Continuity Correction Calculator

What Is a Continuity Correction?

A continuity correction is an adjustment you make when approximating a discrete probability distribution with a continuous one. The most common situation is approximating a binomial or Poisson distribution using the normal distribution.

The reason an adjustment is needed comes down to how probability is represented in each type of distribution. A discrete distribution, like the binomial, assigns probability to individual integer values: P(X = 10), P(X = 11), and so on. A continuous distribution, like the normal, assigns probability to intervals through area under a curve. A single point on a continuous distribution has zero area, so translating P(X = 10) directly to a normal approximation without any correction loses precision.

The fix: when approximating P(X = 10) with a normal distribution, treat the discrete value 10 as the interval from 9.5 to 10.5. This "bar" of width 1 corresponds to the rectangle at X = 10 in a histogram of the discrete distribution. Extending this idea gives you the full set of continuity correction rules for all event types.

How Continuity Correction Works

Picture a bar chart of the binomial distribution. Each bar sits over an integer value and has a width of 1, so the bar at X = 10 runs from 9.5 to 10.5. When you draw a normal curve over that bar chart, the area under the curve from 9.5 to 10.5 approximates the bar's height, which represents P(X = 10).

The same logic extends to cumulative events. For P(X ≤ 10), you want the area of all bars up to and including X = 10, meaning from −∞ up to the right edge of the bar at 10, which is 10.5. So the corrected event is P(Y < 10.5). For P(X ≥ 10), you want all bars from X = 10 rightward, meaning from the left edge of the bar at 10 (which is 9.5) to ∞. The corrected event becomes P(Y > 9.5).

Continuity Correction Rules

The table below shows how each discrete event converts to its continuity-corrected continuous form. These rules assume the discrete variable takes consecutive integer values with spacing of 1.

Discrete Event	Description	Continuity-Corrected Event
X = k	Exact value	k − 0.5 < Y < k + 0.5
X ≤ k	At most k	Y < k + 0.5
X < k	Strictly less than k	Y < k − 0.5
X ≥ k	At least k	Y > k − 0.5
X > k	Strictly greater than k	Y > k + 0.5
a ≤ X ≤ b	Closed range	a − 0.5 < Y < b + 0.5
a < X ≤ b	Right-closed range	a + 0.5 < Y < b + 0.5
a ≤ X < b	Left-closed range	a − 0.5 < Y < b − 0.5

Notice a pattern: boundary values that are included in the event move outward by 0.5 (away from the interior of the interval). Boundary values that are excluded move inward by 0.5. This correctly captures whether the bar at the boundary is part of the probability being approximated.

Continuity Correction Formula

For a binomial distribution X ~ Binomial(n, p), the normal approximation uses:

After applying the continuity correction to get a corrected boundary x_c, the z-score is:

The corrected boundary x_c depends on the event type as shown in the table above. The z-score is then used with the standard normal cumulative distribution function (CDF) to find the approximate probability.

Worked Examples

Continuity Correction and Z-Scores

The continuity correction must happen before computing the z-score. You are correcting the boundary first, then standardizing. Getting this order wrong is one of the most common mistakes.

For P(X ≤ 60) with μ = 50 and σ = 5:

These two z-scores give different normal probabilities. The second, using the continuity correction, is the better approximation of the true binomial probability.

Normal Approximation to the Binomial Distribution

The normal approximation to the binomial relies on the Central Limit Theorem. When the sample size is large enough, the sum of independent Bernoulli trials converges to a normal distribution. Continuity correction improves how well the continuous approximation matches the discrete probabilities.

For X ~ Binomial(n, p), the conditions for a reasonable normal approximation vary by source, but a common teaching threshold checks whether:

If either quantity is small, the binomial distribution is substantially skewed, and the normal approximation becomes less reliable regardless of whether you apply continuity correction. In those cases, an exact binomial calculation is preferable.

