Calculating Expected Systematic Error in a Pendulum Experiment I am a little confused by part c of problem 4.28 of Taylor's Introduction to Error Analysis book. A student measures the acceleration due to gravity by using a steel ball suspended by a light string. He records five different lengths (51.2, 59.7, 68.2, 79.7, 88.3)(all in cm) and five different periods (1.448, 1.566, 1.669, 1.804, 1.896)(all in seconds). He uses the formula
$$g=\frac{4\pi^2 l}{T^2}$$
To calculate the mean and SDOM of the acceleration due to gravity, which I calculated to be $965.6\pm 1.5\,cm/s^2$. I have double checked this value, and am pretty sure of the result. The student is concerned that the accepted value $g=979.6\,cm/s^2$ is not contained within the calculated uncertainty, and looks for systematic error. 
The question: how large would a systematic error in length l have to be so that the margins of the total error just include the accepted value of g? What I did was set
$$Avg(\frac{4\pi^2(l+\Delta l)}{T^2})+SDOM(g)=979.6$$
$$\rightarrow Avg(\frac{4\pi^2l}{T^2})+Avg(\frac{4\pi^2\Delta l}{T^2})+SDOM(g)=979.6$$
$$Avg(\frac{4\pi^2\Delta l}{T^2})=12.5$$
$$\rightarrow \Delta l=0.894$$
This is about 1.3% of the average length. However, in the book, it says that my answer should be approximately 1.5%. Am I doing something wrong? Is my procedure or calculation wrong, or I am over-analyzing the discrepancy between my value and the book's?  
 A: Maybe this problem solve like this
Look at the Sec. 4.6 (Systematic Errors)
In part (c) Problem 4.28
We must find the systematic error of length of the pendulum ($\delta l_{sys}$).
Given that the value of $\delta g_{tot}$ is $|g_{accepted} - \bar{g}|$.
Using the above argument and Eq. 4.26 $\left[ \delta g_{tot} = \sqrt{\delta g_{sys}^2 + \delta g_{ran}^2} \,\,\right]$, 
$$
\begin{align}
    \delta g_{tot} &= \sqrt{\delta g_{sys}^2 + \delta g_{ran}^2} \\
    |g_{accepted} - \bar{g}| &= \sqrt{\delta g_{sys}^2 + \sigma_{\bar{g}}^{\,\,\,2}} \\
    \delta g_{sys} &= \sqrt{\mathrm{discrepancy}^2 - \sigma_{\bar{g}}^{\,\,\,2}} 
\end{align}
$$
and propagation error techniques for the equation $g = 4\pi l/T^2$ gives the formula for $\delta g_{sys}$
$$
\begin{align}
    \frac{\delta g_{sys}}{\bar{g}} = \sqrt{\left(\frac{\delta l_{sys}}{\bar{l}}\right)^2 + \left(2\frac{\delta{T_{sys}}}{\bar{T}}\right)^2}
\end{align}
$$
Since there was no problem with the measurement of the period $T$ ($\delta T_{sys} = 0$), we have the propagation error for $\delta g_{sys}$
$$
\begin{align}
    \frac{\delta g_{sys}}{\bar{g}} = \frac{\delta l_{sys}}{\bar{l}}
\end{align}
$$
Therefore, the systematic of the length $l$ (in percentage unit) is
$$
    \frac{\delta l_{sys}}{\bar{l}} = \frac{\sqrt{\mathrm{discrepancy}^2 - \sigma_{\bar{g}}}}{\bar{g}} \cdot 100\%
$$
Substituting all the value has given and obtained from the previous question, we have the final result
$$
\begin{align}
    \frac{\delta l_{sys}}{\bar{l}} &= 1.43793098102868 \% \\
    &\approx 1.4\%
\end{align}
$$





