Question

determining a missing value
which value should be added to this data set to produce a mean of 13?
10 11 12 13 14 15 16 17 18 19 20
should be added to the data set to produce a mean of 13.

Explanation:

Step1: Calculate sum of original data set

The original data - set is \(10,11,12,13,14,15,16,17,18,19,20\). The sum of an arithmetic series \(S_n=\sum_{i = 1}^{n}a_i\), and here \(n = 11\), \(a_i\) is the \(i\) - th number in the set. Using the formula for the sum of an arithmetic series \(S=\frac{n(a_1 + a_n)}{2}\), where \(a_1 = 10\), \(a_n=20\), \(n = 11\), so \(S=\frac{11\times(10 + 20)}{2}=11\times15 = 165\).

Step2: Let the missing value be \(x\)

We want the new mean of the data - set with \(n'=12\) (the original 11 values plus the new value \(x\)) to be 13. The formula for the mean \(\bar{y}=\frac{\text{Sum of all values}}{\text{Number of values}}\). So \(13=\frac{165 + x}{12}\).

Step3: Solve for \(x\)

Multiply both sides of the equation \(13=\frac{165 + x}{12}\) by 12: \(13\times12=165 + x\). Then \(156=165 + x\). Subtract 165 from both sides: \(x=156 - 165=-9\).

Answer:

\(-9\)