Exact Probability vs Normal Approximation

Continuity correction improves the normal approximation, but it does not make it exact. The two methods answer slightly different questions:

Consideration	Exact Binomial	Normal Approx (with CC)
Accuracy	✓ Always exact	Good when n is large
Works with small n	✓ Yes	✗ Can be inaccurate
Works when p near 0 or 1	✓ Yes	✗ Less reliable
Computational ease	Hard for large n	✓ Fast, closed-form
Requires tables or software	For large n	✓ Just a z-table
Pedagogical use	Gold standard	✓ Taught in most intro courses

In practice, use the exact binomial whenever your software makes it straightforward. The normal approximation with continuity correction is most useful in hand calculations, in exams, or when n is large enough that exact computation is cumbersome.

Continuity Correction for Poisson Distributions

The continuity correction applies to any integer-valued discrete distribution, including the Poisson. For X ~ Poisson(λ), the normal approximation uses:

The Poisson distribution becomes approximately symmetric when λ is large (commonly, λ ≥ 10 is taught as a rough threshold). The same event-correction rules apply:

For example, if X ~ Poisson(25) and you want P(X ≤ 30):

Common Continuity Correction Mistakes

• Always adding 0.5. The direction depends on the event type. For P(X ≥ k), you subtract 0.5 to get the lower boundary at k − 0.5. Adding 0.5 there would be wrong.
• Always subtracting 0.5. The same mistake in reverse. For P(X ≤ k), the corrected boundary is k + 0.5, not k − 0.5.
• Skipping the correction entirely. Using z = (60 − 50) / 5 instead of z = (60.5 − 50) / 5 for P(X ≤ 60). The error is small for large n but consistent.
• Confusing X ≤ k with X < k. For integer-valued X, P(X ≤ 20) and P(X < 21) are the same discrete event, but their corrections differ: the first uses k + 0.5 = 20.5, the second uses k − 0.5 = 20.5 as well (since k = 21). For P(X < 20), the corrected boundary is 19.5.
• Applying the correction after the z-score. The correction adjusts the boundary before computing z. You cannot apply it afterward as a final tweak.
• Using 0.5 when the spacing is not 1. If the discrete variable takes values 0, 2, 4, 6, … (spacing = 2), the half-spacing correction is 1.0, not 0.5.
• Calling the result exact. The normal approximation with continuity correction is still an approximation, not the true binomial probability.
• Using the wrong standard deviation. The binomial standard deviation is √(np(1−p)), not √(np). These differ whenever p ≠ 1.
• Applying continuity correction when it is not needed. If you are working with a continuous variable to begin with, no correction is needed. Continuity correction is only for discrete-to-continuous approximations.

When to Use Continuity Correction

Apply continuity correction whenever you approximate a discrete probability distribution with a continuous one and the discrete variable is integer-valued with unit spacing. The two main applications are:

• Binomial to normal: X ~ Binomial(n, p), approximated by N(np, np(1−p)). Use when np and n(1−p) are both reasonably large.
• Poisson to normal: X ~ Poisson(λ), approximated by N(λ, λ). Use when λ is large enough for the distribution to be roughly symmetric.

You do not need continuity correction when:

• Working directly with a continuous distribution (no approximation involved).
• Computing exact binomial or Poisson probabilities using software or tables.
• The discrete variable does not take integer values with unit spacing.

Why Does the 0.5 Correction Work?

Each integer value k in a discrete distribution can be thought of as the center of a bar of width 1, running from k − 0.5 to k + 0.5. This corresponds to a histogram where bars are centered on each integer and have width 1. The probability at X = k equals the area of that bar.

When you overlay a normal curve on that histogram, the area under the curve from k − 0.5 to k + 0.5 approximates the bar's area. The correction effectively assigns continuous regions to each discrete value by locating the boundaries halfway between adjacent integers. There is nothing arbitrary about 0.5: it is exactly half the distance between consecutive integers.

Related Statistical Topics

Continuity correction sits at the intersection of several important ideas. These pages cover the concepts you need before and after applying a continuity correction:

Frequently Asked Questions