Exponential Distribution Calculator
Last reviewed 8 September 2026
Notes the diagnostic fingerprint: an exponential has standard deviation exactly equal to its mean, so a coefficient of variation far from 1 rules the model out in one line.
Geometric Distribution Calculator
Last reviewed 8 September 2026
States the planning consequence the mean hides: budgeting 1/p attempts succeeds only about 63% of the time, and roughly 3/p is needed for 95% confidence.
Chi-Square Calculator
Last reviewed 8 September 2026
Shows each cell's contribution to the statistic, because chi-square is an omnibus test that finds a departure without saying which cell caused it — and prints Cramér's V, since the statistic itself scales with sample size and cannot be compared across studies.
Hypergeometric Distribution Calculator
Last reviewed 8 September 2026
Draws only the achievable support, so the lower bound people miss is visible: drawing 50 cards from a deck guarantees at least 2 aces, and fewer has probability exactly zero rather than merely small.
Poisson Distribution Calculator
Last reviewed 8 September 2026
Includes the rule of three: observing zero events in n trials bounds the rate at about 3/n, which is the honest answer to 'we saw none, so it is safe'.
Binomial Distribution Calculator
Last reviewed 8 September 2026
Names independence as the condition that actually fails, almost always because the sampling was done without replacement, which is a different distribution rather than a small error.
Normal Distribution Calculator
Last reviewed 8 September 2026
States the central limit theorem correctly: it is about the distribution of the MEAN, not about the data. Being cited as though it licensed assuming the data is normal is why the normal is over-assumed.
z-test Calculator
Last reviewed 8 September 2026
Uses p₀ in the one-proportion standard error and the pooled rate in the two-proportion one — both because the null determines the variance — and explains why a confidence interval for the same difference correctly does neither.
t-test Calculator
Last reviewed 8 September 2026
The suite checks the Welch–Satterthwaite degrees of freedom against a hand computation and against the boundary case where Welch must collapse exactly onto Student — equal variances with equal n, where the two are algebraically identical.
Critical Value Calculator
Last reviewed 8 September 2026
Explains why chi-square and F have no ± form — they are non-negative and one-tailed by construction — and shows the two-sided pair separately, which is for a variance interval rather than a test.
Sample Size Calculator
Last reviewed 8 September 2026
Every figure quoted in prose is asserted in the suite, including the finite-population corrections and the 73% uplift from 95% to 99% confidence.
Standard Error Calculator
Last reviewed 8 September 2026
Notes that s/√n is the standard error OF A MEAN specifically, and gives the different forms for a proportion, a difference, a regression slope and a median.
Margin of Error Calculator
Last reviewed 8 September 2026
Corrects the folklore figure: ±5 points needs 385 responses, not the 384 that circulates, because the exact multiplier is 1.959964 rather than 1.96. The first draft repeated the folklore.
Confidence Interval Calculator
Last reviewed 8 September 2026
The suite asserts the duality the page rests on: a hypothesised value falls inside the 95% interval exactly when a two-tailed test fails to reject it at 0.05, across a grid of sample sizes and null values.
p-value Calculator
Last reviewed 8 September 2026
Tail areas come from survival functions rather than 1 − CDF. The first version reported p = 0 for the n = 2500 row, because Φ rounds to 1.0 past about z = 8 and the subtraction has nothing left.
Map Scale Converter
Last reviewed 8 September 2026
Verified by 81 automated cases, asserting that a smaller denominator classifies as a larger scale across the whole ladder, that area is the square of the linear ratio at several pairs, that 1:63,360 is exactly one inch to the mile from 63,360 written out, and that the ground-to-map ratio equals the denominator whatever unit was measured in.
kWp to kWh Converter
Last reviewed 8 September 2026
Verified by 112 automated cases, asserting that a missing specific yield returns null and that no export offers a location-free annual figure, that the sunniest listed location beats the dullest by more than double, that the temperature factor is exactly one at 25 degrees and falls monotonically above it, and that the performance ratio equals the product of its components.
TPI to mm Pitch Converter
Last reviewed 8 September 2026
Verified by 161 automated cases, asserting that pitch times threads per inch is exactly 25.4 at every size in every table, that nearest-metric returns a literal false for compatibility for all thirteen unified sizes, that a 55-degree and a 60-degree thread are refused even at identical diameter and pitch, and that every BSP thread measures more than its nominal name.
