Question

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

Explanation:

Step1: Calculate the sum of the 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 \(S_n=\frac{n(a_1 + a_n)}{2}\), where \(n = 11\), \(a_1=10\), \(a_n = 20\).
\(S=\frac{11\times(10 + 20)}{2}=11\times15=165\)

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

The new number of data points is \(n'=12\), and the new mean \(\bar{x}=13\).
We know that the formula for the mean \(\bar{x}=\frac{S + x}{n'}\).
Substitute \(\bar{x}=13\) and \(n' = 12\) into the formula: \(13=\frac{165+x}{12}\)

Step3: Solve the equation for \(x\)

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

Answer:

\(-9\)