Steam reforming

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https://en.wikipedia.org/wiki/Steam_reforming 

Illustrating inputs and outputs of steam reforming of natural gas, a process to produce hydrogen and CO₂ greenhouse gas that may be captured with CCS

Steam reforming or steam methane reforming is a method for producing syngas (hydrogen and carbon monoxide) by reaction of hydrocarbons with water. Commonly natural gas is the feedstock. The main purpose of this technology is hydrogen production. The reaction is represented by this equilibrium:

${\displaystyle CH_{4}+H_{2}O\rightleftharpoons CO+3H_{2}}$

The reaction is strongly endothermic (ΔH_SR = 206 kJ/mol).

Hydrogen produced by steam reforming is termed 'grey hydrogen' when the waste carbon dioxide is released to the atmosphere and 'blue hydrogen' when the carbon dioxide is (mostly) captured and stored geologically - see carbon capture and storage. (Zero carbon 'green' hydrogen is produced by electrolysis using low- or zero-carbon electricity. Zero carbon emissions 'turquoise' hydrogen is produced by one-step methane pyrolysis of natural gas.)

Steam reforming of natural gas produces most of the world's hydrogen. Hydrogen is used in the industrial synthesis of ammonia and other chemicals.

Reactions

Steam reforming reaction kinetics, in particular using nickel-alumina catalysts, have been studied in detail since the 1950s.

Pre-reforming

Depiction of the general process flow of a typical steam reforming plant. (PSA = Pressure swing adsorption, NG = Natural gas)

The purpose of pre-reforming is to break down higher hydrocarbons such as propane, butane or naphta into methane (CH₄), which allows for more efficient reforming downstream.

Steam reforming

The name-giving reaction is the steam reforming (SR) reaction and is expressed by the equation:

${\displaystyle [1]\qquad CH_{4}+H_{2}O\rightleftharpoons CO+3H_{2}\qquad \Delta H_{SR}=206\ kJ/mol}$

Via the water-gas shift reaction (WGSR), additional hydrogen is released by reaction of water with the carbon monoxide generated according to equation [1]:

${\displaystyle [2]\qquad CO+H_{2}O\rightleftharpoons CO_{2}+H_{2}\qquad \Delta H_{WGSR}=-41\ kJ/mol}$

Some additional reactions occurring within steam reforming processes have been studied. Commonly the direct steam reforming (DSR) reaction is also included:

${\displaystyle [3]\qquad CH_{4}+2H_{2}O\rightleftharpoons CO_{2}+4H_{2}\qquad \Delta H_{DSR}=165\ kJ/mol}$

As these reactions by themselves are highly endothermic (apart from WGSR, which is mildly exothermic), a large amount of heat needs to be added to the reactor to keep a constant temperature. Optimal SMR reactor operating conditions lie within a temperature range of 800 °C to 900 °C at medium pressures of 20-30 bar. High excess of steam is required, expressed by the (molar) steam-to-carbon (S/C) ratio. Typical S/C ratio values lie within the range 2.5:1 - 3:1.

Industrial practice

Global Hydrogen Production by Method

The reaction is conducted in multitubular packed bed reactors, a subtype of the plug flow reactor category. These reactors consist of an array of long and narrow tubes which are situated within the combustion chamber of a large industrial furnace, providing the necessary energy to keep the reactor at a constant temperature during operation. Furnace designs vary, depending on the burner configuration they are typically categorized into: top-fired, bottom-fired, and side-fired. A notable design is the Foster-Wheeler terrace wall reformer.

Inside the tubes, a mixture of steam and methane are put into contact with a nickel catalyst. Catalysts with high surface-area-to-volume ratio are preferred because of diffusion limitations due to high operating temperature. Examples of catalyst shapes used are spoked wheels, gear wheels, and rings with holes (see: Raschig rings). Additionally, these shapes have a low pressure drop which is advantageous for this application.

Steam reforming of natural gas is 65–75% efficient.

The United States produces 9–10 million tons of hydrogen per year, mostly with steam reforming of natural gas. The worldwide ammonia production, using hydrogen derived from steam reforming, was 144 million tonnes in 2018. The energy consumption has been reduced from 100 GJ/tonne of ammonia in 1920 to 27 GJ by 2019.

