<p>The value of <span class="math-container">$H_0$</span> as measured by the Planck Satellite is 67.8 km/s/Mpc (<a href="https://sci.esa.int/web/planck/-/60504-measurements-of-the-hubble-constant" rel="nofollow noreferrer">https://sci.esa.int/web/planck/-/60504-measurements-of-the-hubble-constant</a>). The standard deviation of the joint analysis is <span class="math-container">$\sigma \approx 1.75$</span> km/s/Mpc. So the difference between the mean value of <span class="math-container">$H_0$</span> in the joint analysis and the value from Planck is <span class="math-container">$3.1\sigma$</span>.</p>
<p>As you mentioned, the probability that the value you measured lies within <span class="math-container">$3 \sigma$</span> is 99.7%, so the probability that the offset between the different <span class="math-container">$H_0$</span> measurements is due to a statistical error is <span class="math-container">$&lt;0.3\%$</span>.</p>
<p>In general, <span class="math-container">$k\sigma$</span> measures the degree of certainty you have in your measurement. If your instrument has a noise level of <span class="math-container">$\sigma$</span> and you measure something above <span class="math-container">$3\sigma$</span>, you can be 99.7% sure that this was actually a signal.</p>

CC BY-SA 4.0
2024-07-05T11:43:06.557Z
Can we determine the value of $k$ from Fisher information matrix?
Wang Yun
CC BY-SA 4.0
2024-07-05T10:38:59.750Z
2024-07-05T10:48:32.937Z
2024-07-05T10:48:32.937Z
Lenard Kasselmann
<p>In statistics, <span class="math-container">$\sigma$</span> or <span class="math-container">$1\sigma$</span> is the standard deviation. For the normal distribution, approximately <span class="math-container">$68\%$</span> of the values lie within <span class="math-container">$1\sigma$</span> range, <span class="math-container">$95\%$</span> within <span class="math-container">$2\sigma$</span>, and <span class="math-container">$99.7\%$</span> within <span class="math-container">$3\sigma$</span>. These are the things I know.</p>
<p>However, as I read papers, I came across two situations, which are:</p>
<ol>
<li><a href="https://link.aps.org/accepted/10.1103/PhysRevLett.126.011301" rel="nofollow noreferrer">Massara2021</a></li>
</ol>
<blockquote>
<p>The errors on the neutrino masses from the marked power spectrum are <span class="math-container">$\sigma_{cb}(M_\nu)=0.5 \ \text{eV}$</span> and <span class="math-container">$\sigma_{m}(M_\nu)=0.017 \ \text{eV}$</span>. The latter indicates that marked power spectra on the total matter density can put a <span class="math-container">$3.5\sigma$</span> constraint on the minimum sum of the neutrino masses.</p>
</blockquote>
<p>I know that the smaller <span class="math-container">$\sigma$</span> is, the tighter the constraints on the parameters. But, what does <span class="math-container">$3.5\sigma$</span> mean? How is it calculated to?</p>
<ol start="2">
<li><a href="https://open.uct.ac.za/server/api/core/bitstreams/76482e50-aaa1-4030-a4bb-5e6f1347d5bb/content" rel="nofollow noreferrer">Houliston2022</a></li>
</ol>
<blockquote>
<p>In the joint analysis, the Hubble constant found in flat <span class="math-container">$\Lambda$</span>CDM is <span class="math-container">$H_0 = 73.3^{+1.7}_{-1.8}\text{km}\cdot \text{s}^{−1} \text{Mpc}^{−1}$</span>. This <span class="math-container">$2.4\%$</span> precision value agrees with the SH0Es 2020 value, and is in <span class="math-container">$3.1\sigma$</span> tension with the Planck CMB value.</p>
</blockquote>
<p>What does <span class="math-container">$3.1\sigma$</span> <strong>tension</strong> mean?</p>

