# Numerical Error Propagation

I'm doing the common experiment of determining $g$ by means of a simple pendulum, and I've decided to do so by measuring the period of the pendulum at variable lengths. I've had no problems propagating uncertainty for each data point, but now I've got the equation for the linear fit, and I'm having a problem trying to figure out the error bounds for the slope of that fit ( which is analytically $\frac{4\pi^2}{g}$ ). I've sifted through a few of the error questions already posted, but there are far too many for me to hit everyone to find and answer. Any help would be appreciated!

I have the analytic expression for the uncertainty in $g(L,T)$ , but I'm not sure what points I would even use, or if I would just average over the points for the uncertainty in $g$.

• I have the analytic expression for the uncertainty What is it?
– user4552
Oct 16, 2014 at 23:41
• $\delta g = g_{0}(\frac{\delta L}{L_0}+2\frac{\delta T}{T_0})$ My issue is what to plug in for the naught values. I've just plugged in averages and what the calculated value of $g_0$ was, and I got a fair value with doing that, but is that the proper method? ( The period has factor $2$ because I used $g = \frac{4\pi^2 L}{T^2}$ Oct 17, 2014 at 0:03
• I recommend using a program for the calculation. There are many program languages, which are free and rather simple to learn. As you will need the skill of using such a program later in your studies anyway, it's a good idea to start early. The program does the heavy lifting for you and estimates the average slope and its uncertainty (or confidence interval). Feb 17, 2023 at 15:11

The knowledge you seek is (1) classical Goodness of Fit Statistical Tests and Regression Analysis.

Just to get you going: standard linear regression (see the "Linear Regression" section in the Wikipedia "Regression Analysis" page) gives you formulas for the standard errors of the estimated parameters.

One of your estimated parameters - the gradient is - $4\pi^2/g$. The distribution of this estimate, given enough samples (roughly 20 to 30) for the Central Limit Theorem to take hold, is a Gaussian normal distribution, whose variance is given by the standard error formula on the Wiki page.

So you can thus get bounds on $4\pi/g^2$. Now simply use the expansion $\Delta (4\pi/g) \approx {\rm d}_g (4\pi/g) \Delta g =-(4 \pi\,\Delta g/g^2)$ to get a rough figure for the variance of the $g$ estimate.

Actually, I would do two regressions. If you can be sure that there is no systematic error for the period measurement, then the linear relationship $T = \beta_1 L + \beta_0$ is actually a linear homogeneous relationship. So you KNOW $\beta_0=0$ and you do not need to statistically estimate it.

So I would work through the first principles derivation of the linear regression formulas, but do it for the model $T = \beta_1 L$ and not $T = \beta_1 L + \beta_0$. That is, you are working with probability distributions conditioned on the knowledge that $\beta_0=0$. I can't recall of the top of my head what standard error formulas this line of reasonings lead to, but this should get you going.

• Great answer and I've definitely bookmarked it, but I'm looking for the method of calculus for this particular problem. Oct 17, 2014 at 0:16
• @DoryanMiller Well, once you've applied the standard error formula for $\beta_1=4π/g^2$, you have its variance in terms of the variance of the residuals, and you know it is normal distribution, centred on the mean. So calculate the standard deviation of the residuals, the mean $L$ ($L$ is the abscissa $x$ variable on the Wiki page) and you've got the standard deviation of the $\beta_1$ estimate. You then use whatever confidence interval - say $±3\,\sigma$ - you think appropriate. Oct 17, 2014 at 0:42