#### sin$^2(\theta _{23})$

The reported limits below correspond to the projection onto the sin$^2(\theta _{23})$ axis of the 90$\%$ CL contours in the sin$^2(\theta _{23})$ $−$ $\Delta {{\mathit m}^{2}}_{{{\mathit 32}}}$ plane presented by the authors. Unless otherwise specified, the limits are 90$\%$ CL and the reported uncertainties are 68$\%$ CL.

If an experiment reports sin$^2(2~\theta _{23})$ we convert the value to sin$^2(~\theta _{23})$.
VALUE DOCUMENT ID TECN  COMMENT
 $\bf{ 0.539 \pm0.022}$ OUR FIT  Error includes scale factor of 1.1.  Assuming inverted mass ordering
 $\bf{ 0.546 \pm0.021}$ OUR FIT  Assuming normal mass ordering
$0.53$ ${}^{+0.03}_{-0.04}$ 1
 2020 F
T2K Both mass orderings
$0.43$ ${}^{+0.20}_{-0.04}$ 2
 2020 A
MINS Normal mass ordering
$0.42$ ${}^{+0.07}_{-0.03}$ 2
 2020 A
MINS Inverted mass ordering
$0.56$ ${}^{+0.04}_{-0.03}$ 3
 2019
NOVA Normal mass order; octant II for ${{\mathit \theta}_{{23}}}$
$0.56$ ${}^{+0.04}_{-0.03}$ 3, 4
 2019
NOVA Inverted mass order; octant II for ${{\mathit \theta}_{{23}}}$
$0.51$ ${}^{+0.07}_{-0.09}$ 5
 2018 A
ICCB Normal mass ordering
$0.588$ ${}^{+0.031}_{-0.064}$ 6
 2018 B
SKAM Normal mass ordering, ${{\mathit \theta}_{{13}}}$ constrained
$0.575$ ${}^{+0.036}_{-0.073}$ 6
 2018 B
SKAM Inverted mass ordering, ${{\mathit \theta}_{{13}}}$ constrained
• • We do not use the following data for averages, fits, limits, etc. • •
$0.51$ ${}^{+0.06}_{-0.07}$ 7
 2021 A
T2K ${{\mathit \nu}_{{\mu}}}$ disappearance
$0.43$ ${}^{+0.21}_{-0.05}$ 7
 2021 A
T2K ${{\overline{\mathit \nu}}_{{\mu}}}$ disappearance
$0.574$ $\pm0.014$ 8
 2021
FIT Normal mass ordering, global fit
$0.578$ ${}^{+0.010}_{-0.017}$ 8
 2021
FIT Inverted mass ordering, global fit
$0.455$ 9
 2020
ICCB For both mass orderings
$0.573$ ${}^{+0.016}_{-0.020}$ 10
 2020 A
FIT Normal mass ordering, global fit
$0.575$ ${}^{+0.016}_{-0.019}$ 10
 2020 A
FIT Inverted mass ordering, global fit
$0.58$ ${}^{+0.04}_{-0.13}$ 11
 2019 C
ICCB
$0.48$ ${}^{+0.04}_{-0.03}$ 3, 4
 2019
NOVA Normal mass order; octant I for ${{\mathit \theta}_{{23}}}$
$0.47$ ${}^{+0.04}_{-0.03}$ 3, 4
 2019
NOVA Inverted mass order; octant I for ${{\mathit \theta}_{{23}}}$
$0.49$ ${}^{+0.30}_{-0.28}$
 2019
OPER
$0.50$ ${}^{+0.20}_{-0.19}$ 12
 2019
ANTR Atmospheric ${{\mathit \nu}}$ , deep sea telescope
$0.587$ ${}^{+0.036}_{-0.069}$ 13
 2018 B
SKAM 3${{\mathit \nu}}$ osc: normal mass ordering, ${{\mathit \theta}_{{13}}}$ free
$0.551$ ${}^{+0.044}_{-0.075}$ 13
 2018 B
SKAM 3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{13}}}$ free
$0.526$ ${}^{+0.032}_{-0.036}$ 14
 2018 G
T2K Normal mass ordering, ${{\mathit \theta}_{{13}}}$ constrained
$0.530$ ${}^{+0.030}_{-0.034}$ 14
 2018 G
T2K Inverted mass ordering, ${{\mathit \theta}_{{13}}}$ constrained
$0.56$ $\pm0.04$ 15
 2018
NOVA Normal mass order; octant II for ${{\mathit \theta}_{{23}}}$
$0.47$ $\pm0.04$ 15
 2018
NOVA Normal mass order; octant I for ${{\mathit \theta}_{{23}}}$
$0.547$ ${}^{+0.020}_{-0.030}$
 2018
FIT Normal mass ordering, global fit
$0.551$ ${}^{+0.018}_{-0.030}$
 2018
FIT Inverted mass order, global fit
$0.532$ ${}^{+0.061}_{-0.087}$ 16
 2017 A