Globally, almost 50% of hydrogen is produced via steam reforming. It is currently the least expensive method for hydrogen production available in terms of its capital cost.

In an effort to decarbonise hydrogen production, carbon capture and storage (CCS) methods are being implemented within the industry, which have the potential to remove up to 90% of CO₂ produced from the process. Despite this, implementation of this technology remains problematic, costly, and increases the price of the produced hydrogen significantly.

Autothermal reforming

Autothermal reforming (ATR) uses oxygen and carbon dioxide or steam in a reaction with methane to form syngas. The reaction takes place in a single chamber where the methane is partially oxidized. The reaction is exothermic due to the high energy of O₂ with its relatively weak double bond. When the ATR uses carbon dioxide, the H₂:CO ratio produced is 1:1; when the ATR uses steam, the H₂:CO ratio produced is 2.5:1. The outlet temperature of the syngas is between 950–1100 °C and outlet pressure can be as high as 100 bar.

In addition to reactions [1] - [3], ATR introduces the following reaction:

${\displaystyle [4]\qquad CH_{4}+0.5O_{2}\rightleftharpoons CO+2H_{2}\qquad \Delta H_{R}=-24.5kJ/mol}$

The main difference between SMR and ATR is that SMR only uses air for combustion as a heat source to create steam, while ATR uses purified oxygen. The advantage of ATR is that the H₂:CO ratio can be varied, which can be useful for producing specialty products. Due to the exothermic nature of some of the additional reactions occurring within ATR, the process can essentially be performed at a net enthalpy of zero (ΔH = 0).

Partial oxidation

Main article: Partial oxidation

Partial oxidation (POX) occurs when a sub-stoichiometric fuel-air mixture is partially combusted in a reformer creating hydrogen-rich syngas. POX is typically much faster than steam reforming and requires a smaller reactor vessel. POX produces less hydrogen per unit of the input fuel than steam reforming of the same fuel.

Steam reforming at small scale

The capital cost of steam reforming plants is considered prohibitive for small to medium size applications. The costs for these elaborate facilities do not scale down well. Conventional steam reforming plants operate at pressures between 200 and 600 psi (14–40 bar) with outlet temperatures in the range of 815 to 925 °C.

For combustion engines

Flared gas and vented volatile organic compounds (VOCs) are known problems in the offshore industry and in the on-shore oil and gas industry, since both release greenhouse gases into the atmosphere. Reforming for combustion engines utilizes steam reforming technology for converting waste gases into a source of energy.

Reforming for combustion engines is based on steam reforming, where non-methane hydrocarbons (NMHCs) of low quality gases are converted to synthesis gas (H₂ + CO) and finally to methane (CH₄), carbon dioxide (CO₂) and hydrogen (H₂) - thereby improving the fuel gas quality (methane number).

For fuel cells

There is also interest in the development of much smaller units based on similar technology to produce hydrogen as a feedstock for fuel cells. Small-scale steam reforming units to supply fuel cells are currently the subject of research and development, typically involving the reforming of methanol, but other fuels are also being considered such as propane, gasoline, autogas, diesel fuel, and ethanol.

Disadvantages

The reformer– the fuel-cell system is still being researched but in the near term, systems would continue to run on existing fuels, such as natural gas or gasoline or diesel. However, there is an active debate about whether using these fuels to make hydrogen is beneficial while global warming is an issue. Fossil fuel reforming does not eliminate carbon dioxide release into the atmosphere but reduces the carbon dioxide emissions and nearly eliminates carbon monoxide emissions as compared to the burning of conventional fuels due to increased efficiency and fuel cell characteristics. However, by turning the release of carbon dioxide into a point source rather than distributed release, carbon capture and storage becomes a possibility, which would prevent the carbon dioxide's release to the atmosphere, while adding to the cost of the process.

The cost of hydrogen production by reforming fossil fuels depends on the scale at which it is done, the capital cost of the reformer, and the efficiency of the unit, so that whilst it may cost only a few dollars per kilogram of hydrogen at an industrial scale, it could be more expensive at the smaller scale needed for fuel cells.

