About

When approximating a discrete probability distribution (binomial, Poisson, hypergeometric) with a continuous normal curve, a systematic error arises at each integer boundary. The continuity correction adjusts the discrete value x by ±0.5 before computing the z-score, compensating for the area lost between histogram bars and the smooth curve. Without this adjustment, tail probabilities can deviate by 5 - 15% from exact values when n is small or p is far from 0.5. The correction is most critical when np or nq falls below 10.

This calculator applies the appropriate ±0.5 shift based on your inequality direction, computes the corrected z-score, and returns both corrected and uncorrected cumulative probabilities using the Abramowitz & Stegun rational approximation to the standard normal CDF (absolute error < 1.5 × 10⁻⁷). It handles six inequality types including point probability P(X = x). Note: the normal approximation itself degrades when the underlying distribution is highly skewed. Always verify that np ≥ 5 and nq ≥ 5 for binomial cases.

Formulas

The continuity correction modifies the observed discrete value x before standardization. The corrected z-score depends on the inequality direction.

General form: z = (x ± 0.5) − μσ

For P(X ≤ x), add 0.5 to x. For P(X ≥ x), subtract 0.5. For point probability P(X = x), compute the area between x − 0.5 and x + 0.5.

Binomial preset: μ = np, σ = √np(1 − p)

CDF approximation: Φ(z) ≈ 1 − φ(z)(a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵)

where t = 11 + 0.2316419|z| and φ(z) is the standard normal PDF. Coefficients a₁ through a₅ are the Abramowitz & Stegun constants (Handbook of Mathematical Functions, formula 26.2.17). Absolute error is bounded by 1.5 × 10⁻⁷.

Where: x = observed discrete value, μ = population or distribution mean, σ = standard deviation, z = standardized score, Φ = cumulative distribution function of standard normal, n = number of trials (binomial), p = probability of success per trial.

Reference Data

Inequality Type	Without Correction	With Correction	Direction of Shift
P(X = x)	0 (continuous)	P(x − 0.5 ≤ Z ≤ x + 0.5)	Both ±0.5
P(X ≤ x)	P(Z ≤ z)	P(Z ≤ z_+0.5)	+0.5
P(X < x)	P(Z < z)	P(Z < z_−0.5)	−0.5
P(X ≥ x)	P(Z ≥ z)	P(Z ≥ z_−0.5)	−0.5
P(X > x)	P(Z > z)	P(Z > z_+0.5)	+0.5
P(a ≤ X ≤ b)	P(z_a ≤ Z ≤ z_b)	P(z_a−0.5 ≤ Z ≤ z_b+0.5)	−0.5 / +0.5

Distribution	Mean (μ)	Std Dev (σ)	Correction Rule of Thumb	Typical n Threshold
Binomial(n, p)	np	√npq	np ≥ 5 and nq ≥ 5	n ≥ 20
Poisson(λ)	λ	√λ	λ ≥ 5	λ ≥ 10
Hypergeometric	nKN	Complex formula	All expected counts ≥ 5	n ≥ 20
Negative Binomial	rp	√rqp	r ≥ 5	r ≥ 10
Chi-Square (Yates)	k	√2k	2×2 tables, any cell < 5	All cells ≥ 5
Sign Test	0.5n	0.5√n	Always recommended	n ≥ 10
Wilcoxon Signed-Rank	n(n+1)4	Complex formula	Always recommended	n ≥ 10
Mann-Whitney U	n₁n₂2	Complex formula	Always recommended	n₁, n₂ ≥ 10
Geometric(p)	1p	√qp	p < 0.1	Large n
Multinomial (approx)	np_i	√np_i(1−p_i)	All np_i ≥ 5	n ≥ 30

Frequently Asked Questions

The correction has the greatest impact when the sample size n is small (below 30) or when the probability parameter p is far from 0.5. In a Binomial(10, 0.5) case, computing P(X ≤ 3) without correction gives approximately 0.1587, while the corrected value is 0.1719 - the exact binomial answer is 0.1719. As n grows past 100, the difference shrinks below 1% and becomes negligible.

A discrete histogram bar at integer x occupies the interval [x − 0.5, x + 0.5]. When computing P(X ≤ x), you need to include the full bar at x, so you extend to x + 0.5. Conversely, P(X ≥ x) requires starting at the left edge x − 0.5. Getting this backwards inverts the correction and worsens accuracy rather than improving it.

Yes. Yates' correction for 2×2 contingency tables subtracts 0.5 from the absolute difference between observed and expected frequencies before squaring: χ² = Σ(|O − E| − 0.5)² / E. It is recommended when any expected cell count falls below 5. For larger tables (3×3 or above), Yates' correction is not standard practice - use Fisher's exact test instead.

No. The correction shifts x by ±0.5 before computing the CDF, and the normal CDF Φ(z) is bounded to [0, 1] by definition. However, for point probabilities P(X = x), the result is Φ(z_upper) − Φ(z_lower), which is always non-negative. If you get a negative result, check that your σ is positive and your inputs are within the distribution's support.

For Poisson(λ), the mean and variance are both λ, so μ = λ and σ = √λ. The continuity correction rules are identical - adjust by ±0.5 based on inequality direction. The approximation is reliable when λ ≥ 10. Below that, the Poisson distribution is noticeably right-skewed and the normal curve underestimates left-tail probabilities even with correction.

This calculator implements formula 26.2.17 from Abramowitz and Stegun's Handbook of Mathematical Functions. It uses a degree-5 rational approximation with polynomial coefficients a₁ = 0.319381530, a₂ = −0.356563782, a₃ = 1.781477937, a₄ = −1.821255978, a₅ = 1.330274429. The maximum absolute error across the entire real line is 7.5 × 10⁻⁸. For practical statistics (probabilities above 0.0001), this exceeds the precision of standard z-tables.