T2K Normal mass ordering
$0.534$ ${}^{+0.061}_{-0.087}$ 16
 2017 A
T2K Inverted mass ordering
$0.51$ ${}^{+0.08}_{-0.07}$
 2017 C
T2K Normal mass ordering with neutrinos
$0.42$ ${}^{+0.25}_{-0.07}$
 2017 C
T2K Normal mass ordering with antineutrinos
$0.52$ ${}^{+0.075}_{-0.09}$
 2017 C
T2K normal mass ordering with neutrinos and antineutrinos
$0.55$ ${}^{+0.05}_{-0.09}$ 16
 2017 F
T2K Normal mass ordering
$0.55$ ${}^{+0.05}_{-0.08}$ 16
 2017 F
T2K Inverted mass ordering
$0.404$ ${}^{+0.022}_{-0.030}$ 17
 2017 A
NOVA Normal mass ordering; octant I for ${{\mathit \theta}_{{23}}}$
$0.624$ ${}^{+0.022}_{-0.030}$ 17
 2017 A
NOVA Normal mass ordering; octant II for ${{\mathit \theta}_{{23}}}$
$0.398$ ${}^{+0.030}_{-0.022}$ 17
 2017 A
NOVA Inverted mass ordering; octant I for ${{\mathit \theta}_{{23}}}$
$0.618$ ${}^{+0.022}_{-0.030}$ 17
 2017 A
NOVA Inverted mass ordering; octant II for ${{\mathit \theta}_{{23}}}$
$0.45$ ${}^{+0.19}_{-0.07}$ 18
 2016 D
T2K 3${{\mathit \nu}}$ osc; normal mass ordering; ${{\overline{\mathit \nu}}}$ beam
$0.38\text{ to }0.65$ 19
 2016 A
NOVA normal mass ordering
$0.37\text{ to }0.64$ 19
 2016 A
NOVA Inverted mass ordering
$0.53$ ${}^{+0.09}_{-0.12}$ 20
 2015 A
ICCB Normal mass ordering
$0.51$ ${}^{+0.09}_{-0.11}$ 20
 2015 A
ICCB Inverted mass ordering
$0.514$ ${}^{+0.055}_{-0.056}$ 21
 2014
T2K 3${{\mathit \nu}}$ osc.; normal mass ordering
$0.511$ $\pm0.055$ 21
 2014
T2K 3${{\mathit \nu}}$ osc.; inverted mass ordering
$0.41$ ${}^{+0.23}_{-0.06}$ 22
 2014
MINS Normal mass ordering
$0.41$ ${}^{+0.26}_{-0.07}$ 22
 2014
MINS Inverted mass ordering
$0.567$ ${}^{+0.032}_{-0.128}$ 23
 2014
FIT Normal mass ordering
$0.573$ ${}^{+0.025}_{-0.043}$ 23
 2014
FIT Inverted mass ordering
$0.452$ ${}^{+0.052}_{-0.028}$ 24
 2014
FIT Normal mass ordering; global fit
$0.579$ ${}^{+0.025}_{-0.037}$ 24
 2014
FIT Inverted mass ordering; global fit
$0.24\text{ to }0.76$ 25
 2013 B
ICCB DeepCore, 2${{\mathit \nu}}$ oscillation
$0.514$ $\pm0.082$ 26
 2013 G
T2K 3${{\mathit \nu}}$ osc.; normal mass ordering
$0.388$ ${}^{+0.051}_{-0.053}$ 27
 2013 B
MINS Beam + Atmospheric; identical ${{\mathit \nu}}$ $\&$ ${{\overline{\mathit \nu}}}$
$0.3\text{ to }0.7$ 28
 2012 A
T2K Off-axis beam
$0.28\text{ to }0.72$ 29
 2012
MINS ${{\overline{\mathit \nu}}}$ beam
$0.25\text{ to }0.75$ 30, 31
 2012 B
MINS MINOS atmospheric
$0.27\text{ to }0.73$ 30, 32
 2012 B
MINS MINOS pure atmospheric ${{\mathit \nu}}$
$0.21\text{ to }0.79$ 30, 32
 2012 B
MINS MINOS pure atmospheric ${{\overline{\mathit \nu}}}$
$0.15\text{ to }0.85$ 33
 2012
ANTR Atmospheric ${{\mathit \nu}}$ with deep see telescope
$0.39\text{ to }0.61$ 34
 2011 C
SKAM Super-Kamiokande
$0.34\text{ to }0.66$
 2011
MINS 2${{\mathit \nu}}$ osc.; maximal mixing
$0.31$ ${}^{+0.10}_{-0.07}$ 35
 2011 B
MINS ${{\overline{\mathit \nu}}}$ beam
$0.41\text{ to }0.59$ 36
 2010
SKAM 3${{\mathit \nu}}$ osc. with solar terms; ${{\mathit \theta}_{{13}}}$ =0
$0.39\text{ to }0.61$ 37
 2010
SKAM 3${{\mathit \nu}}$ osc.; normal mass ordering
$0.37\text{ to }0.63$ 38
 2010
SKAM 3${{\mathit \nu}}$ osc.; inverted mass ordering
$0.31\text{ to }0.69$
 2008 A
MINS MINOS
$0.05\text{ to }0.95$ 39
