About Mean Vs Median Vs Mode
Definition of Mean, Mode and Median
We will look at each of the definitions in detail.
- Mean is the average, which is the ratio of the sum of all the given data entities, upon the total number of data entities.
\(\text{Mean}~=~\frac{\text{Sum of the given data entities}}{\text{Total number of data entities}}\)
- Median is the middle value of the given data; the data is first arranged in either ascending or descending order, and then we find the middle value.
We have different methods of finding the mode for either even or an odd number of observations.
If n represents the number of observations, then:
For an odd number of observation: \(\text{Median}~=~\left ( \frac{n+1}{2} \right )^{th} \) term, which is simply the middle term of the data set.
For an even number of observation: \(\text{Median}~=~\left [ \left ( \frac{n}{2} \right )^{th}~+~\left ( \frac{n}{2}+1 \right )^{th} \right ]\) term, which is the average of the two middle values of the data set.
- Mode is the most occurring value in the given observations.
Let us have a look at the differences between mean, median, and mode.
Mean | Median | Mode |
|---|---|---|
The ratio of the sum of all the data entities to the total number of data entities in a data set, is called mean. | The middle value in a data set is called the median. | The most accurate value in a data set is called the mode. |
The formula for obtaining mean does not get affected if the total entities are odd or even. | There are different formulas for finding median, when,
1) the total number of observations are odd \(\text{Median}~=~\left ( \frac{n~+~1}{2} \right )^{th} \) term2) the total number of observations are even \(\text{Median}~=~\left [ \left ( \frac{n}{2} \right )^{th}~+\left ( \frac{n}{2}+1 \right )^{th} \right ] \) term | The method for finding mode remains the same when the total number of observations are odd or even. |
There can be only one mean for a particular data set. | There can be only one median value for a particular data set. | There can be one mode, more than one mode, for a particular data set. |
We do not need to arrange the data in ascending or descending order to find the mean. | Data need to be arranged in a ascending or descending order before finding the median. | It is advisable to arrange the data in ascending or descending form, so that the mode can be obtained easily. |
Solved Mean, Median, Mode Examples
Example 1: Find the mean and median of the given dataset: 15, 20, 25, 16, 14, 17, 23, 7, 17, 6, 10.
Solution:
Given data: 15, 20, 25, 16, 14, 17, 23, 7, 17, 6.
i) \(\text{Mean}~=~\frac{\text{Sum of the given data entities}}{\text{Total number of data entities}} \)
\(\text{Mean}~=~\frac{15~+~20~+25~+26~+~14~+17~+23~+~7~+17~+~6}{10} \) [Simplify]
\(=~\frac{160}{10} \)
\(\text{Mean}~=~16 \)
ii) Mode is the most frequently occurring value.
In the given data number 17 is occurring twice.
Therefore, the mode = 17
Hence, the mean and mode of the given data are 16 and 17 respectively.
Example 2: Find the mode of 6, 16, 9, 17, 27, 7, 48.
Solution:
Given data: 6, 16, 9, 17, 27, 7, 48.
Here in the given data, no value is getting repeated.
So, the data has no mode.
Example 3: Find the median of the given data 15, 28, 6, 10, 19, 30, 45, 22, 35, 31, 40.
Solution:
Given data: 15, 28, 6, 10, 19, 30, 45, 22, 35, 31, 40.
We will first arrange the data in ascending order and then find the middle value.
Data arranged in ascending order is,
6, 10, 15, 19, 22, 28, 30, 31, 35, 40, 45.
The number of observations in this data set = 11, which is odd, hence the median will be the middle value.
Here, the middle value is 28.
Therefore, the median of the given data is 28.
The three measures of center are: Mean, Median and Mode.
The middle value of a data set is called the median.
Median is the middle value of the given observations in a data set. To find the middle value, first arrange the given data in ascending or descending order, and then find the median.
A data set where no value is getting repeated, results in a no mode situation.