Probit

Key takeaways

Probit belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Probit to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Probit from memory before moving on to harder problems.

Reference excerpt

In statistics, the probit function converts a probability (a number between 0 and 1) into a score. This score indicates how many standard deviations a value from a standard normal distribution (or "bell curve") is from the mean. For example, a probability of 0.5 (50%) represents the exact middle of the distribution, so its probit score is 0. A smaller probability like 0.025 (2.5%) is far to the left on the curve, corresponding to a probit score of approximately −1.96. The function is widely used in probit models, a type of regression analysis for binary outcomes (e.g., success/failure or pass/fail). It was first developed in toxicology to analyze dose-response relationships, such as how the percentage of pests killed by a pesticide changes with its concentration. The probit function is also used to create Q–Q plots, a graphical tool for assessing whether a dataset is normally distributed. Mathematically, the probit function is the quantile function (the inverse of the cumulative distribution function (CDF)) associated with the standard normal distribution. If the CDF is denoted by ⁠ Φ ( z ) {\displaystyle \Phi (z)} ⁠, then the probit function is defined as:

probit ⁡ ( p ) = Φ − 1 ( p ) for p ∈ ( 0 , 1 ) . {\displaystyle \operatorname {probit} (p)=\Phi ^{-1}(p)\quad {\text{for}}\quad p\in (0,1).}

This means that for any probability ⁠ p {\displaystyle p} ⁠, the probit function finds the value z {\displaystyle z} such that the area under the standard normal curve to the left of z {\displaystyle z} is equal to ⁠ p {\displaystyle p} ⁠.

Conceptual development The idea of the probit function was published by Chester Ittner Bliss in a 1934 article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" (or "probit") which was linearly related to the modern definition (he defined it arbitrarily as equal to 0 for 0.0001 and 1 for 0.9999): He included a table to aid other researchers to convert their kill percentages to his probit, which they could then plot against the logarithm of the dose and thereby, it was hoped, obtain a more or less straight line. Such a so-called probit model is still important in toxicology, as well as other fields. The approach is justified in particular if response variation can be rationalized as a lognormal distribution of tolerances among subjects on test, where the tolerance of a particular subject is the dose just sufficient for the response of interest. The method introduced by Bliss was carried forward in Probit Analysis, an important text on toxicological applications by D. J. Finney. Values tabled by Finney can be derived from probits as defined here by adding a value of 5. This distinction is summarized by Collett Probit methodology, including numerical optimization for fitting of probit functions, was introduced before widespread availability of electronic computing. When using tables, it was convenient to have probits uniformly positive. Common areas of application do not require positive probits.

Symmetries Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics. If we consider the familiar fact that the standard normal distribution places 95% of probability between −1.96 and 1.96 and is symmetric around zero, it follows that

Φ ( − 1.96 ) = 0.025 = 1 − Φ ( 1.96 ) . {\displaystyle \Phi (-1.96)=0.025=1-\Phi (1.96).}

The probit function gives the 'inverse' computation, generating a value of a standard normal random variable, associated with specified cumulative probability. Continuing the example,

probit ⁡ ( 0.025 ) = − 1.96 = − probit ⁡ ( 0.975 ) . {\displaystyle \operatorname {probit} (0.025)=-1.96=-\operatorname {probit} (0.975).}

In general,

Φ ( probit ⁡ ( p ) ) = p , {\displaystyle \Phi (\operatorname {probit} (p))=p,} and

probit ⁡ ( Φ ( z ) ) = z . {\displaystyle \operatorname {probit} (\Phi (z))=z.}

Diagnosing deviation of a distribution from normality

In addition to providing a basis for important types of regression, the probit function is useful in statistical analysis for diagnosing deviation from normality, according to the method of Q–Q plotting. If a set of data is actually a sample of a normal distribution, a plot of the values against their probit scores will be approximately linear. Specific deviations from normality such as asymmetry, heavy tails, or bimodality can be diagnosed based on detection of specific deviations from linearity. While the Q–Q plot can be used for comparison to any distribution family (not only the normal), the normal Q–Q plot is a relatively standard exploratory data analysis procedure because the assumption of normality is often a starting point for analysis.

Computation The normal distribution CDF and its inverse are not available in closed form, and computation requires careful use of numerical procedures. However, the functions are widely available in software for statistics and probability modeling, and in spreadsheets. In computing environments where numerical implementations of the inverse error function are available, the probit function may be obtained as

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Probit

Start with the simplest possible case. Write down what Probit claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Probit before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Probit ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Probit

In research: Probit appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Probit in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Probit is common in secondary-school and first-year university syllabi. It links to neighbouring topics Biostatistics, Normal distribution, Single-equation methods (econometrics), so understanding it makes those chapters shorter.
In everyday life: Look for Probit outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Probit in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Probit means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Probit out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Probit in simple terms?

In statistics, the probit function converts a probability (a number between 0 and 1) into a score. This score indicates how many standard deviations a value from a standard normal distribution (or "bell curve") is from the mean.

Why does Probit matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Probit?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Probit.