Item Response Theory (IRT 1PL Rasch, 2PL & 3PL) Probability & Fisher Information Calculator

Evaluate IRT 1PL/2PL/3PL Item Characteristic Curve P(θ) and Fisher Item Information I(θ).

Probability of Correct Response P(θ)
83.902 %
Fisher Item Information I(θ) 0.646 info
Probability at Inflection Point (θ = b) 60 %
1PL Rasch Probability (a=1, c=0) 64.566 %
Inputs
⚡ Scenarios:
θ logits
Examinee latent ability trait θ on the standard normal logit scale (−3.0 to +3.0).
b logits
Item difficulty location parameter b (−3.0 easy to +3.0 hard).
a slope
Item discrimination slope parameter a (1.0 for 1PL Rasch; 0.5–2.5 for 2PL/3PL).
c
Lower guessing asymptote c (0.0 for free-response; 0.20 for 5-choice; 0.25 for 4-choice).
Results & Graph
Probability of Correct Response P(θ)
83.902 %
Fisher Item Information I(θ)
0.646 info
Probability at Inflection Point (θ = b)
60 %
1PL Rasch Probability (a=1, c=0)
64.566 %

Probability of Correct Response P(θ)

Graph 1 • IRT 3PL / Rasch Item Curve • Domain Architecture & Live Telemetry Maps Examinee Ability Theta, Item Difficulty B, Item Discrimination A through IRT 3PL / Rasch Item Curve Formula.
Item Response Theory (IRT 1PL Rasch, 2PL & 3PL) Probability & Fisher Information Calculator • Custom Domain Schematic (Psychometrics & Assessment) examineeitemDiffIRT 3PL / Rasch Item C Examinee Ability Theta: 0.800 θ logits • Item Difficulty B: 0.200 b logits PROBABILITY OF CORRECT RES 83.902 % • Fisher Item Informatio: 0.646 info • Probability at Inflect: 60 % Eq: P(θ) = c + (1 − c) / [ 1 + exp(−
Graph 2 • Probability of Correct Response P(θ) Across Examinee Ability Theta Sweep (40% → 160%) Evaluates P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]; I(θ) = 2.8968 across 7 operating points (current input highlighted).
0 % 51.60 % 103.2 % 65.48 0.320 72.44 0.480 78.66 0.640 83.90 0.800 88.12 0.960 91.38 1.12 93.83 1.28
Graph 3 • Joint Sensitivity: Examinee Ability Theta × Item Difficulty B Compares Probability of Correct Response P(θ) when Item Difficulty B is shifted +25% (solid) vs. −25% (dashed).
0 % 49.96 % 99.92 % --- Item Difficulty B @ 75% 73.67 -30% 78.30 -15% 82.37 +0% 85.87 +15% 88.80 +30%
Summary: For examinee ability θ = 0.8 on an item with difficulty b = 0.2, discrimination a = 1.35, and guessing c = 0.2, correct response probability P(θ) is 83.9% and Fisher item information I(θ) is 0.646.
Summary: For examinee ability θ = 0.8 on an item with difficulty b = 0.2, discrimination a = 1.35, and guessing c = 0.2, correct response probability P(θ) is 83.9% and Fisher item information I(θ) is 0.646.
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Formula & Mathematical Methodology: IRT 3PL / Rasch Item Curve Formula

P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]
I(θ) = 2.8968 × a² × [ (P − c)² / (1 − c)² ] × [ (1 − P) / P ]
P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]; I(θ) = 2.8968 × a² × [ (P − c)² / (1 − c)² ] × [ (1 − P) / P ]
  • examineeAbilityTheta: Examinee latent ability trait θ on the standard normal logit scale (−3.0 to +3.0).
  • itemDifficultyB: Item difficulty location parameter b (−3.0 easy to +3.0 hard).
  • itemDiscriminationA: Item discrimination slope parameter a (1.0 for 1PL Rasch; 0.5–2.5 for 2PL/3PL).
  • guessingAsymptoteC: Lower guessing asymptote c (0.0 for free-response; 0.20 for 5-choice; 0.25 for 4-choice).

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: IRT 3PL / Rasch Item Curve Worked Example

Evaluate IRT 1PL/2PL/3PL Item Characteristic Curve P(θ) and Fisher Item Information I(θ).

  1. Enter the input measurement, system configuration, or astronomical parameter.
  2. Apply the standard conversion or analytical equation: P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]; I(θ) = 2.8968 × a² × [ (P − c)² / (1 − c)² ] × [ (1 − P) / P ].
  3. Inspect the primary metric, secondary equivalents, and 3 real-time SVG graphs.
Result: Computes IRT 3PL / Rasch Item Curve via P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]; I(θ) = 2.8968 × a² × [ (P − c)² / (1 − c)² ] × [ (1 − P) / P ].

Frequently Asked Questions

How does the Item Response Theory (IRT 1PL Rasch, 2PL & 3PL) Probability & Fisher Information Calculator perform the calculation?

It evaluates P(θ) = c + (1 − c) / [ 1 + exp(−1.702 × a × (θ − b)) ]; I(θ) = 2.8968 × a² × [ (P − c)² / (1 − c)² ] × [ (1 − P) / P ] directly in your browser and renders three custom SVG charts in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this tool work offline?

Yes. All calculations run 100% locally in your browser with full offline PWA support.

Education · Psychometrics & Assessment

Item Response Theory (IRT 1PL Rasch, 2PL & 3PL) Probability & Fisher Information Calculator: inputs, method, and how to read the result

Evaluate IRT 1PL/2PL/3PL Item Characteristic Curve P(θ) and Fisher Item Information I(θ).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Examinee Ability Theta: Examinee latent ability trait θ on the standard normal logit scale (−3.0 to +3.0).
  • Item Difficulty B: Item difficulty location parameter b (−3.0 easy to +3.0 hard).
  • Item Discrimination A: Item discrimination slope parameter a (1.0 for 1PL Rasch; 0.5–2.5 for 2PL/3PL).
  • Guessing Asymptote C: Lower guessing asymptote c (0.0 for free-response; 0.20 for 5-choice; 0.25 for 4-choice).

What the calculation includes

With the default values shown, Probability of Correct Response P(θ) is 83.902 %. The calculation follows IRT 3PL / Rasch Item Curve Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Examinee Ability Theta from 0.8 θ logits to 0.88 θ logits changes Probability of Correct Response P(θ) from 83.902 % to 86.136 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses official NIST Special Publication 811, IEEE/RFC systems engineering, APA psychometric, and USNO astronomical formulas.
  • Empirical metallurgy and calendar conversions represent standard reference-condition approximations.

Scope: Global NIST, IEEE, APA & Astronomical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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