Other pages in this series (6)

Surprise against a reference

Each estimate's distance from a reference value divided by its own error: Planck 2018 and Riess et al. 2022 side by side

Takeaway

Against Planck 2018, 25.9% of the 548 estimates in Huchra's list lie beyond ±2σ (normal: 4.6%) and the surprise index (|z| > 2.326) is 20.4% (2%). Against Riess et al. 2022: 31.9% and 26.5%.
Of the 163 rows of Table A2, 25.9% lie beyond ±2σ against Planck and 23.9% against Riess. Under either reference, in both compilations, the reference falls inside the quoted error less often than a normal distribution predicts.
The metric does not decide which side of the tension is right: change the reference and different estimates become the surprising ones.

Definition: For each estimate z = (reference − x)/u_x, and the shares with |z| > 1 and |z| > 2 (misses), |z| < 0.674 (interquartile index) and |z| > 2.326 (surprise index). Two references are shown side by side instead of one: Planck 2018 (67.36 ± 0.54) and Riess et al. 2022 (73.04 ± 1.04). Under a normal distribution the expectations are 31.7%, 4.6%, 50% and 2%.
Formula: z = (reference − x) / u_x. The reference's uncertainty does not enter z. Asymmetric errors enter as the mean.
Denominator: Huchra's list: 548 rows from 1929 to 2010 (543 from 1950). Table A2: 163 rows from 1976 to 2019; against Planck, 162 rows, the table's 2018 row "67.4 ± 0.5" (the rounded Planck 2018 value) being left out. 95% intervals: cluster bootstrap by publication year, 10,000 resamples, seed 2026.
How to read it: Against Planck 2018, Huchra's list misses at ±1σ in 51.8% (normal: 31.7%), at ±2σ in 25.9% (4.6%), with a surprise index of 20.4% (2%) and an interquartile index of 33.9% (50%). Against Riess et al. 2022: 54.7%, 31.9%, 26.5% and 34.9%. Table A2 against Planck: 46.9%, 25.9%, 24.1% and 42.0%; against Riess: 47.2%, 23.9%, 18.4% and 39.3%. In every cell the intervals of the miss and surprise shares lie above the normal expectation and the interquartile index below it. The ±2σ miss rate of the physical-constants chapter (M3) is 29%.
Caveats: Which side counts as "surprising" depends on the reference; that is why two are shown. These shares count whether quoted errors covered the distance to a reference; they do not say which reference is right or whether the tension is real. The references are measurements too, not true values, and the Planck value holds within the ΛCDM model. Estimates that share a calibrator (Cepheid zero point, LMC distance) or the same data are not independent, and the year-cluster bootstrap handles only part of that dependence. Neither compilation is complete.

Planck 2018Riess et al. 2022Normal expectation

Filled dots: Huchra’s list (from 1929); open dots: Table A2 (1976–2019). Vertical bars: 95% year-cluster bootstrap intervals.

z = (reference − x)/u_x for each Table A2 row. Top: against Planck 2018 (the 2018 rounded-Planck row left out); bottom: against Riess et al. 2022. Triangles: clipped beyond ±8.

Series	Start	Reference	n	Left out	Values below the reference	1σ miss %	2σ miss %	Interquartile %	Surprise %	Median \|z\|
Huchra's list	from 1929	Planck 2018	548	0	265	51.8 [46–60%]	25.9 [20–33%]	33.9 [25–41%]	20.4 [16–26%]	1.05
Huchra's list	from 1929	Riess et al. 2022	548	0	398	54.7 [49–61%]	31.9 [28–37%]	34.9 [29–40%]	26.5 [22–32%]	1.12
Huchra's list	from 1950	Planck 2018	543	0	265	51.4 [46–59%]	25.4 [20–33%]	34.3 [26–41%]	19.9 [16–25%]	1.04
Huchra's list	from 1950	Riess et al. 2022	543	0	398	54.5 [49–61%]	31.5 [27–37%]	35.2 [29–40%]	26.0 [22–31%]	1.12
Table A2	1976–2019	Planck 2018	162	1	68	46.9 [38–58%]	25.9 [18–34%]	42.0 [32–50%]	24.1 [17–32%]	0.93
Table A2	1976–2019	Riess et al. 2022	163	0	112	47.2 [38–57%]	23.9 [17–31%]	39.3 [30–48%]	18.4 [12–25%]	0.94
Merged (sensitivity)	from 1929	Planck 2018	641	1	299	50.9 [45–58%]	26.5 [22–33%]	35.9 [28–42%]	21.5 [18–27%]	1.04
Merged (sensitivity)	from 1929	Riess et al. 2022	642	0	460	53.3 [48–60%]	30.5 [26–35%]	35.5 [30–40%]	25.2 [21–30%]	1.07
Merged (sensitivity)	from 1950	Planck 2018	636	1	299	50.5 [45–57%]	26.1 [21–32%]	36.2 [28–42%]	21.1 [17–26%]	1.04
Merged (sensitivity)	from 1950	Riess et al. 2022	637	0	460	53.1 [48–59%]	30.1 [26–35%]	35.8 [30–41%]	24.8 [21–30%]	1.06
Huchra's list, cluster = type code (sensitivity)	from 1929	Planck 2018	548	0	265	51.8 [42–60%]	25.9 [17–34%]	33.9 [25–42%]	20.4 [14–26%]	1.05
Huchra's list, cluster = type code (sensitivity)	from 1929	Riess et al. 2022	548	0	398	54.7 [45–62%]	31.9 [21–39%]	34.9 [29–42%]	26.5 [16–34%]	1.12

Download CSV: h0_h3_summary.csv

Sources: J. Huchra, Estimates of the Hubble Constant (hubble.plot.dat, last updated 2010-10-07), Harvard-Smithsonian Center for Astrophysics; assembled as part of the NASA/HST Key Project on the Extragalactic Distance Scale. Aggregates only. · Faerber, T.; López-Corredoira, M. A Chi-Squared Analysis of the Measurements of Two Cosmological Parameters over Time. Universe 2020, 6, 114, doi:10.3390/universe6080114, Appendix Table A2 (read from arXiv:2008.04035v1). CC BY 4.0.