Continuity Correction Calculator

Continuity Correction Calculator

Convert discrete probability questions into corrected normal approximation bounds and calculate the adjusted probability.

Adjust a discrete boundary before using the normal curve

A continuity correction shifts each whole-number cutoff by 0.5 before applying a normal approximation.

A continuity correction calculator improves normal approximations to discrete probability distributions. Enter the discrete value, population parameters, and distribution type. The calculator applies a ±0.5 adjustment before calculating probabilities, which increases accuracy when approximating binomial or Poisson distributions with a normal distribution.

Enter a mean, standard deviation, event type, and cutoff value. The calculator returns the corrected bound, z-score, normal probability, and a comparison with the uncorrected approximation.

This is commonly used for binomial, Poisson, and other count-based distributions when approximating a discrete probability with a continuous normal distribution.

More event types

Supports exact counts, tails, strict inequalities, and inclusive ranges.

Flexible inputs

Use manual mean and standard deviation, or fill them from binomial or Poisson parameters.

Exact comparison

Compare the corrected normal estimate with exact binomial or Poisson probability when available.

Show your work

See the corrected event, z-score, percent, and step-by-step calculation.

Normal approximation inputs

Use the mean and standard deviation of the discrete distribution.

Boundary shift: 0.5

Mean of X, written as mu.

Standard deviation of X, written as sigma.

Discrete event

Choose the probability statement you want to approximate.

Discrete X to continuous Y

The correction depends on the inequality.

Whole-number cutoff, k.

Distribution shortcuts

Fill mean and standard deviation from binomial or Poisson parameters, then compare with the exact discrete probability.

Binomial distribution

Use when there are fixed trials with the same success probability.

Binomial n.

Binomial p, from 0 to 1.

Sets mu = n p and sigma = sqrt(n p (1 - p)).

Poisson distribution

Use when counts occur over a fixed time, area, distance, or exposure window.

Lambda, the expected count.

Sets mu = lambda and sigma = sqrt(lambda).

Left tail

P(X <= k) -> P(Y <= k + 0.5)

Right tail

P(X >= k) -> P(Y >= k - 0.5)

Exact count

P(X = k) -> P(k - 0.5 <= Y <= k + 0.5)

What Is Continuity Correction?

Continuity correction is a 0.5 adjustment used when a continuous normal distribution approximates a discrete distribution. Because the normal curve covers every decimal value, each whole-number count is represented by the interval from 0.5 below to 0.5 above the count.

When To Use It

Use continuity correction for normal approximations to counts, such as binomial or Poisson probabilities. It is especially useful when you are calculating tail, exact, or interval probabilities from a discrete random variable.

Continuity Correction Rules

Use these conversions before calculating a normal approximation.

Discrete event Corrected normal event Why
P(X <= k) P(Y <= k + 0.5) Include all of count k.
P(X < k) P(Y <= k - 0.5) Stop before count k begins.
P(X >= k) P(Y >= k - 0.5) Include all of count k.
P(X > k) P(Y >= k + 0.5) Start after count k ends.
P(X = k) P(k - 0.5 <= Y <= k + 0.5) Use the full width of count k.
P(a <= X <= b) P(a - 0.5 <= Y <= b + 0.5) Include both endpoint counts.

How To Use This Calculator

Follow this workflow to turn a discrete probability question into a corrected normal approximation.

1. Match the event to the right boundary

Choose whether your probability is less than, greater than, exactly equal to, or within a range. The calculator then decides whether to move the boundary left or right by 0.5.

2. Enter the distribution parameters

Use the mean and standard deviation for the discrete distribution. For a binomial distribution, the shortcut can calculate them from sample size, success probability, and failure probability.

3. Read the corrected z-score

The corrected bound is standardized with z = (x - mean) / standard deviation. For a range or exact count, the calculator gives two z-scores and subtracts the two normal areas.

4. Compare corrected, uncorrected, and exact results

The comparison result shows how much the 0.5 adjustment changed the probability. If you use a binomial or Poisson shortcut, the calculator also shows the exact discrete probability and the approximation error.

