Take a few data points, calculate average and standard deviation. Common practice and useful to make a simple assessment. However 5 data points do not tell you anything about about what is going on. In an experiment the goal is to estimate the true value of some parameter.
Mathematically from maximum livelihood estimators the best estimator or expected value for a normal distribution is the arithmetic mean. It is not necessarily true for all distributions. For a normal distribution the arithmetic mean is the best estimator of the actual value.
So, taking 5 data points and computing average and SD knowing nothing more is called distribution free or non parametric methods. It means you look at data and try to make sense out of it.
There is small sample and large sample theory, and the obituary is roughly between 10 and 20 samples. I have seen it born out in actual analysis and it can be demonstrated by simulation. If you have a large data set take the first two points and average, then the first three and so on. As N gets large the sample means vary about the true mean by smaller deviations.
From experience I go with 20 as a min sample size. With that I can do a histogram and cumulative distribution to look for an underlying distribution.
When you are in the large sample region the Central Limit Theorem applies. It says that regardless of any underlying distribution random sample means will be normally distributed. It is the basis of practical statistics,. For random sampling parameters tend to be approximated by normal distributions.
For small samples I use probability plotting. 5 data plots makes for a straight line on a plot. I plot the ordered data points vs median ranks. If the least squares line looks good on the plot estimates of the mean and standard deviation can be read. In an old incarnation as a reliability engineer I used Weibul plots.
In one case I had 5 or 6 sample parts from a manufacturer. To estimate the variability of a parameter I used a probability plot.
There is a lot you can do with 5 data points depending on what you are trying to do and the situation.
In an experiment like the paper an error estimate can be complicated. The variability in a number of test runs does not tell you accuracy or resolution. Some error sources are added algebraically and some are added Root Sum Square. And which is which can be a source of contention.
Following is Sciab script. Given a random vector of a normal distribution, compute sample means for the first two samples, the first three samples and so on. Vary the standard deviation and see how many samples are needed before for the sample mean to converge on the true mean.
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clear;
n_points = 1000;
_mean = 100;
stand_dev = 2;
r = grand(1,n_points,"nor",_mean,stand_dev);
_index = 1;
for i = 1:100;
_sum = 0;
n = 0;
for j = 1:_index;
n = n + 1;
_sum = _sum + r(i + j);
end
samp_mean(1,j) = j;
samp_mean(2,j) = _sum/n;
sm(j) = _sum/n;
_index = _index + 1;
end
figure(1);
histplot(20,r)
figure(2);
plot2d(sm);
mean(r) // true mean
samp_mean
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Y sample means, X number of samples. Mean 100, standard deviations of 2 followed by 10.
