# Elementary questions about error estimation

1. Oct 8, 2016

### jonjacson

1. The problem statement, all variables and given/known data

We measure the temperature with a thermometer and we get these results:

3.3 +- 0.1 ºC
3.5 +- 0.1 ºC
3.6 +- 0.1 ºC

What is the temperature?

2. Relevant equations

Average = Sum of values/amount of measurementes

3. The attempt at a solution

Well, I calculate the average and I get 3.4666666666 period.

The sensibility of the thermometer is 0.1 ºC so I can't give more than 1 decimal number so I guess the solution is:

3.4 +- 0.1 ºC ( I just rounded the number 3.4666 to the first decimal place)

Is this correct?

2. Oct 8, 2016

### haruspex

The question is a bit awkward because it quotes non-overlapping ranges, implying either that the temperature is changing or that the measurements are not as accurate as claimed!
Leaving that aside, the next question is the error distribution. With thermometer readings, the precision is limited by the scale markings, so presumably the error distribution is uniform over a .2C range. But by something of a fudge, most authorities seem to pretend its Gaussian. (Pet peeve of mine.)
Now, if you have N samples from a Gaussian distribution that has an inherent standard deviation of σ, what is the standard error in your estimate of the mean?

3. Oct 8, 2016

### Ray Vickson

You should round 3.466.... to the nearest 0.1, so up to 3.5, not down to 3.4 as you did; basically, 66 is closer to 100 than to 0. The only time this is troublesome is in a case like 3.45, which is equidistant between 3.4 and 3.5, and in such a case people can (and do) argue about the best way to proceed. There are some published "rules", but they seem a bit arbitrary to me.

4. Oct 8, 2016

### jonjacson

Well the formula I found in wikipedia is:

https://en.wikipedia.org/wiki/Standard_error

The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean. (It can also be viewed as the standard deviation of the error in the sample mean with respect to the true mean, since the sample mean is an unbiased estimator.) SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample):

SE = s / √n
where

s is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
n is the size (number of observations) of the sample.

Yes, you are right.