Flight Level to Feet Converter
Last reviewed 8 September 2026
Verified by 75 automated cases, asserting that a flight level and a true altitude coincide exactly at the standard datum, that low pressure puts the aircraft lower and high pressure higher across a sweep of settings, that feet per hectopascal rises monotonically and more than doubles between sea level and FL200, and that the cold error has the unsafe sign.
dBm to Watts Converter
Last reviewed 8 September 2026
Verified by 86 automated cases, asserting that each referenceless scale returns null and carries a stated reason, that 0 dBm into 600 ohms is exactly 0 dBu, that the professional-consumer gap is 11.78 rather than 14, and that the power ratio is the amplitude ratio squared at every level.
Baker’s Percentage Calculator
Last reviewed 8 September 2026
Verified by 82 automated cases, asserting that the baker's percentages sum well above a hundred while the shares of the dough sum to exactly a hundred, that percentages and weights round-trip at every flour weight, that a levain at a given hydration is the right fraction of flour and moves the dough hydration in the correct direction, and that a recipe without flour is refused rather than dividing by zero.
CC to HP Converter
Last reviewed 8 September 2026
Verified by 66 automated cases, asserting that the displacement-to-power function returns null and that no export offers that answer under another name, that the full class spread exceeds a factor of ten, that the three horsepowers match their definitions of 550 ft-lbf/s, 75 kgf-m/s and exactly 746 W, and that power and torque are numerically equal exactly at 33000/2pi rpm.
Gauss to Tesla Converter
Last reviewed 8 September 2026
Verified by 91 automated cases, asserting that every cross-quantity pair returns null AND names both quantities in its refusal, that one oersted gives exactly one gauss in vacuum and two thousand in ferrite, that the oersted and MGOe factors match 1000/4pi and 100/4pi rather than rounded decimals, and that no export produces a field from a magnet grade.
NPK Fertilizer Converter
Last reviewed 8 September 2026
Verified by 106 automated cases, asserting the factors against atomic masses rather than decimals, that a 10-10-10 is emphatically not balanced as elements and that its middle number more than halves, that the oxide and elemental ratio triples genuinely differ, and that a grade declaring more than a full bag of nutrient is refused.
Time Unit Converter
Last reviewed 8 September 2026
Verified by 70 automated cases, asserting that months, years and quarters are absent from the ladder structurally rather than as a special case, that month spans come from the real calendar and always bracket their own average, that the four years are distinct and exactly six hours apart at the extremes, and that the sidereal gap is between 3.9 and 4.0 minutes.
Mesh to Micron Converter
Last reviewed 8 September 2026
Verified by 141 automated cases, asserting that the same mesh count with two wire diameters gives openings more than 20 micrometres apart, that US 140 and Tyler 150 resolve to one aperture while their numbers differ, that aperture ascends while mesh number descends on every adjacent pair in the table, and that a wire as thick as the pitch is refused.
Gas m³ to kWh Converter
Last reviewed 8 September 2026
Verified by 82 automated cases, asserting that the bridge refuses a zero, negative, missing or non-finite calorific value while still returning zero energy for a zero reading, that volume-to-energy and energy-to-volume invert each other through the correction factor, that the 11.2 shortcut is exact at exactly one calorific value and signed correctly either side of it, and that every network's typical value lies inside its own published range.
Pace Converter
Last reviewed 8 September 2026
Verified by 55 automated cases, asserting that pace times speed is 3600 at every speed, that each additional km/h saves strictly less than the one before, that the naive mean is correct for equal distances and wrong for equal times, and that a segment supplying both a distance and a time is refused.
px to pt Converter
Last reviewed 8 September 2026
Verified by 55 automated cases, asserting the specification chain at ZERO tolerance rather than a floating-point budget, that every relative unit refuses without a base, and that the device pixel area ratio is the square of the linear one at every ratio.
Cord of Wood Calculator
Last reviewed 8 September 2026
Verified by 57 automated cases, asserting that the face cord stores no fixed volume and refuses conversion in both directions, that its fraction is exactly a quarter, a third and a half at the common lengths, and that the moisture calculation carries both the dilution and the evaporation term.
Bushel to kg Converter
Last reviewed 8 September 2026
Verified by 48 automated cases, asserting that the bushel volume is crop-independent while the weights span 2.14 times, cross-checking every derived yield factor against the published agronomic one, and requiring that drying conserves dry matter at every moisture pair.
