Question

For the data 2, 4, 6, 8, 9, 19, 8, 2, 4, 6, 8, 9 then
A) \[{\text{median}} = {\text{mode}}\]
B) \[{\text{median}} > {\text{mode}}\]
C) \[{\text{median}} > {\text{mode}} = {\text{mean}}\]
D) None of these

Answer 1

Hint:
Here we will use the basic concept of the mean, mode, and median. We will first rearrange the data in ascending order and find the mean using the formula. We will then find the mode using the definition of mode. To find the median we will first count the total number of terms in the data and apply suitable formulas. We will then compare the result and find which relation is correct.

Complete step by step solution:
The given data is 2, 4, 6, 8, 9, 19, 8, 2, 4, 6, 8, 9.
First, we will arrange the given data in the proper form i.e. in the increasing order to simplify the given data.
So, the data will be 2, 2, 4, 4, 6, 6, 8, 8, 8, 9, 9, 19.
Now we will find the mean of the given data. We know that mean is equal to the ratio of the sum of the numbers to the total count of the numbers. Therefore, we get
\[{\text{mean}} = \] sum of the numbers \[ \div \] total observation
Substituting the values in the above equation, we get
\[ \Rightarrow {\text{mean}} = \dfrac{{2 + 2 + 4 + 4 + 6 + 6 + 8 + 8 + 8 + 9 + 9 + 19}}{{12}} = \dfrac{{85}}{{12}}\]
Now we will calculate the mode of the given data. We know that mode is the most repeating term of a given data. Now, we can see the number 8 is repeated for a maximum number of times in the above data. Therefore, the mode of the given data is equal to 8.
\[{\text{mode}} = 8\]
Median is the middle value of the given list of numbers or the value which is separating the data into two halves i.e. upper half and lower half.
Here, the total number of observation is 12, which is even so we will use the formula \[{\text{median}} = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}}}{2}\] observation.
Now substituting \[n = 12\] in the formula of the median, we get
\[ \Rightarrow {\text{median}} = \dfrac{{6 + 8}}{2} = 7\]
Therefore, we can clearly see that mode is greater than the median of the given data.
\[{\text{mode}} > {\text{median}}\]
Hence none of the given options is true.

So, option D is the correct option.

Note:
Here, we should keep in mind that we must arrange the given data in either ascending or descending before finding the mean and median of the data. If we will not arrange the data then we will get the wrong answer. Now we know that the formula of the median for an even number of data and an odd number of data is different. We can make a mistake if we do not count the total number of observations and use the wrong formula. This will give us the wrong answer.