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I have a data set with 11 points. The smallest point is 60, the largest is 122, and the average is 92.36. This is all that is known about the data set. I add one point (unknown size), such that there are now 12 data points. The smallest and largest points in the set remain the same, but the average is now 92.5. Are there any methods to estimate what the value of the added data point is?

mattk62586
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2 Answers2

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The sum of the original $11$ datapoints is $92.36 \cdot 11$ and the sum of all $12$ datapoints is $92.5 \cdot 12$. So, the new datapoint is $92.5 \cdot 12 - 92.36 \cdot 11 = 94.04$.

EDIT: If you know from context that all the datapoints are integers, then the average of the original $11$ datapoints is actually $92.363636\ldots$, and the new datapoint is $94$.

JimmyK4542
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We have $\frac{\sum_{i=1}^{11}x_{i}}{11}=92.36$ and:

$92.5=\frac{\sum_{i=1}^{12}x_{i}}{12}=\frac{11}{12}\frac{\sum_{i=1}^{11}x_{i}}{11}+\frac{x_{12}}{12}=\frac{11}{12}(92.36)+\frac{1}{12}x_{12}$.

We can now solve for the twelfth data point.

user71352
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