 2006
MINS Atmospheric ${{\mathit \nu}}$ with far detector
$0.18\text{ to }0.82$ 40
 2006 A
K2K KEK to Super-K
$0.23\text{ to }0.77$ 41
 2006
MINS MINOS
$0.18\text{ to }0.82$ 42
 2005
K2K KEK to Super-K
$0.18\text{ to }0.82$ 43
 2005
SOU2
$0.36\text{ to }0.64$ 44
 2005
SKAM Super-Kamiokande
$0.28\text{ to }0.72$ 45
 2004
MCRO MACRO
$0.34\text{ to }0.66$ 46
 2004
SKAM L/E distribution
$0.08\text{ to }0.92$ 47
 2003
K2K KEK to Super-K
$0.13\text{ to }0.87$ 48
 2003
MCRO MACRO
$0.26\text{ to }0.74$ 49
 2003
MCRO MACRO
$0.15\text{ to }0.85$ 50
 2003
SOU2 Soudan-2 Atmospheric
$0.28\text{ to }0.72$ 51
 2001
MCRO Upward ${{\mathit \mu}}$
$0.29\text{ to }0.71$ 52
 2001
MCRO Upward ${{\mathit \mu}}$
$0.13\text{ to }0.87$ 53
 1999 C
SKAM Upward ${{\mathit \mu}}$
$0.23\text{ to }0.77$ 54
 1999 D
SKAM Upward ${{\mathit \mu}}$
$0.08\text{ to }0.92$ 55
 1999 D
SKAM Stop ${{\mathit \mu}}$ $/$ through
$0.29\text{ to }0.71$ 56
 1998 C
SKAM Super-Kamiokande
$0.08\text{ to }0.92$ 57
 1998
KAMI Kamiokande
$0.24\text{ to }0.76$ 58
 1998
KAMI Kamiokande
$0.20\text{ to }0.80$ 59
 1994
KAMI Kamiokande
 1 ABE 2020F results are based on data collected between 2009 and 2018 in (anti)neutrino mode and include a neutrino beam exposure of $1.49 \times 10^{21}$ ($1.64 \times 10^{21}$) protons on target. Supersedes ABE 2018G.
 2 ADAMSON 2020A uses the complete dataset from MINOS and MINOS+ experiments. The data were collected using a total exposure of $23.76 \times 10^{20}$ protons on target and 60.75 kton$\cdot{}$yr exposure to atmospheric neutrinos. Supersedes ADAMSON 2014 .
 3 ACERO 2019 is based on a sample size of $12.33 \times 10^{20}$ protons on target. The fit combines both antineutrino and neutrino data to extract the oscillation parameters. The results favor the normal mass ordering by 1.9 ${{\mathit \sigma}}$ and $\theta _{23}$ values in octant II by 1.6 ${{\mathit \sigma}}$ . Supersedes ACERO 2018 .
 4 Errors are from normal mass ordering and ${{\mathit \theta}_{{13}}}$ octant II fits.
 5 AARTSEN 2018A uses three years (April 2012 $-$ May 2015) of neutrino data from full sky with reconstructed energies between 5.6 and 56 GeV, measured with the low-energy subdetector DeepCore of the IceCube neutrino telescope. AARTSEN 2018A also reports the best fit result for the inverted mass ordering as $\Delta$m${}^{2}_{32}$ = $-2.32 \times 10^{-3}$ eV${}^{2}$ and sin$^2({{\mathit \theta}_{{23}}} )$ = 0.51. Uncertainties for the inverted mass ordering fits were not provided. Supersedes AARTSEN 2015A.
 6 ABE 2018B uses 328 kton$\cdot{}$years of Super-Kamiokande I-IV atmospheric neutrino data to obtain this result. The fit is performed over the three parameters, $\Delta$m${}^{2}_{32}$, sin$^2({{\mathit \theta}_{{23}}} )$, and $\delta$, while the solar parameters and sin$^2({{\mathit \theta}_{{13}}} )$ are fixed to $\Delta$m${}^{2}_{21}$= ($7.53$ $\pm0.18$) $\times 10^{-5}$ eV${}^{2}$, sin$^2({{\mathit \theta}_{{12}}} )$ = $0.304$ $\pm0.014$, and sin$^2({{\mathit \theta}_{{13}}} )$ = $0.0219$ $\pm0.0012$.
 7 ABE 2021A results are based on $1.49 \times 10^{21}$ POT in neutrino mode and $1.64 \times 10^{21}$ POT in antineutrino mode.
