For the data $1, 5, 7, x + 1, 9, x - 2, 3$ , if mean is $4$ , find the value of $x$ . Also, find the mode of the data.

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Solution

Step 1: Find the value of $x$
Given: The observations are $1, 5, 7, x + 1, 9, x - 2, 3$ . The mean of the given observations is $4$ .
Mean = $\frac{S u m o f a l l o b s e r v a t i o n s}{t o t a l n u m b e r o f o b s e r v a t i o n s}$
$∴ 4 = \frac{1 + 5 + 7 + x + 1 + 9 + x - 2 + 3}{7}$
$\Rightarrow 28 = 24 + 2 x$
$\Rightarrow 2 x = 4$
$\Rightarrow x = 2$
Step 2: Find the mode
Now, the observations are $1, 5, 7, 3, 9, 0, 3$
Mode is that value that occurs most frequently.
Therefore the mode is $3$
Hence, the value of $x$ is $2$ and the mode of the given observations is $3$ .

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