CC BY-SA 4.0
2024-07-05T12:15:31.240Z
The exact relation is $P(\le \! k\sigma) = \mathrm{erf}(k/\sqrt{2})$, where $\mathrm{erf}(x) \equiv \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}\,dt$ is the error function. Thus, $P(\le \!1\sigma) = 0.683$, and $P(\le \!3.1\sigma) = 0.998$.
pela
CC BY-SA 4.0
2024-07-05T12:36:39.863Z
@pela I understand these relations, but I do not understand k=3.1 comes from.
Wang Yun
CC BY-SA 4.0
2024-07-06T13:56:17.153Z
Saying "the tension is at 3.1σ" is another way of saying "we're 99.8% sure both results cannot be true". The number is calculated from the known random and systematic uncertainties, as seen in Lenard Kasselmann's answer. But is your question perhaps how exactly to perform this calculation? If so, I wonder if the question is more appropriate for [stats.SE](https://stats.stackexchange.com) (although I still think it's fine here).
pela
<astrophysics><cosmology><universe><hubble-constant><neutrinos>
What exactly does $k\sigma$ mean in astrophysics and cosmology?
Wang Yun
# What exactly does $k\sigma$ mean in astrophysics and cosmology?

(Asked by: username_0 on 2024-07-04)

In statistics, $\sigma$ or $1\sigma$ is the standard deviation. For the normal distribution, approximately $68\%$ of the values lie within $1\sigma$ range, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$. These are the things I know.

However, as I read papers, I came across two situations, which are:

1. [Massara2021](https://link.aps.org/accepted/10.1103/PhysRevLett.126.011301)

> The errors on the neutrino masses from the marked power spectrum are $\sigma\_{cb}(M\_\nu)=0.5 \ \text{eV}$ and $\sigma\_{m}(M\_\nu)=0.017 \ \text{eV}$. The latter indicates that marked power spectra on the total matter density can put a $3.5\sigma$ constraint on the minimum sum of the neutrino masses.

I know that the smaller $\sigma$ is, the tighter the constraints on the parameters. But, what does $3.5\sigma$ mean? How is it calculated to?

2. [Houliston2022](https://open.uct.ac.za/server/api/core/bitstreams/76482e50-aaa1-4030-a4bb-5e6f1347d5bb/content)

> In the joint analysis, the Hubble constant found in flat $\Lambda$CDM is $H\_0 = 73.3^{+1.7}\_{-1.8}\text{km}\cdot \text{s}^{−1} \text{Mpc}^{−1}$. This $2.4\%$ precision value agrees with the SH0Es 2020 value, and is in $3.1\sigma$ tension with the Planck CMB value.

What does $3.1\sigma$ **tension** mean?

**Tags:** astrophysics, cosmology, universe, hubble-constant, neutrinos

> **Comments:**
> **username_1**: The exact relation is $P(\le \! k\sigma) = \mathrm{erf}(k/\sqrt{2})$, where $\mathrm{erf}(x) \equiv \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}\,dt$ is the error function. Thus, $P(\le \!1\sigma) = 0.683$, and $P(\le \!3.1\sigma) = 0.998$.  
> **username_0**: @username_1 I understand these relations, but I do not understand k=3.1 comes from.  
> **username_1**: Saying "the tension is at 3.1σ" is another way of saying "we're 99.8% sure both results cannot be true". The number is calculated from the known random and systematic uncertainties, as seen in username_2 username_2's answer. But is your question perhaps how exactly to perform this calculation? If so, I wonder if the question is more appropriate for [stats.SE](https://stats.stackexchange.com) (although I still think it's fine here).  

---

## 1 answers

### Answer by username_2 on 2024-07-05. Score: 3

The value of $H\_0$ as measured by the Planck Satellite is 67.8 km/s/Mpc (<https://sci.esa.int/web/planck/-/60504-measurements-of-the-hubble-constant>). The standard deviation of the joint analysis is $\sigma \approx 1.75$ km/s/Mpc. So the difference between the mean value of $H\_0$ in the joint analysis and the value from Planck is $3.1\sigma$.

As you mentioned, the probability that the value you measured lies within $3 \sigma$ is 99.7%, so the probability that the offset between the different $H\_0$ measurements is due to a statistical error is $<0.3\%$.

In general, $k\sigma$ measures the degree of certainty you have in your measurement. If your instrument has a noise level of $\sigma$ and you measure something above $3\sigma$, you can be 99.7% sure that this was actually a signal.

> **Comments:**
> **username_0**: Can we determine the value of $k$ from Fisher information matrix?  

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