# Sampling error vs. measurement error

I have a dataset involving sensor measurements (GPS trajectories). I am using this dataset to estimate aggregate statistics such as the total distance travelled and related quantities. My understanding is that I can calculate confidence intervals for these quantities based on the sampling error using the CLT, which would give me a CI for the true population statistic.

However does this interval take into account measurement error, i.e. random error due to the uncertainty in sensor measurements? It seems to me that this would not be the case, if not how would I account for this?

## 2 Answers

Since I feel you like to go more advanced, I edit my answer.

Let's assume you have data for one week $$\{y_1, y_2, \ldots, y_7\}$$. As you pointed out, each data-point consist of two components: the "true" traveled distance per day $$d_i$$ and measurement error $$m_i$$. Thus, our model is $$Y = D + M$$, where $$D$$ and $$M$$ are random variables and $$d_i$$ and $$m_i$$ are realisations of those for the $$i^{th}$$ day.

In order to obtain inside, you have to make assumptions about the random variables. The standard assumptions are that $$D_i \sim N(\mu, \sigma_d)$$ and $$M \sim N(0, \sigma_m)$$. What you are interested in are the contributions $$\mu$$ and $$\sigma_d$$. However, only the point-estimator $$\hat{\mu} = \textrm{E}[Y] = \frac{1}{7}\sum_{i=1}^7 y_i$$ is "simple" to obtain. In order to estimate the two so called variance components $$\{\sigma^2_d, \sigma^2_m\}$$ you need to define your set-up first. There exists extensive literature on measurement system analysis and you might want to look into that.

Once you estimated $$\sigma_d$$ the 95% confidence interval for the distance travelled per day is given by $$\hat{\mu} \pm 1.96 \hat{\sigma_d}$$.

• Would it also be valid to estimate the distribution on a per vehicle basis also and calculate CIs from this? And do you have any intuitive way of explaining how this captures all sources of error, not just sampling error, as I'm not sure I can see this
– CJO
Commented Dec 29, 2019 at 9:13
• I edited my answer. Commented Dec 29, 2019 at 11:14
• Thanks, I think I see more clearly now. So essentially, the sampling error from the overall measurement $Y$ captures the measurement error because this is already included in every data point?
– CJO
Commented Dec 30, 2019 at 0:10

I think some of this is already covered by @Semoi's answer, but adding another perspective here.

If by "take into account", you mean "is partly determined by measurement error", then yes, I would say confidence intervals are partly a function of measurement error, even if only indirectly. This is because measurement error exists at the level measurements, which are of course the basis of any statistic we would calculate, meaning the variance, mean, and even individual data points are tainted by measurement error. What effect measurement error has, exactly, on the behavior of confidence intervals is, alas, above my pay grade! But this resource could be of interest for exploring the consequences of measurement error on estimation.

If, however, by "take into account measurement error", you mean partitions out measurement error to give you a confidence interval that only quantifies sampling error and nothing else, then no, the confidence interval does not take measurement error into account.

With respect to this second point: in my line of work, we use e.g., structural equation models to separate out variance attributable to the constructs under evaluation from measurement error. This is taken care of by the "measurement model", which is estimated simultaneously alongside the "structural model" estimating relationships among your variables (or whatever estimates you want, which could be means). The measurement model generates new values (factor scores) for your variables, which are taken to be error-free measurements the now exist at the level of the construct (in contrast to the original values, which are at the level of the measuring instrument). Confidence intervals around estimates using factor scores would then take measurement error into account in the sense they are free of measurement error.

If you happen to have repeated measures of the same variable – for instance, in your case, maybe you have the distance from point A to point B measure 5 times or so, and similarly for other routes – you can actually quantify your measurement error by calculating the SEM, or Standard Error of the Measurement. The SEM is interpreted as "the number of unit increases (or decreases) on the measuring instrument I would need to detect a true difference in the underlying construct". I don't know if this has applications outside of human subjects research, which is the area I see it in, but just letting you know in case maybe there is an analogue or use case for this in your field or similar ones.

Hope this was helpful!