EC to TDS Converter
Last reviewed 8 September 2026
Verified by 47 automated cases, asserting the spread across the three scales rather than any one factor, that decisiemens per metre equals millisiemens per centimetre at zero tolerance, and that compensation moves cold readings up and warm ones down at every degree.
Moles to Grams Converter
Last reviewed 8 September 2026
Verified by 52 automated cases, asserting that grams and moles refuse without a molar mass, that the molarity-molality gap widens monotonically across five measured densities rather than being checked at one point, and that v/v returns nothing.
Gold Purity Converter
Last reviewed 8 September 2026
Verified by 62 automated cases, including absolute mass assertions rather than ratios alone: during development the grain was briefly defined in kilograms, every ratio in the engine stayed correct, and only an absolute case caught it.
Tons to BTU Converter
Last reviewed 8 September 2026
Verified by 51 automated cases, asserting the ton against its definition rather than a copied decimal, that every seasonal metric refuses to become a COP, and that at every realistic efficiency the power drawn is under half the heat removed.
Z-Score Calculator
Last reviewed 7 September 2026
Separates the two assumptions usually bundled together: normality, which the percentile needs, and a known sigma, which is what Student's t exists to relax.
Mean, Median and Mode Calculator
Last reviewed 7 September 2026
Treats the mean-median gap as a result: signed, scaled by the standard deviation into Pearson's second skewness coefficient, and judged against a threshold relative to the spread so a gap of 2 is not called skew on salaries and ignored on exam marks.
Variance Calculator
Last reviewed 7 September 2026
States two things courses routinely omit: Bessel's correction leaves the standard deviation biased even though it fixes the variance, and the Sigma-x-squared shortcut formula loses precision catastrophically on data far from zero.
Percentile Calculator
Last reviewed 7 September 2026
Exposes the three tie-handling rules for percentile rank instead of choosing one silently, since on a dataset with repeated values they can differ by thirty points.
Five Number Summary Calculator
Last reviewed 7 September 2026
Draws whiskers to the most extreme observation still inside the fence, as Tukey defined them, and says plainly that this differs from the min-to-max whiskers some software draws — the commonest box-plot misreading.
IQR Calculator
Last reviewed 7 September 2026
States where the fence rule misleads: it is symmetric and most real data is not, so on skewed distributions the upper fence flags ordinary observations, and the adjusted boxplot of Hubert and Vandervieren is cited as the fix rather than left out.
Quartile Calculator
Last reviewed 7 September 2026
Corrected the worked example before publishing: the first draft claimed an even count makes every convention agree. It does not — it only makes the two hinge rules agree with each other, and the page now shows the exam-marks preset giving 59.5 by hand and 60.25 in the tool, with the reason.
Multiplying Binomials Calculator
Last reviewed 7 September 2026
Identical brackets are recognised as a power and the coefficients matched against the corresponding row of Pascal's triangle.
Inequality to Interval Notation Calculator
Last reviewed 7 September 2026
When the variable sits only on the right the engine turns the statement round rather than moving the variable leftwards, so it never reports a reversal the reader would not have made.
Graphing Inequalities on a Number Line
Last reviewed 7 September 2026
The AND and OR rows are drawn from the same interval sets that produce the printed notation, so the picture and the brackets cannot drift apart.
Fraction Calculator
Last reviewed 7 September 2026
Multiplication cross-cancels before multiplying, so 4/9 × 3/8 never builds 12/72; the decimal beside every answer is the exact expansion with its repeating block in brackets.
Simplify Fractions Calculator
Last reviewed 7 September 2026
A negative denominator is moved onto the numerator first and the page says so, because cross multiplication silently reverses without that step.
Decimal to Fraction Calculator
Last reviewed 7 September 2026
The answer is expanded back to a decimal and compared with the input, so a rounded decimal is reported as rounded instead of being silently guessed at.
Fraction to Decimal Calculator
Last reviewed 7 September 2026
Rounded forms are shown separately from the exact decimal and each is labelled exact or approximate at that width.
Comparing Fractions Calculator
Last reviewed 7 September 2026
Cross multiplication is stated with the condition the schoolbook rule omits: it only holds while both denominators are positive, so a negative denominator is normalised first.
Mixed Number Calculator
Last reviewed 7 September 2026
Multiplication converts first and shows what part-by-part would have given: 2 1/2 × 3 1/2 is 8 3/4, not the 6 1/4 the cross terms would have lost.
