3 (Mean − Median)
Soln:-
= Mean − Mode = 3 Mean − 3 Median
= 3 Median − Mode = 3 Mean − Mean
= 3 Median − Mode = 2 Mean
∴ Mode = 3 Median − 2 Mean
Please note:-
- that empirical relation between mean median and mode for normal distribution (or symmetrical distribution) is mean = median = mode. on the other hand the mean median and mode are equal to each other for normal distribution (or symmetrical distribution).
- The empirical relation between mean median and mode for asymmetrical distribution is mean ≠ mode ≠ median. on the other hand the mean median and mode are not equal to each other for asymmetrical distribution.
Derivation of relation between mean median and mode
Mean − Mode = 3 (Mean − Median)
= Mean − Mode = 3 Mean − 3 Median
= 3 Median = 3 Mean − Mean + Mode
= 3 Median = 2 Mean + Mode
= Median = 2/3 Mean + 1/3 Mode
Hence, the formula of relation between mean median and mode is Median = 2/3 Mean + 1/3 Mode.
Please note that you can also understand mean median and mode by the following examples.
mean median and mode examples
Suppose we want to compare the wage distribution of workers in two factories and determine which factory pays more to its workers. If we compare on the basis of individual workers, we cannot conclude anything. However, if for the given data, we get a representative value that signifies the characteristics of the data, the comparison become easy.
A certain value representative of the whole data and signifying its characteristics is called an average of the data.
Three types of averages are useful for analysing data. They are:-
- Mean
- Median
- Mode
The relation between mean median and mode is a very famous relation and it is also called the empirical relation between mean median and mode as we know above. here is also a complete information about mean median and mode. we all are studying mean median and mode from class 10. but In class 10 we have a little bit idea that what is mean median and mode. here we will discuss about the actual meaning of mean median and mode. like what is mean? what is median? what is mode? in detail. so lets start with Mean.
What is mean?
Mean is that value of a given static data or for a data set, which is the sum of the values divided by the number of values. the arithmetic mean, also called the expected value or average value of any set number.
Mean can be calculated by various method like Direct method, Assumed-Mean method, Step-Deviation method, etc. every student study all these types of method in class 10.
What is Median?
Median is defined as the middle value for a data set. In statistics the median is the value separating the higher half from the lower half of a given data.
What is Mode?
It is that value of a variate which occurs most often. More precisely, mode is that value of the variable at which the concentration of the data is maximum.