Challenges with reformers supplying fuel cells

There are several challenges associated with this technology:

• The reforming reaction takes place at high temperatures, making it slow to start up and requiring costly high-temperature materials.
• Sulfur compounds in the fuel will poison certain catalysts, making it difficult to run this type of system from ordinary gasoline. Some new technologies have overcome this challenge with sulfur-tolerant catalysts.
• Coking would be another cause of catalyst deactivation during steam reforming. High reaction temperatures, low steam-to-carbon ratio (S/C), and the complex nature of sulfur-containing commercial hydrocarbon fuels make coking especially favorable. Olefins, typically ethylene, and aromatics are well-known carbon-precursors, hence their formation must be reduced during steam reforming. Additionally, catalysts with lower acidity were reported to be less prone to coking by suppressing dehydrogenation reactions. H₂S, the main product in the reforming of organic sulfur, can bind to all transition metal catalysts to form metal–sulfur bonds and subsequently reduce catalyst activity by inhibiting the chemisorption of reforming reactants. Meanwhile, the adsorbed sulfur species increases the catalyst acidity, and hence indirectly promotes coking. Precious metal catalysts such as Rh and Pt have lower tendencies to form bulk sulfides than other metal catalysts such as Ni. Rh and Pt are less prone to sulfur poisoning by only chemisorbing sulfur rather than forming metal sulfides.
• Low temperature polymer fuel cell membranes can be poisoned by the carbon monoxide (CO) produced by the reactor, making it necessary to include complex CO-removal systems. Solid oxide fuel cells (SOFC) and molten carbonate fuel cells (MCFC) do not have this problem, but operate at higher temperatures, slowing start-up time, and requiring costly materials and bulky insulation.
• The thermodynamic efficiency of the process is between 70% and 85% (LHV basis) depending on the purity of the hydrogen product.

p-value

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/P-value

In null hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Reporting p-values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p-value is hard to grasp, misuse is widespread and has been a major topic in metascience.

Basic concepts

In statistics, every conjecture concerning the unknown probability distribution of a collection of random variables representing the observed data ${\displaystyle X}$ in some study is called a statistical hypothesis. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis is tenable, but not to investigate other specific hypotheses, then such a test is called a null hypothesis test.

As our statistical hypothesis will, by definition, state some property of the distribution, the null hypothesis is the default hypothesis under which that property does not exist. The null hypothesis is typically that some parameter (such as a correlation or a difference between means) in the populations of interest is zero. Note that our hypothesis might specify the probability distribution of ${\displaystyle X}$ precisely, or it might only specify that it belongs to some class of distributions. Often, we reduce the data to a single numerical statistic, e.g., ${\displaystyle T}$, whose marginal probability distribution is closely connected to a main question of interest in the study.

The p-value is used in the context of null hypothesis testing in order to quantify the statistical significance of a result, the result being the observed value of the chosen statistic ${\displaystyle T}$.  The lower the p-value is, the lower the probability of getting that result if the null hypothesis were true. A result is said to be statistically significant if it allows us to reject the null hypothesis. All other things being equal, smaller p-values are taken as stronger evidence against the null hypothesis

Loosely speaking, rejection of the null hypothesis implies that there is sufficient evidence against it.

As a particular example, if a null hypothesis states that a certain summary statistic ${\displaystyle T}$ follows the standard normal distribution N(0,1), then the rejection of this null hypothesis could mean that (i) the mean of ${\displaystyle T}$ is not 0, or (ii) the variance of ${\displaystyle T}$ is not 1, or (iii) ${\displaystyle T}$ is not normally distributed. Different tests of the same null hypothesis would be more or less sensitive to different alternatives. However, even if we do manage to reject the null hypothesis for all 3 alternatives, and even if we know the distribution is normal and variance is 1, the null hypothesis test does not tell us which non-zero values of the mean are now most plausible. The more independent observations from the same probability distribution one has, the more accurate the test will be, and the higher the precision with which one will be able to determine the mean value and show that it is not equal to zero; but this will also increase the importance of evaluating the real-world or scientific relevance of this deviation.

