The mean of $13$ observations is $14$. If the mean of the first $7$ observations is $12$ and that of the last $7$ observations is $16$, then the $7^{th}$ observation is ___________.

Using the formula for the arithmetic mean, the total sum of all 13 observations is 13 * 14 = 182. The sum of the first 7 observations is 7 * 12 = 84, and the sum of the last 7 observations is 7 * 16 = 112. Adding the sums of the first and last seven gives 84 + 112 = 196. Since this combined sum counts the 7th observation twice, the 7th observation is 196 - 182 = 14.

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