"# What is meant with error/uncertainty on a measured quantity?"
]
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"If we quote $a = 1 \\pm 0.5$, we usually mean that the probability for the *true* value of $a$ is Gaussian $G(a, \\mu, \\sigma)$ distributed with $\\mu = 1$ and $\\sigma = 0.5$. "
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"# How often can/should the measurement be outside one sigma?"
"print(\"fraction outside one sigma:\", sum(is_outside)/len(pseudo_a)) "
]
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"# Why is it a Gaussian"
]
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"Central limit theorem:\n",
"\n",
"\"let $X_{1},X_{2},\\dots ,X_{n}$ denote a statistical sample of size $n$ from a population with expected value (average) $\\mu$ and finite positive variance $\\sigma ^{2}$, and let $\\bar {X_{n}}$ denote the sample mean (which is itself a random variable). Then the limit as $n\\to \\infty$ of the distribution of $\\frac {({\\bar {X}}_{n}-\\mu )}{\\frac {\\sigma }{\\sqrt {n}}}$, is a normal distribution with mean 0 and variance 1.\""
# What is meant with error/uncertainty on a measured quantity?
%% Cell type:markdown id:d369a48b tags:
If we quote $a = 1 \pm 0.5$, we usually mean that the probability for the *true* value of $a$ is Gaussian $G(a, \mu, \sigma)$ distributed with $\mu = 1$ and $\sigma = 0.5$.
%% Cell type:markdown id:a56aa4e8 tags:
# How often can/should the measurement be outside one sigma?
"let $X_{1},X_{2},\dots ,X_{n}$ denote a statistical sample of size $n$ from a population with expected value (average) $\mu$ and finite positive variance $\sigma ^{2}$, and let $\bar {X_{n}}$ denote the sample mean (which is itself a random variable). Then the limit as $n\to \infty$ of the distribution of $\frac {({\bar {X}}_{n}-\mu )}{\frac {\sigma }{\sqrt {n}}}$, is a normal distribution with mean 0 and variance 1."