Definition and interpretation

General

Consider an observed test-statistic ${\displaystyle t}$ from unknown distribution ${\displaystyle T}$. Then the p-value ${\displaystyle p}$ is what the prior probability would be of observing a test-statistic value at least as "extreme" as ${\displaystyle t}$ if null hypothesis ${\displaystyle H_{0}}$ were true. That is:

• ${\displaystyle p=\Pr(T\geq t\mid H_{0})}$ for a one-sided right-tail test,
• ${\displaystyle p=\Pr(T\leq t\mid H_{0})}$ for a one-sided left-tail test,
• ${\displaystyle p=2\min\{\Pr(T\geq t\mid H_{0}),\Pr(T\leq t\mid H_{0})\}}$ for a two-sided test. If distribution ${\displaystyle T}$ is symmetric about zero, then ${\displaystyle p=\Pr(|T|\geq |t|\mid H_{0})}$

If the p-value is very small, then either the null hypothesis is false or something unlikely has occurred. In a formal significance test, the null hypothesis ${\displaystyle H_{0}}$ is rejected if the p-value is less than a pre-defined threshold value ${\displaystyle \alpha }$, which is referred to as the alpha level or significance level. The value of ${\displaystyle \alpha }$ is instead set by the researcher before examining the data. ${\displaystyle \alpha }$ defines the proportion of the distribution, ${\displaystyle T}$, that is said to define such a narrow range of all the possible outcomes of ${\displaystyle T}$ that if ${\displaystyle t}$'s value is within that range its value is unlikely to have occurred by chance. Intuitively, this means that if ${\displaystyle \alpha }$ is set to be 0.10, only 1/10th of the distribution of ${\displaystyle T}$ is defined by ${\displaystyle \alpha }$, so if ${\displaystyle t}$ falls within that range it is already occurring over a number of outcomes that happen a rare 1/10th of the time, thus suggesting this is unlikely to occur randomly. By convention, ${\displaystyle \alpha }$ is commonly set to 0.05, though lower alpha levels are sometimes used.

The p-value is a function of the chosen test statistic ${\displaystyle T}$ and is therefore a random variable. If the null hypothesis fixes the probability distribution of ${\displaystyle T}$ precisely, and if that distribution is continuous, then when the null-hypothesis is true, the p-value is uniformly distributed between 0 and 1. Thus, the p-value is not fixed. If the same test is repeated independently with fresh data (always with the same probability distribution), one will obtain a different p-value in each iteration. If the null-hypothesis is composite, or the distribution of the statistic is discrete, the probability of obtaining a p-value less than or equal to any number between 0 and 1 is less than or equal to that number, if the null-hypothesis is true. It remains the case that very small values are relatively unlikely if the null-hypothesis is true, and that a significance test at level ${\displaystyle \alpha }$ is obtained by rejecting the null-hypothesis if the significance level is less than or equal to ${\displaystyle \alpha }$.

Different p-values based on independent sets of data can be combined, for instance using Fisher's combined probability test.

Distribution

When the null hypothesis is true, if it takes the form ${\displaystyle H_{0}:\theta =\theta _{0}}$, and the underlying random variable is continuous, then the probability distribution of the p-value is uniform on the interval [0,1]. By contrast, if the alternative hypothesis is true, the distribution is dependent on sample size and the true value of the parameter being studied.

The distribution of p-values for a group of studies is sometimes called a p-curve. A p-curve can be used to assess the reliability of scientific literature, such as by detecting publication bias or p-hacking.

For composite hypothesis

In parametric hypothesis testing problems, a simple or point hypothesis refers to a hypothesis where the parameter's value is assumed to be a single number. In contrast, in a composite hypothesis the parameter's value is given by a set of numbers. For example, when testing the null hypothesis that a distribution is normal with a mean less than or equal to zero against the alternative that the mean is greater than zero (variance known), the null hypothesis does not specify the probability distribution of the appropriate test statistic. In the just mentioned example that would be the Z-statistic belonging to the one-sided one-sample Z-test. For each possible value of the theoretical mean, the Z-test statistic has a different probability distribution. In these circumstances (the case of a so-called composite null hypothesis) the p-value is defined by taking the least favourable null-hypothesis case, which is typically on the border between null and alternative.