 8 SALAS 2021 reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.
 9 AARTSEN 2020 uses the data taken between May 2012 and April 2014 with the low-energy subdetector DeepCore of the IceCube neutrino telescope. The reconstructed energy range is between 4 (5) and 90 (80) GeV for the main (confirmatory) analysis. Though the observed best-fit is in the lower octant for both mass orderings, a substantial range of sin$^2({{\mathit \theta}_{{23}}} )$ $>$ 0.5 is still compatible with the observed data for both mass orderings.
 10 ESTEBAN 2020A reports results of a global fit to neutrino oscillation data available at the time of the Neutrino2020 conference.
 11 AARTSEN 2019C uses three years (April 2012 $-$ May 2015) of neutrino data from full sky with reconstructed energies between 5.6 and 56 GeV, measured with the low-energy subdetector DeepCore of the IceCube neutrino telescope. AARTSEN 2019C adopts looser event selection criteria to prioritize the efficiency of selecting neutrino events, different from tighter event selection criteria which closely follow the criteria used by AARTSEN 2018A to measure the ${{\mathit \nu}_{{\mu}}}$ disappearance.
 12 ALBERT 2019 measured the oscillation parameters of atmospheric neutrinos with the ANTARES deep sea neutrino telescope using the data taken from 2007 to 2016 (2830 days of total live time). Supersedes ADRIAN-MARTINEZ 2012 .
 13 ABE 2018B uses 328 kton$\cdot{}$years of Super-Kamiokande I-IV atmospheric neutrino data to obtain this result. The fit is performed over the four parameters, $\Delta$m${}^{2}_{32}$, sin$^2{{\mathit \theta}_{{23}}}$, sin$^2{{\mathit \theta}_{{13}}}$, and $\delta$, while the solar parameters are fixed to $\Delta$m${}^{2}_{21}$= ($7.53$ $\pm0.18$) $\times 10^{-5}$ eV${}^{2}$ and sin$^2{{\mathit \theta}_{{12}}}$ = $0.304$ $\pm0.014$.
 14 ABE 2018G data prefers normal mass ordering is with a posterior probability of 87$\%$. Supersedes ABE 2017F.
 15 ACERO 2018 performs a joint fit to the data for ${{\mathit \nu}_{{\mu}}}$ disappearance and ${{\mathit \nu}_{{e}}}$ appearance. The overall best fit favors normal mass ordering and ${{\mathit \theta}_{{23}}}$ in octant II. No 1$\sigma$ confidence intervals are presented for the inverted mass ordering scenarios. Superseded by ACERO 2019 .
 16 Errors are from the projections of the 68$\%$ contour on 2D plot of $\Delta$m${}^{2}$ versus sin$^2({{\mathit \theta}_{{23}}} )$. ABE 2017F supersedes ABE 2017A. Superseded by ABE 2018G.
 17 Superseded by ACERO 2018 .
 18 ABE 2016D reports oscillation results using ${{\overline{\mathit \nu}}_{{\mu}}}$ disappearance in an off-axis beam.
 19 ADAMSON 2016A obtains sin$^2({{\mathit \theta}_{{23}}} )$ in the 68$\%$ C.L. range [0.38, 0.65] ([0.37, 0.64]), with two statistically degenerate best-fit values of 0.44 and 0.59 (0.44 and 0.59) for normal (inverted) mass ordering. Superseded by ADAMSON 2017A.
 20 AARTSEN 2015A obtains this result by a three-neutrino oscillation analysis using $10 - 100$ GeV muon neutrino sample from a total of 953 days of measurement with the low-energy subdetector DeepCore of the IceCube neutrino telescope. Superseded by AARTSEN 2018A.
 21 ABE 2014 results are based on ${{\mathit \nu}_{{\mu}}}$ disappearance using three-neutrino oscillation fit. The confidence intervals are derived from one dimensional profiled likelihoods. Superseded by ABE 2017A.
 22 ADAMSON 2014 uses a complete set of accelerator and atmospheric data. The analysis combines the ${{\mathit \nu}_{{\mu}}}$ disappearance and ${{\mathit \nu}_{{e}}}$ appearance data using three-neutrino oscillation fit. The fit results are obtained for normal and inverted mass ordering assumptions. The best fit is for first ${{\mathit \theta}_{{23}}}$ octant and inverted mass ordering.
 23 FORERO 2014 performs a global fit to neutrino oscillations using solar, reactor, long-baseline accelerator, and atmospheric neutrino data.
 24 GONZALEZ-GARCIA 2014 result comes from a frequentist global fit. The corresponding Bayesian global fit to the same data results are reported in BERGSTROM 2015 as 68$\%$ CL intervals of $0.433 - 0.496$ or $0.530 - 0.594$ for normal and $0.514 - 0.612$ for inverted mass ordering.