GCF Calculator
Last reviewed 7 September 2026
For a pair the page also gives the least common multiple, since GCF × LCM = the product means finding one gives the other for free.
LCM Calculator
Last reviewed 7 September 2026
The page reports how many multiples counting would have taken on the reader's own numbers, which is the honest argument for using the table instead.
Prime Factorization Calculator
Last reviewed 7 September 2026
The divisor count is derived from the exponents rather than by enumerating, and the suite checks every claimed prime factor is genuinely prime, since a factorisation into composites would still pass a product check.
Factor Calculator
Last reviewed 7 September 2026
Each number is classified perfect, abundant or deficient from the sum of its proper factors, computed from the factorisation rather than by adding the list.
Prime Number Calculator
Last reviewed 7 September 2026
Carmichael numbers, which defeat the naive Fermat test, are correctly rejected — 561 is a preset for exactly that reason.
Divisibility Test Calculator
Last reviewed 7 September 2026
Nine of the thirteen rules are shown to be one idea — what remainder ten leaves modulo the divisor — rather than thirteen things to memorise.
Square Root Calculator
Last reviewed 7 September 2026
A negative radicand is answered in terms of i rather than refused, since 'no real value' and 'no value' are different statements.
Cube Root Calculator
Last reviewed 7 September 2026
Simplification needs THREE of a prime to release one, where a square root needs two; the page shows that division per prime.
Simplify Radicals Calculator
Last reviewed 7 September 2026
Rationalising the denominator is included because it is always the next question, and every rationalised form is checked numerically against the original in the suite.
Exponent Calculator
Last reviewed 7 September 2026
Whole-number powers run on integers of unlimited size, so 2 to the 200 shows all sixty-one digits where a double would show sixteen and an exponent.
Log Calculator
Last reviewed 7 September 2026
Log of zero and of a negative are refused with the reason, which is a fact about exponentials rather than a limitation of the tool.
Factorial Calculator
Last reviewed 7 September 2026
0! = 1 is answered twice — by the recurrence and by the counting argument — rather than called a convention, because 'convention' is not an answer to the commonest question about factorials.
Weighted Average Calculator
Last reviewed 7 September 2026
The unweighted mean is computed alongside, so the difference the weighting made is visible instead of implied.
Geometric Mean Calculator
Last reviewed 7 September 2026
A zero or a negative value is refused with the reason rather than silently dropped, since the product would be meaningless either way.
Harmonic Mean Calculator
Last reviewed 7 September 2026
A zero rate has no reciprocal and the page refuses rather than returning something small, matching the reading that a leg travelled at zero speed never finishes.
Root Mean Square Calculator
Last reviewed 7 September 2026
The RMS of the deviations from the mean is computed alongside, so the identity with standard deviation is visible on the page rather than claimed in prose.
Rounding Calculator
Last reviewed 7 September 2026
Rounding to the nearest arbitrary multiple is included, which covers quarter-hour timesheets and five-cent prices without a second tool.
Significant Figures Calculator
Last reviewed 7 September 2026
Both propagation rules are given and distinguished, since using the significant-figures rule on a sum is the commonest error a lab report contains.
Modulo Calculator
Last reviewed 7 September 2026
A wrap-around strip shows where the answer lands on a circle of n, which makes the negative case obvious instead of confusing.
Remainder Calculator
Last reviewed 7 September 2026
A negative dividend gets both defensible answers rather than one, with the identity each satisfies.
Modular Exponentiation Calculator
Last reviewed 7 September 2026
Fermat's little theorem and Euler's theorem are surfaced when they apply, since reducing the exponent modulo n instead of modulo φ(n) is the commonest mistake in this area.
Modular Inverse Calculator
Last reviewed 7 September 2026
Results are checked against an exhaustive search over all pairs below sixty, refusals included, so the absence of an inverse is verified rather than assumed.
Chinese Remainder Theorem Calculator
Last reviewed 7 September 2026
A contradictory system is reported with the two residues that disagree, rather than as a bare failure.
Coprime Calculator
Last reviewed 7 September 2026
Euler's totient of the first value is given, since that count is exactly how many residues have a modular inverse.
Arithmetic Sequence Calculator
Last reviewed 7 September 2026
The recursive form is shown beside the closed one, because different courses teach different ones and the page should not assume which.