This definition ensures the complementarity of p-values and alpha-levels. If we set the significance level alpha to 0.05, and only reject the null hypothesis if the p-value is less than or equal to 0.05, then our hypothesis test will indeed have significance level (maximal type 1 error rate) 0.05. As Neyman wrote: “The error that a practising statistician would consider the more important to avoid (which is a subjective judgment) is called the error of the first kind. The first demand of the mathematical theory is to deduce such test criteria as would ensure that the probability of committing an error of the first kind would equal (or approximately equal, or not exceed) a preassigned number α, such as α = 0.05 or 0.01, etc. This number is called the level of significance”; Neyman 1976, p. 161 in "The Emergence of Mathematical Statistics: A Historical Sketch with Particular Reference to the United States","On the History of Statistics and Probability", ed. D.B. Owen, New York: Marcel Dekker, pp. 149-193. See also "Confusion Over Measures of Evidence (p's) Versus Errors (a's) in Classical Statistical Testing", Raymond Hubbard and M. J. Bayarri, The American Statistician, August 2003, Vol. 57, No 3, 171--182 (with discussion). For a concise modern statement see Chapter 10 of "All of Statistics: A Concise Course in Statistical Inference", Springer; 1st Corrected ed. 20 edition (September 17, 2004). Larry Wasserman.

Usage

The p-value is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing. In this method, before conducting the study, one first chooses a model (the null hypothesis) and the alpha level α (most commonly .05). After analyzing the data, if the p-value is less than α, that is taken to mean the observed data is sufficiently inconsistent with the null hypothesis for the null hypothesis to be rejected. However, that does not prove that the null hypothesis is false. The p-value does not, in itself, establish probabilities of hypotheses. Rather, it is a tool for deciding whether to reject the null hypothesis.

Misuse

Main article: Misuse of p-values

According to the ASA, there is widespread agreement that p-values are often misused and misinterpreted. One practice that has been particularly criticized is accepting the alternative hypothesis for any p-value nominally less than .05 without other supporting evidence. Although p-values are helpful in assessing how incompatible the data are with a specified statistical model, contextual factors must also be considered, such as "the design of a study, the quality of the measurements, the external evidence for the phenomenon under study, and the validity of assumptions that underlie the data analysis". Another concern is that the p-value is often misunderstood as being the probability that the null hypothesis is true.

Some statisticians have proposed abandoning p-values and focusing more on other inferential statistics, such as confidence intervals, likelihood ratios, or Bayes factors, but there is heated debate on the feasibility of these alternatives. Others have suggested to remove fixed significance thresholds and to interpret p-values as continuous indices of the strength of evidence against the null hypothesis. Yet others suggested to report alongside p-values the prior probability of a real effect that would be required to obtain a false positive risk (i.e. the probability that there is no real effect) below a pre-specified threshold (e.g. 5%).

Calculation

Usually, ${\displaystyle T}$ is a test statistic. A test statistic is the output of a scalar function of all the observations. This statistic provides a single number, such as a t-statistic or an F-statistic. As such, the test statistic follows a distribution determined by the function used to define that test statistic and the distribution of the input observational data.

For the important case in which the data are hypothesized to be a random sample from a normal distribution, depending on the nature of the test statistic and the hypotheses of interest about its distribution, different null hypothesis tests have been developed. Some such tests are the z-test for hypotheses concerning the mean of a normal distribution with known variance, the t-test based on Student's t-distribution of a suitable statistic for hypotheses concerning the mean of a normal distribution when the variance is unknown, the F-test based on the F-distribution of yet another statistic for hypotheses concerning the variance. For data of other nature, for instance categorical (discrete) data, test statistics might be constructed whose null hypothesis distribution is based on normal approximations to appropriate statistics obtained by invoking the central limit theorem for large samples, as in the case of Pearson's chi-squared test.