 25 AARTSEN 2013B obtained this result by a two-neutrino oscillation analysis using $20 - 100$ GeV muon neutrino sample from a total of 318.9 days of live-time measurement with the low-energy subdetector DeepCore of the IceCube neutrino telescope.
 26 The best fit value is sin${}^{2}({{\mathit \theta}_{{23}}}$ ) = $0.514$ $\pm0.082$. Superseded by ABE 2014 .
 27 ADAMSON 2013B obtained this result from ${{\mathit \nu}_{{\mu}}}$ and ${{\overline{\mathit \nu}}_{{\mu}}}$ disappearance using ${{\mathit \nu}_{{\mu}}}$ ($10.71 \times 10^{20}$ POT) and ${{\overline{\mathit \nu}}_{{\mu}}}$ ($3.36 \times 10^{20}$ POT) beams, and atmospheric (37.88kton-years) data from MINOS The fit assumed two-flavor neutrino hypothesis and identical ${{\mathit \nu}_{{\mu}}}$ and ${{\overline{\mathit \nu}}_{{\mu}}}$ oscillation parameters. Superseded by ADAMSON 2014 .
 28 ABE 2012A obtained this result by a two-neutrino oscillation analysis. The best-fit point is sin${}^{2}(2{{\mathit \theta}_{{23}}}$ ) = 0.98.
 29 ADAMSON 2012 is a two-neutrino oscillation analysis using antineutrinos. The best fit value is sin${}^{2}(2{{\mathit \theta}_{{23}}}$ ) = $0.95$ ${}^{+0.10}_{-0.11}$ $\pm0.01$.
 30 ADAMSON 2012B obtained this result by a two-neutrino oscillation analysis of the L/E distribution using 37.9 kton$\cdot{}$yr atmospheric neutrino data with the MINOS far detector.
 31 The best fit point is $\Delta$m${}^{2}$ = 0.0019 eV${}^{2}$ and sin$^22\theta$ = 0.99. The 90$\%$ single-parameter confidence interval at the best fit point is sin$^22\theta$ $>$ 0.86.
 32 The data are separated into pure samples of ${{\mathit \nu}}$ s and ${{\overline{\mathit \nu}}}$ s, and separate oscillation parameters for ${{\mathit \nu}}$ s and ${{\overline{\mathit \nu}}}$ s are fit to the data. The best fit point is ($\Delta$m${}^{2}$, sin$^22\theta$) = (0.0022 eV${}^{2}$, 0.99) and ($\Delta \bar m{}^{2}$, sin$^22{{\overline{\mathit \theta}}}$) = (0.0016 eV${}^{2}$, 1.00). The quoted result is taken from the 90$\%$ C.L. contour in the ($\Delta$m${}^{2}$, sin$^22\theta$) plane obtained by minimizing the four parameter log-likelihood function with respect to the other oscillation parameters.
 33 ADRIAN-MARTINEZ 2012 measured the oscillation parameters of atmospheric neutrinos with the ANTARES deep sea neutrino telescope using the data taken from 2007 to 2010 (863 days of total live time). Superseded by ALBERT 2019 .
 34 ABE 2011C obtained this result by a two-neutrino oscillation analysis using the Super-Kamiokande-I+II+III atmospheric neutrino data. ABE 2011C also reported results under a two-neutrino disappearance model with separate mixing parameters between ${{\mathit \nu}}$ and ${{\overline{\mathit \nu}}}$ , and obtained sin$^22{{\mathit \theta}} >$ 0.93 for ${{\mathit \nu}}$ and sin$^22{{\mathit \theta}} >$ 0.83 for ${{\overline{\mathit \nu}}}$ at 90$\%$ C.L.
 35 ADAMSON 2011B obtained this result by a two-neutrino oscillation analysis of antineutrinos in an antineutrino enhanced beam with $1.71 \times 10^{20}$ protons on target. This results is consistent with the neutrino measurements of ADAMSON 2011 at 2$\%$ C.L.
 36 WENDELL 2010 obtained this result (sin$^2\theta _{23}$ = $0.407 - 0.583$) by a three-neutrino oscillation analysis using the Super-Kamiokande-I+II+III atmospheric neutrino data, assuming $\theta _{13}$ = 0 but including the solar oscillation parameters $\Delta$m${}^{2}_{21}$ and sin$^2\theta _{12}$ in the fit.
 37 WENDELL 2010 obtained this result (sin$^2\theta _{23}$ = $0.43 - 0.61$) by a three-neutrino oscillation analysis with one mass scale dominance ($\Delta$m${}^{2}_{21}$ = 0) using the Super-Kamiokande-I+II+III atmospheric neutrino data, and updates the HOSAKA 2006A result.
 38 WENDELL 2010 obtained this result (sin$^2\theta _{23}$ = $0.44 - 0.63$) by a three-neutrino oscillation analysis with one mass scale dominance ($\Delta$m${}^{2}_{21}$ = 0) using the Super-Kamiokande-I+II+III atmospheric neutrino data, and updates the HOSAKA 2006A result.