Geometric Sequence Calculator
Last reviewed 7 September 2026
The awkward ratios are handled explicitly: 1 uses the n × a₁ form rather than dividing by zero, and −1 is named as Grandi's series.
Fibonacci Calculator
Last reviewed 7 September 2026
The nature claims are separated into the ones that hold up — seed-head spirals, from the golden angle — and the ones that do not, since a page repeating the nautilus story would be less trustworthy about the parts that are true.
Pascal’s Triangle Calculator
Last reviewed 7 September 2026
The history is given accurately: Al-Karaji had it around 1000 CE and Yang Hui published it in 1261, so Pascal was several centuries late to a triangle that carries his name in the West.
Collatz Conjecture Calculator
Last reviewed 7 September 2026
Tao's 2019 result is cited as the strongest partial progress rather than the problem being left as folklore.
Nth Term Calculator
Last reviewed 7 September 2026
A list with no standard pattern returns 'no rule found' with the tests that were tried, since a degree-4 polynomial through any five points would be technically correct and practically worthless.
Reference Angle Calculator
Last reviewed 7 September 2026
Checked against an independently written loop-and-lookup routine on all 2,161 whole degrees from −1080° to 1080°.
Complementary and Supplementary Angles Calculator
Last reviewed 7 September 2026
The explement is included because every angle has one and it is the quantity behind the two readings of a clock-hand angle.
Clock Angle Calculator
Last reviewed 7 September 2026
Checked against the closed form |30H − 5.5M| at every one of the 720 distinct times on a twelve-hour face, which is exhaustive rather than sampled.
Central Angle Calculator
Last reviewed 7 September 2026
The segment formula is left to handle angles past 180° on its own — sin θ turns negative there, so the triangle is added rather than subtracted, and the segment is correctly LARGER than the sector.
Angle Between Two Vectors Calculator
Last reviewed 7 September 2026
Checked against the law of cosines applied to the triangle the two vectors and their difference form, on three thousand random pairs.
Ratio Calculator
Last reviewed 7 September 2026
Restores a slug that had been answering 410 Gone since an old content purge, with a page that earns it.
Proportion Calculator
Last reviewed 7 September 2026
Names the cases where it is the wrong model — a fixed cost, an area, an inverse relationship — because a proportion assumes strict linearity through the origin.
Average Percentage Calculator
Last reviewed 7 September 2026
Covers Simpson's paradox, where weighting does not merely shift the answer but reverses it.
Percentage Point Calculator
Last reviewed 7 September 2026
A change from zero returns no relative figure at all, because dividing by zero is undefined rather than infinite.
Percent to Goal Calculator
Last reviewed 7 September 2026
States that a steady rate is an assumption: seasonal targets are not linear, and the page reports pace rather than predicting an outcome.
Exponential Growth Calculator
Last reviewed 7 September 2026
Says plainly that nothing checks whether the quantity really grows exponentially — populations meet limits and adoption saturates, so a smooth projection is an assumption rather than a finding.
Triangle Calculator
Last reviewed 7 September 2026
Areas use Heron's numerically stable rearrangement, because every reported height is derived from the area and inherits its error.
Pythagorean Theorem Calculator
Last reviewed 7 September 2026
Restores a slug that had been answering 410 Gone since an old content purge.
Heron’s Formula Calculator
Last reviewed 7 September 2026
Degenerate lengths return exactly zero and are labelled as such, rather than returning NaN, because measured survey data lands there.
Special Right Triangles Calculator
Last reviewed 7 September 2026
Names the error the ratio invites: in a 30-60-90 the HYPOTENUSE is twice the short leg, while the long leg is √3 times it — about 13% out, which no sanity check catches.
Circle Calculator
Last reviewed 7 September 2026
Reports the semicircle perimeter as the arc PLUS the diameter rather than half the circumference, which under-orders edging by the whole diameter.
Trigonometric Functions Calculator
Last reviewed 7 September 2026
Names the trap in the reciprocal names — cosecant is 1/sin and secant is 1/cos, and the 'co' does not point where it looks like it does.
Inverse Trigonometric Functions Calculator
Last reviewed 7 September 2026
A value outside −1 to 1 is refused with the reason rather than returning NaN — in a triangle problem it usually means the measurements describe no triangle.
Unit Circle Calculator
Last reviewed 7 September 2026
Reports x² + y² so the Pythagorean identity is visible as a number on the page rather than quoted in prose.