Thus computing a p-value requires a null hypothesis, a test statistic (together with deciding whether the researcher is performing a one-tailed test or a two-tailed test), and data. Even though computing the test statistic on given data may be easy, computing the sampling distribution under the null hypothesis, and then computing its cumulative distribution function (CDF) is often a difficult problem. Today, this computation is done using statistical software, often via numeric methods (rather than exact formulae), but, in the early and mid 20th century, this was instead done via tables of values, and one interpolated or extrapolated p-values from these discrete values. Rather than using a table of p-values, Fisher instead inverted the CDF, publishing a list of values of the test statistic for given fixed p-values; this corresponds to computing the quantile function (inverse CDF).

Example

Main article: Checking whether a coin is fair

As an example of a statistical test, an experiment is performed to determine whether a coin flip is fair (equal chance of landing heads or tails) or unfairly biased (one outcome being more likely than the other).

Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips. The full data ${\displaystyle X}$ would be a sequence of twenty times the symbol "H" or "T". The statistic on which one might focus, could be the total number ${\displaystyle T}$ of heads. The null hypothesis is that the coin is fair, and coin tosses are independent of one another. If a right-tailed test is considered, which would be the case if one is actually interested in the possibility that the coin is biased towards falling heads, then the p-value of this result is the chance of a fair coin landing on heads at least 14 times out of 20 flips. That probability can be computed from binomial coefficients as

${\displaystyle {\begin{aligned}&\Pr(14{\text{ heads}})+\Pr(15{\text{ heads}})+\cdots +\Pr(20{\text{ heads}})\\&={\frac {1}{2^{20}}}\left[{\binom {20}{14}}+{\binom {20}{15}}+\cdots +{\binom {20}{20}}\right]={\frac {60,\!460}{1,\!048,\!576}}\approx 0.058\end{aligned}}}$

This probability is the p-value, considering only extreme results that favor heads. This is called a one-tailed test. However, one might be interested in deviations in either direction, favoring either heads or tails. The two-tailed p-value, which considers deviations favoring either heads or tails, may instead be calculated. As the binomial distribution is symmetrical for a fair coin, the two-sided p-value is simply twice the above calculated single-sided p-value: the two-sided p-value is 0.115.

In the above example:

• Null hypothesis (H₀): The coin is fair, with Pr(heads) = 0.5
• Test statistic: Number of heads
• Alpha level (designated threshold of significance): 0.05
• Observation O: 14 heads out of 20 flips; and
• Two-tailed p-value of observation O given H₀ = 2 × min(Pr(no. of heads ≥ 14 heads), Pr(no. of heads ≤ 14 heads)) = 2 × min(0.058, 0.978) = 2*0.058 = 0.115.

Note that the Pr (no. of heads ≤ 14 heads) = 1 - Pr(no. of heads ≥ 14 heads) + Pr (no. of head = 14) = 1 - 0.058 + 0.036 = 0.978; however, symmetry of the binomial distribution makes it an unnecessary computation to find the smaller of the two probabilities. Here, the calculated p-value exceeds .05, meaning that the data falls within the range of what would happen 95% of the time were the coin in fact fair. Hence, the null hypothesis is not rejected at the .05 level.

However, had one more head been obtained, the resulting p-value (two-tailed) would have been 0.0414 (4.14%), in which case the null hypothesis would be rejected at the .05 level.

History

John Arbuthnot

 

Pierre-Simon Laplace

 

Karl Pearson

 

Ronald Fisher

Computations of p-values date back to the 1700s, where they were computed for the human sex ratio at birth, and used to compute statistical significance compared to the null hypothesis of equal probability of male and female births. John Arbuthnot studied this question in 1710, and examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 1/2⁸², or about 1 in 4,836,000,000,000,000,000,000,000; in modern terms, the p-value. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." In modern terms, he rejected the null hypothesis of equally likely male and female births at the p = 1/2⁸² significance level. This and other work by Arbuthnot is credited as "… the first use of significance tests …" the first example of reasoning about statistical significance, and "… perhaps the first published report of a nonparametric test …", specifically the sign test; see details at Sign test § History.

