How this instrument works
Continuity correction is a small adjustment applied whenever a discrete distribution — one that only takes whole-number values, like the binomial or Poisson — is approximated using a continuous distribution, almost always the normal curve. A discrete variable can only equal exactly 50, 51, 52 and so on, with nothing in between; a continuous normal curve, by contrast, assigns zero probability to any single exact point and only measures probability over a range. Continuity correction bridges that mismatch by treating each whole number as if it occupies the half-unit interval centred on it.
In practice, this means shifting the target value by 0.5 in the direction that matches your inequality before looking it up on the normal curve. For P(X < x) or P(X <= x-1) — anything at or below a value — you use x - 0.5. For P(X > x) or P(X >= x+1) — anything at or above a value — you use x + 0.5. Getting the direction right matters: it's the difference between including or excluding the boundary value itself in the approximation.
The correction matters most when the number of trials is relatively small; as n grows very large, the half-unit shift becomes a vanishingly small fraction of the distribution's spread and its effect shrinks toward negligible. It's still good practice to apply it whenever you're approximating a binomial or Poisson distribution with a normal curve, and many textbooks recommend checking that both np and n(1-p) are comfortably large (often a rule of thumb of at least 5, sometimes a stricter 10) before trusting the normal approximation at all, correction or not.
- Enter the whole-number value you're correcting for into Target value (x).
- Enter the number of trials into Number of trials (n) and the probability of success on each trial into Probability of success (p) — these describe the discrete distribution being approximated.
- Read Corrected value for P(X < x) / P(X <= x-1) — use x - 0.5 when your inequality looks at values at or below x.
- Read Corrected value for P(X > x) / P(X >= x+1) — use x + 0.5 when your inequality looks at values at or above x.
- Feed the corrected value, not the raw x, into the normal distribution's z-score formula for a more accurate approximation of the discrete probability.
Worked example — approximating Binomial(n=100, p=0.5) at x = 50
Enter 50 into Target value (x), 100 into Number of trials (n), and 0.5 into Probability of success (p) — describing a variable X that counts successes across 100 independent trials, each with a 50% chance of success. Corrected value for P(X < x) / P(X <= x-1) reads 49.5, and Corrected value for P(X > x) / P(X >= x+1) reads 50.5.
To approximate P(X <= 50) using the normal curve, you'd standardize 49.5 rather than 50 itself: the underlying binomial has mean np = 100 x 0.5 = 50 and standard deviation sqrt(np(1-p)) = sqrt(25) = 5, so the corrected z-score uses (49.5 - 50) / 5 rather than (50 - 50) / 5 — a small shift that measurably improves the accuracy of the normal approximation, especially for a moderate trial count like 100.
Questions
Why shift by exactly 0.5, and not some other amount?
Because a discrete whole number like 50 is treated as representing the entire half-unit interval from 49.5 to 50.5 on a continuous scale — the midpoint between 50 and its neighbours, 49 and 51. Shifting by exactly half a unit is what makes the continuous approximation align with the discrete value it's standing in for, neither overincluding nor underincluding it.
How do I know whether to use x - 0.5 or x + 0.5?
It depends on which direction your inequality points. Use x - 0.5 for 'less than x' or 'at most x-1' type probabilities, since you want to capture everything up to and including x itself. Use x + 0.5 for 'greater than x' or 'at least x+1' type probabilities, capturing everything from x itself upward. Getting the direction backwards silently excludes the boundary value you meant to include.
Does this apply to distributions other than the binomial?
Yes — continuity correction applies to any discrete distribution being approximated by a continuous one, most commonly the binomial or Poisson distribution approximated with the normal curve. The same x ± 0.5 logic holds regardless of which discrete distribution you started from, since the correction addresses the discrete-to-continuous mismatch itself, not anything specific to the binomial.
Why do the Number of trials and Probability of success fields matter if the correction itself is just x ± 0.5?
The correction values themselves don't depend on n or p at all — they're simple arithmetic on x. But n and p determine whether applying the normal approximation makes sense in the first place: many texts recommend checking that both np and n(1-p) are reasonably large (a common rule of thumb is at least 5, some sources use a stricter 10) before trusting a normal approximation to the binomial, corrected or not.
Does continuity correction matter for very large sample sizes?
Less so. As the number of trials n grows large, the standard deviation of the distribution grows too, and a fixed 0.5-unit shift becomes a smaller and smaller fraction of that spread — so its effect on the final probability shrinks toward negligible. It remains good practice to include it regardless, since it never hurts accuracy and matters most exactly when n is small or moderate.