 39 ADAMSON 2006 obtained this result by a two-neutrino oscillation analysis of the L/E distribution using 4.54 kton yr atmospheric neutrino data with the MINOS far detector.
 40 Supercedes ALIU 2005 .
 42 The best fit is for maximal mixing.
 43 ALLISON 2005 result is based upon atmospheric neutrino interactions including upward-stopping muons, with an exposure of 5.9 kton yr. From a two-flavor oscillation analysis the best-fit point is $\Delta \mathit m{}^{2}$ = 0.0017 eV${}^{2}$ and sin$^2(2\theta )$ = 0.97.
 44 ASHIE 2005 obtained this result by a two-neutrino oscillation analysis using 92 kton yr atmospheric neutrino data from the complete Super-Kamiokande I running period.
 45 AMBROSIO 2004 obtained this result, without using the absolute normalization of the neutrino flux, by combining the angular distribution of upward through-going muon tracks with ${{\mathit E}_{{\mu}}}$ $>$ 1 GeV, N$_{low}$ and N$_{high}$, and the numbers of InDown + UpStop and InUp events. Here, N$_{low}$ and N$_{high}$ are the number of events with reconstructed neutrino energies $<$ 30 GeV and $>$ 130 GeV, respectively. InDown and InUp represent events with downward and upward-going tracks starting inside the detector due to neutrino interactions, while UpStop represents entering upward-going tracks which stop in the detector. The best fit is for maximal mixing.
 46 ASHIE 2004 obtained this result from the L(flight length)/E(estimated neutrino energy) distribution of ${{\mathit \nu}_{{\mu}}}$ disappearance probability, using the Super-Kamiokande-I 1489 live-day atmospheric neutrino data.
 47 There are several islands of allowed region from this K2K analysis, extending to high values of $\Delta \mathit m{}^{2}$. We only include the one that overlaps atmospheric neutrino analyses. The best fit is for maximal mixing.
 48 AMBROSIO 2003 obtained this result on the basis of the ratio R = N$_{low}/N_{high}$, where N$_{low}$ and N$_{high}$ are the number of upward through-going muon events with reconstructed neutrino energy $<$ 30 GeV and $>$ 130 GeV, respectively. The data came from the full detector run started in 1994. The method of FELDMAN 1998 is used to obtain the limits.
 49 AMBROSIO 2003 obtained this result by using the ratio R and the angular distribution of the upward through-going muons. R is given in the previous note and the angular distribution is reported in AMBROSIO 2001 . The method of FELDMAN 1998 is used to obtain the limits. The best fit is to maximal mixing.
 50 SANCHEZ 2003 is based on an exposure of 5.9 kton yr. The result is obtained using a likelihood analysis of the neutrino L/E distribution for a selection ${{\mathit \mu}}$ flavor sample while the ${{\mathit e}}$ -flavor sample provides flux normalization. The method of FELDMAN 1998 is used to obtain the allowed region. The best fit is sin$^2(2{{\mathit \theta}} )$ = 0.97.
 51 AMBROSIO 2001 result is based on the angular distribution of upward through-going muon tracks with ${{\mathit E}_{{\mu}}}$ $>$ 1 GeV. The data came from three different detector configurations, but the statistics is largely dominated by the full detector run, from May 1994 to December 2000. The total live time, normalized to the full detector configuration is 6.17 years. The best fit is obtained outside the physical region. The method of FELDMAN 1998 is used to obtain the limits. The best fit is for maximal mixing.
 52 AMBROSIO 2001 result is based on the angular distribution and normalization of upward through-going muon tracks with ${{\mathit E}_{{\mu}}}$ $>$ 1 GeV. See the previous footnote.
 53 FUKUDA 1999C obtained this result from a total of 537 live days of upward through-going muon data in Super-Kamiokande between April 1996 to January 1998. With a threshold of ${{\mathit E}_{{\mu}}}$ $>$ 1.6 GeV, the observed flux is ($1.74$ $\pm0.07$ $\pm0.02$) $\times 10^{-13}$ cm${}^{-2}$s${}^{-1}$sr${}^{-1}$. The best fit is sin$^2(2{{\mathit \theta}} )$ = 0.95.
 54 FUKUDA 1999D obtained this result from a simultaneous fitting to zenith angle distributions of upward-stopping and through-going muons. The flux of upward-stopping muons of minimum energy of 1.6 GeV measured between April 1996 and January 1998 is ($0.39$ $\pm0.04$ $\pm0.02$) $\times 10^{-13}$ cm${}^{-2}$s${}^{-1}$sr${}^{-1}$. This is compared to the expected flux of ($0.73$ $\pm0.16$ (theoretical error))${\times }10^{-13}$ cm${}^{-2}$s${}^{-1}$sr${}^{-1}$. The best fit is to maximal mixing.
