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Question
The numbers 10, 12, 14, 16, 17 and x are in ascending order. If the mean and median of these observations are same, the value of x is ______.
Options
16
14
54
21
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Solution
The numbers 10, 12, 14, 16, 17 and x are in ascending order. If the mean and median of these observations are same, the value of x is 21.
Explanation:
Numbers in ascending order:
10, 12, 14, 16, 17 and x.
\[\text{Mean} = \frac{\text{Sum of observations}}{\text{No. of observations}}\]
Substituting values we get:
\[\text{Mean} = \frac{10 + 12 + 14 + 16 + 17 + x}{6}\]
\[{} = \frac{69 + x}{6}.\]
No. of terms (n) = 6, which is even.
\[\text{Median} = \frac{\left(\frac{n}{2}\right) \text{ th term} + \left(\frac{n}{2} + 1\right) \text{ th term}}{2}\]
Substituting values we get:
\[\text{Median} = \frac{\left(\frac{6}{2}\right) \text{ th term} + \left(\frac{6}{2} + 1\right) \text{ th term}}{2}\]
\[{} = \frac{3\text{rd term} + 4\text{th term}}{2}\]
\[{} = \frac{14 + 16}{2}\]
\[{} = \frac{30}{2}\]
\[{} = 15.\]
Given,
\[\text{Mean} = \text{Median}\]
\[\therefore \frac{69 + x}{6} = 15\]
\[\Rightarrow 69 + x = 15 \times 6\]
\[\Rightarrow 69 + x = 90\]
\[\Rightarrow x = 90 - 69 = 21.\]