The same question was later addressed by Pierre-Simon Laplace, who instead used a parametric test, modeling the number of male births with a binomial distribution:

In the 1770s Laplace considered the statistics of almost half a million births. The statistics showed an excess of boys compared to girls. He concluded by calculation of a p-value that the excess was a real, but unexplained, effect.

The p-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test, using the chi-squared distribution and notated as capital P. The p-values for the chi-squared distribution (for various values of χ² and degrees of freedom), now notated as P, were calculated in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII).

The use of the p-value in statistics was popularized by Ronald Fisher, and it plays a central role in his approach to the subject. In his influential book Statistical Methods for Research Workers (1925), Fisher proposed the level p = 0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, and applied this to a normal distribution (as a two-tailed test), thus yielding the rule of two standard deviations (on a normal distribution) for statistical significance (see 68–95–99.7 rule).

He then computed a table of values, similar to Elderton but, importantly, reversed the roles of χ² and p. That is, rather than computing p for different values of χ² (and degrees of freedom n), he computed values of χ² that yield specified p-values, specifically 0.99, 0.98, 0.95, 0,90, 0.80, 0.70, 0.50, 0.30, 0.20, 0.10, 0.05, 0.02, and 0.01. That allowed computed values of χ² to be compared against cutoffs and encouraged the use of p-values (especially 0.05, 0.02, and 0.01) as cutoffs, instead of computing and reporting p-values themselves. The same type of tables were then compiled in (Fisher & Yates 1938), which cemented the approach.

As an illustration of the application of p-values to the design and interpretation of experiments, in his following book The Design of Experiments (1935), Fisher presented the lady tasting tea experiment, which is the archetypal example of the p-value.

To evaluate a lady's claim that she (Muriel Bristol) could distinguish by taste how tea is prepared (first adding the milk to the cup, then the tea, or first tea, then milk), she was sequentially presented with 8 cups: 4 prepared one way, 4 prepared the other, and asked to determine the preparation of each cup (knowing that there were 4 of each). In that case, the null hypothesis was that she had no special ability, the test was Fisher's exact test, and the p-value was ${\displaystyle 1/{\binom {8}{4}}=1/70\approx 0.014,}$ so Fisher was willing to reject the null hypothesis (consider the outcome highly unlikely to be due to chance) if all were classified correctly. (In the actual experiment, Bristol correctly classified all 8 cups.)

Fisher reiterated the p = 0.05 threshold and explained its rationale, stating:

It is usual and convenient for experimenters to take 5 per cent as a standard level of significance, in the sense that they are prepared to ignore all results which fail to reach this standard, and, by this means, to eliminate from further discussion the greater part of the fluctuations which chance causes have introduced into their experimental results.

He also applies this threshold to the design of experiments, noting that had only 6 cups been presented (3 of each), a perfect classification would have only yielded a p-value of ${\displaystyle 1/{\binom {6}{3}}=1/20=0.05,}$ which would not have met this level of significance. Fisher also underlined the interpretation of p, as the long-run proportion of values at least as extreme as the data, assuming the null hypothesis is true.

In later editions, Fisher explicitly contrasted the use of the p-value for statistical inference in science with the Neyman–Pearson method, which he terms "Acceptance Procedures". Fisher emphasizes that while fixed levels such as 5%, 2%, and 1% are convenient, the exact p-value can be used, and the strength of evidence can and will be revised with further experimentation. In contrast, decision procedures require a clear-cut decision, yielding an irreversible action, and the procedure is based on costs of error, which, he argues, are inapplicable to scientific research.

Related indices

The E-value corresponds to the expected number of times in multiple testing that one expects to obtain a test statistic at least as extreme as the one that was actually observed if one assumes that the null hypothesis is true. The E-value is the product of the number of tests and the p-value.

The q-value is the analog of the p-value with respect to the positive false discovery rate. It is used in multiple hypothesis testing to maintain statistical power while minimizing the false positive rate.

The Probability of Direction (pd) is the Bayesian numerical equivalent of the p-value. It corresponds to the proportion of the posterior distribution that is of the median's sign, typically varying between 50% and 100%, and representing the certainty with which an effect is positive or negative.