Reference for binomial mean and standard deviation: NIST e-Handbook: Binomial Distribution.

Worked Examples

These examples show how the corrected boundary changes common binomial and Poisson-style normal approximations.

Binomial at most

Suppose X is binomial with n = 100 and p = 0.50. To estimate P(X <= 54), use mean = 50 and standard deviation = 5.

Correction: P(X <= 54) becomes P(Y <= 54.5).

Z-score: z = (54.5 - 50) / 5 = 0.90.

Probability: Phi(0.90) is about 0.8159.

Poisson-style at least

If a Poisson count has lambda = 20, the normal approximation uses mean = 20 and standard deviation = sqrt(20), or about 4.4721.

Correction: P(X >= 25) becomes P(Y >= 24.5).

Z-score: z = (24.5 - 20) / 4.4721 = 1.0062.

Probability: 1 - Phi(1.0062) is about 0.1572.

Inclusive range

For a count with mean = 50 and standard deviation = 5, a range probability such as P(45 <= X <= 54) needs both endpoints corrected.

Correction: Use P(44.5 <= Y <= 54.5).

Z-scores: z1 = -1.10 and z2 = 0.90.

Probability: Phi(0.90) - Phi(-1.10) is about 0.6802.

More worked context: Penn State STAT 414: Normal Approximation to Binomial.

When The Approximation Is Reliable

Continuity correction improves the fit, but it does not make every normal approximation accurate. The central limit theorem explains why larger count distributions become more normal-shaped; these checks help you decide whether that approximation is strong enough for the final probability.

Situation Good sign Use caution when
Binomial distribution Both n p and n(1 - p) are at least 5, and preferably at least 10. The sample size is small or success probability is very close to 0 or 1.
Poisson counts The mean lambda is large enough that the count distribution is not strongly skewed. Lambda is small, or the probability is deep in the left or right tail.
Exact or narrow interval The standard deviation is large enough that one count is a small slice of the curve. The event covers only one or two counts and an exact probability formula is available.
Boundary near the mean The cutoff is near the center of the distribution, where the normal curve fits best. The cutoff is far from the mean, where small tail-area errors can matter more.

Practical rule: use this calculator to understand the corrected normal method, then check the exact comparison card when you use the binomial or Poisson shortcut. Prefer an exact binomial or Poisson calculation when the sample is small, the distribution is highly skewed, or the decision depends on a very small tail probability.

Textbook background: OpenStax Statistics: Using the Central Limit Theorem.

Frequently Asked Questions

Why does continuity correction move a boundary by 0.5?

In statistics, a whole-number result from a discrete distribution, such as 54 successes, represents the interval from 53.5 to 54.5 on the continuous distribution used for the normal approximation. The 0.5 shift helps the calculator assign the right probability area to that count.

Does continuity correction always improve a normal approximation?

It usually improves a normal approximation for discrete probability questions, especially around cutoff values. The gain is most noticeable when the event uses an inequality such as less than, greater than, at most, or at least, because the corrected boundary better matches the area under the continuous curve.

Can I use this calculator for a binomial distribution?

Yes. For a binomial distribution, use the sample size and success probability to find the mean n p and standard deviation sqrt(n p (1 - p)); the failure probability is 1 - p. Then apply the corrected bound and z-score to estimate the probability for the event you need.

How should I enter a range probability?

For a range such as P(a <= X <= b), enter the lower and upper whole-number values. The calculator converts that discrete interval to P(a - 0.5 <= Y <= b + 0.5), then uses the mean and standard deviation to calculate the matching normal probability.

Why can the exact probability differ from the corrected normal result?

The corrected normal result is still an approximation. It uses a smooth continuous curve to estimate a discrete distribution, so the answer can differ from the exact binomial or Poisson probability when the sample is small, the boundary is far from the mean, or the event covers a very narrow interval.

Disclaimer

This calculator is for educational and informational use. It provides normal approximation estimates and, when available, exact comparison values, but results should be checked against your course instructions, statistical software, or a qualified professional before being used for research, grading, business, medical, legal, or financial decisions.

Last updated: June 2, 2026