 55 FUKUDA 1999D obtained this result from the zenith dependence of the upward-stopping/through-going flux ratio. The best fit is to maximal mixing.
 56 FUKUDA 1998C obtained this result by an analysis of 33.0 kton yr atmospheric neutrino data. The best fit is for maximal mixing.
 57 HATAKEYAMA 1998 obtained this result from a total of 2456 live days of upward-going muon data in Kamiokande between December 1985 and May 1995. With a threshold of ${{\mathit E}_{{\mu}}}$ $>$ 1.6 GeV, the observed flux of upward through-going muons is ($1.94$ $\pm0.10$ ${}^{+0.07}_{-0.06}$) $\times 10^{-13}$ cm${}^{-2}$s${}^{-1}$sr${}^{-1}$. This is compared to the expected flux of ($2.46$ $\pm0.54$ (theoretical error))${\times }10^{-13}$ cm${}^{-2}$s${}^{-1}$sr${}^{-1}$. The best fit is for maximal mixing.
 58 HATAKEYAMA 1998 obtained this result from a combined analysis of Kamiokande contained events (FUKUDA 1994 ) and upward going muon events. The best fit is sin$^2(2{{\mathit \theta}} )$ = 0.95.
 59 FUKUDA 1994 obtained the result by a combined analysis of sub- and multi-GeV atmospheric neutrino events in Kamiokande. The best fit is for maximal mixing.
References:
 ABE 2021A
PR D103 L011101 T2K measurements of muon neutrino and antineutrino disappearance using $3.13\times 10^{21}$ protons on target
 SALAS 2021
JHEP 2102 071 2020 global reassessment of the neutrino oscillation picture
 AARTSEN 2020
EPJ C80 9 Development of an analysis to probe the neutrino mass ordering with atmospheric neutrinos using three years of IceCube DeepCore data
 ABE 2020F
NAT 580 339 Constraint on the matter?antimatter symmetry-violating phase in neutrino oscillations
 Also
PR D103 112008 Improved constraints on neutrino mixing from the T2K experiment with $\mathbf{3.13\times10^{21}}$ protons on target
PRL 125 131802 Precision Constraints for Three-Flavor Neutrino Oscillations from the Full MINOS+ and MINOS Dataset
 ESTEBAN 2020A
JHEP 2009 178 The fate of hints: updated global analysis of three-flavor neutrino oscillations
 AARTSEN 2019C
PR D99 032007 Measurement of Atmospheric Tau Neutrino Appearance with IceCube DeepCore
 ACERO 2019
PRL 123 151803 First Measurement of Neutrino Oscillation Parameters using Neutrinos and Antineutrinos by NOvA
 AGAFONOVA 2019
PR D100 051301 Final results on neutrino oscillation parameters from the OPERA experiment in the CNGS beam
 ALBERT 2019
JHEP 1906 113 Measuring the atmospheric neutrino oscillation parameters and constraining the 3+1 neutrino model with ten years of ANTARES data
 AARTSEN 2018A
PRL 120 071801 Measurement of Atmospheric Neutrino Oscillations at 6?56 GeV with IceCube DeepCore
 ABE 2018B
PR D97 072001 Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV
 ABE 2018G
PRL 121 171802 Search for CP Violation in Neutrino and Antineutrino Oscillations by the T2K Experiment with $2.2\times10^{21}$ Protons on Target
 ACERO 2018
PR D98 032012 New constraints on oscillation parameters from $\nu_e$ appearance and $\nu_\mu$ disappearance in the NOvA experiment
 DE-SALAS 2018
PL B782 633 Status of neutrino oscillations 2018: 3$\sigma$ hint for normal mass ordering and improved CP sensitivity
 ABE 2017C
PR D96 011102 Updated T2K Measurements of Muon Neutrino and Antineutrino Disappearance using $1.5 \times 10^{21}$ Protons on Target
 ABE 2017F
PR D96 092006 Measurement of Neutrino and Antineutrino Oscillations by the T2K Experiment Including a New Additional Sample of ${{\mathit \nu}_{{e}}}$ Interactions at the Far Detector
 Also
PR D98 019902 (errat.) Measurement of neutrino and antineutrino oscillations by the T2K experiment including a new additional sample of $\nu_e$ interactions at the far detector
 ABE 2017A
PRL 118 151801 Combined Analysis of Neutrino and Antineutrino Oscillations at T2K
PRL 118 151802 Measurement of the Neutrino Mixing Angle $\mathit \theta _{23}$ in NOvA
 ABE 2016D
PRL 116 181801 Measurement of Muon Antineutrino Oscillations with an Accelerator-Produced Off-Axis Beam
PR D93 051104 First easurement of Muon-Neutrino Disappearance in NOvA
