Correct Answer - Option 4 : Only Mode
Concept:
Mean = \(\rm \frac{\sum x}{n}\)
Median
Case 1: If number of observation (n) is even
Median = \(\rm \frac12\)(Value of \(\rm \frac n2\)th observation + Value of \(\rm (\frac n2 + 1)\)th observation )
Case 2: If number of observation (n) is odd
Median = Value of \(\rm \frac {n+1}{2}\)th observation.
Mode: Mode is that value of the observation which occurs maximum number of times.
Calculation:
Given data: 8, 15, 18, 8, 15, 20, 14, 18, 10, 8.
Let us arrange the following data in ascending order to find out the median and mode,
8, 8, 8, 10, 14, 15, 15, 18, 18, 20
Total number of observation = 10
Mean = \(\rm \frac{\sum x}{n}\) = \(\rm \frac{8+8+8+10+14+15+15+18+18+20}{10}=\frac{134}{10}\)
Here, number of observation (n) is even
∴ Median = \(\rm \frac12\)(Value of \(\rm \frac n2\)th observation + Value of \(\rm (\frac n2 + 1)\)th observation )
\(=\frac{14+15}{2}=\frac{29}{2}\)
From above observation 8 occurs maximum number of times.
∴ Mode = 8
But, one of the values is wrongly written as 18 instead of 8 (means we will remove 18 and add 8)
∴ Corrected data: 8, 8, 8, 10, 14, 15, 15, 8, 18, 20
⇒ 8, 8, 8, 8, 10, 14, 15, 15, 18, 20
Mean = \(\rm \frac{\sum x}{n}\) = \(\rm \frac{8+8+8+8+10+14+15+15+18+20}{10}=\frac{124}{10}\)
∴ Median \(=\frac{10+14}{2}=\frac{24}{2}=12\)
From above observation 8 occurs maximum number of times.
∴ Mode = 8
So, we can see that only mode is unchanged.
Hence, option (4) is correct.
