Question

for an ungrouped data 10,15,20,25,15,x median and mode are equal.find x and also mean for the givendata

Answer 1

Concept we will be using:
i) Mode of a data set is the value with the highest frequency. A data set can have more than one mode.
ii) Median  of a data set is the value of the middle term. A data set can have only one median.
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Solution:
The given data set is 10, 15, 20, 25, 15 and x

Step1 : Arrange the data set in ascending order excluding x.
10, 15, 15, 20, 25

Step 2: Identify the mode for which the given relation, median and mode are equal.
In the given data set 15 occurs 2 times.

Case 1: If x=10, then 10 will also occur 2 times and in that case 10 and 15 will be the mode, which is not possible because mode and median are equal and a data set can have only one median.
Therefore, we reject case 1.

Case 2: If x=15, then 15 will occur 3 times( which is the highest frequency) and in that case 15 will be the mode.
Then, the data set will be 10, 15, 15, 15, 20 and 25 ( in ascending order)

Median of a data set is the value of the middle term.
If the number of observation is even, then
$Median= \frac{\text{Value of } \frac{n}{2}\text{th term} +\text{value of } (\frac{n}{2}+1)\text{th term}}{2}$ where n= number of observations
Here, number of observation is,n= 6 which is even.
Therefore,
[tex]Median\\ = \frac{\text{Value of } \frac{n}{2}\text{th term} +\text{value of } (\frac{n}{2}+1)\text{th term}}{2}\\ = \frac{\text{Value of } \frac{6}{2}\text{th term} +\text{value of } (\frac{6}{2}+1)\text{th term}}{2}\\ =\frac{\text{Value of 3rd term} +\text{value of 4th term}}{2}\\ =\frac{15+15}{2}\\ =\frac{30}{2}\\ =15[/tex]

Therefore, Median = 15 and Mode =15
Both, Median and Mode are equal.

Case 3:  For any other value of x, the data set will have unequal median and mode. 


Therefore, x=15.


[tex]Mean\\ = \frac{\text{Sum of the values}}{\text{Number of observations} }\\ =\frac{10+15+15+15+20+25}{6}\\ =\frac{100}{6}\\ =16.67[/tex]


Answer : x=15 and mean=16.67

Answer 2

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