 AARTSEN 2015A
PR D91 072004 Determining Neutrino Oscillation Parameters from Atmospheric Muon Neutrino Disappearance with Three Years of IceCube DeepCore Data
 ABE 2014
PRL 112 181801 Precise Measurement of the Neutrino Mixing Parameter ${{\mathit \theta}_{{23}}}$ from Muon Neutrino Disappearance in an Off-Axis Beam
 Also
PR D91 072010 Measurements of Neutrino Oscillation in Appearance and Disappearance Channels by the T2K Experiment with $6.6 \times 10^{20}$ Protons on Target
PRL 112 191801 Combined Analysis of ${{\mathit \nu}_{{\mu}}}$ Disappearance and ${{\mathit \nu}_{{\mu}}}$ $\rightarrow{{\mathit \nu}_{{e}}}$ Appearance in MINOS using Accelerator and Atmospheric Neutrinos
 FORERO 2014
PR D90 093006 Neutrino Oscillations Refitted
 GONZALEZ-GARCIA 2014
JHEP 1411 052 Updated Fit to Three Neutrino Mixing: Status of Leptonic $\mathit CP$ Violation
 AARTSEN 2013B
PRL 111 081801 Measurement of Atmospheric Neutrino Oscillations with IceCube
 ABE 2013G
PRL 111 211803 Measurement of Neutrino Oscillation Parameters from Muon Neutrino Disappearance with an Off-Axis Beam
PRL 110 251801 Measurement of Neutrino and Antineutrino Oscillations Using Beam and Atmospheric Data in MINOS
 ABE 2012A
PR D85 031103 First Muon-Neutrino Disappearance Study with an Off-Axis Beam
PR D86 052007 Measurements of Atmospheric Neutrinos and Antineutrinos in the MINOS Far Detector
PRL 108 191801 Improved Measurement of Muon Antineutrino Disappearance in MINOS
PL B714 224 Measurement of Atmospheric Neutrino Oscillations with the ANTARES Neutrino Telescope
 ABE 2011C
PRL 107 241801 Search for Differences in Oscillation Parameters for Atmospheric Neutrinos and Antineutrinos at Super-Kamiokande
PRL 107 021801 First Direct Observation of Muon Antineutrino Disappearance
PRL 106 181801 Measurement of the Neutrino Mass Splitting and Flavor Mixing by MINOS
 WENDELL 2010
PR D81 092004 Atmospheric Neutrino Oscillation Analysis with Subleading Effects in Super-Kamiokande I, II, and III
PRL 101 131802 Measurement of Neutrino Oscillations with the MINOS Detectors in the NuMI Beam
PR D73 072002 First Observations of Separated Atmospheric ${{\mathit \nu}_{{\mu}}}$ and ${{\overline{\mathit \nu}}_{{\mu}}}$ Events in the MINOS Detector
 AHN 2006A
PR D74 072003 Measurement of Neutrino Oscillation by the K2K Experiment
 MICHAEL 2006
PRL 97 191801 Observation of Muon Neutrino Disappearance with the MINOS Detectors in the NuMI Neutrino Beam
 ALIU 2005
PRL 94 081802 Evidence for Muon Neutrino Oscillation in an Accelerator-Based Experiment
 ALLISON 2005
PR D72 052005 Neutrino Oscillation Effects in Soudan 2 Upward-Stopping Muons
 ASHIE 2005
PR D71 112005 Measurement of Atmospheric Neutrino Oscillation Parameters by Super-Kamiokande I
 AMBROSIO 2004
EPJ C36 323 Measurements oa Atmospheric Muon Neutrino Oscillations, Global Analysis of the Data Collected with MACRO Detector
 ASHIE 2004
PRL 93 101801 Evidence for an Oscillatory Signature in Atmospheric Neutrino Oscillation
 AHN 2003
PRL 90 041801 Indications of Neutrino Oscillation in a 250 km Long Baseline Experiment
 AMBROSIO 2003
PL B566 35 Atmospheric Neutrino Oscillations from Upward through Going Muon Multiple Scattering in MACRO
 SANCHEZ 2003
PR D68 113004 Measurement of the Distributions of Atmospheric Neutrinos in SOUDAN2 and their Interpretation as Neutrino Oscillations
 AMBROSIO 2001
PL B517 59 Matter Effects in Upward Going Muons and Sterile Neutrino Oscillations
 FUKUDA 1999D
PL B467 185 Neutrino Induced Upward Stopping Muons in Super-Kamiokande
 FUKUDA 1999C
PRL 82 2644 Measurement of the Flux and Zenith Angle Distribution of Upward Through Going Muons by Super-Kamiokande
 FUKUDA 1998C
PRL 81 1562 Evidence for Oscillation of Atmospheric Neutrinos
 HATAKEYAMA 1998
PRL 81 2016 Measurement of the Flux and Zenith Angle Distribution of Upward Through-Going Muons in Kamiokande II + III
 FUKUDA 1994
PL B335 237 Atmospheric ${{\mathit \nu}_{{\mu}}}$ /${{\mathit \nu}_{{e}}}$ Ratio in the Multi-GeV Energy Range