Exercise $4$: Uncertainty estimation comparison

The file gauss$\_$data.txt contains $100$ random numbers generated according to a Gaussian

distribution with $=10$ and $=3.$ Your task is to estimate the uncertainty of these

parameters using the data in three different ways.

$($a$)$ First, calculate the mean and standard deviation $(S_x)$ of the data.

$($b$)$ Calculate the uncertainty of these estimates using the standard formulae $2.$

$($c$)$ Use Monte Carlo to generate repetitions of the dataset $N_{\text{e}\text{x}\text{p}\text{t}}=1000$ times $($i$.$e$.1000100$

data points$)$ according to the underlying Gaussian distribution. Calculate the mean and

standard deviation for each generated dataset and use the RMS of those values to es$-$

timate the uncertainties of the mean and standard deviation of the original dataset in

the text file.

$($d$)$ Bootstrap the dataset $1000$ times by resampling the dataset with replacement. Cal$-$

culate the mean and standard deviation for each bootstrapped dataset and use the

variance of those values to estimate the uncertainties of the mean and standard devia$-$

tion of the original dataset in the text file.

$($e$)$ Which uncertainty technique do you think can be trusted in most situations?

${?}^2(\stackrel{}{x})=\frac{S_x}{\sqrt[\phantom{2}]{N}},(S_x)=\frac{S_x}{\sqrt[\phantom{2}]{2N-2}}$

gauss$\_$data.txt:

$7\mathrm{.}457080557496150064$e$+00$

$1\mathrm{.}625682342005246994$e$+01$

$1\mathrm{.}312910669193915680$e$+01$

$9\mathrm{.}597667829927662098$e$+00$

$1\mathrm{.}424465355823167023$e$+01$

$9\mathrm{.}161716345535273476$e$+00$

$7\mathrm{.}586665829661322746$e$+00$

$...$