Continuity Correction Calculator

Normal Approximation to Binomial

This tool helps you approximate a discrete binomial probability using a continuous normal distribution by applying the continuity correction factor.

Examples

See how to use the calculator with real-world scenarios.

At Most 45 Heads in 100 Coin Flips

basic

Calculate the probability of getting 45 or fewer heads when flipping a fair coin 100 times.

n: 100, p: 0.5, x: 45

Inequality: le

More Than 25 Defective Items

advanced

A factory produces items with a 10% defect rate. In a batch of 200, what's the probability that more than 25 are defective?

n: 200, p: 0.1, x: 25

Inequality: gt

Exactly 60% Pass Rate

real-world

In a class of 50 students, where the probability of passing an exam is 0.7, what is the probability that exactly 30 students (60%) pass?

n: 50, p: 0.7, x: 30

Inequality: eq

Less Than 5 Successes (Approximation Warning)

edge-case

Demonstrating a scenario where the normal approximation might be less accurate (np < 5). Probability of getting less than 5 successes in 20 trials with p=0.1.

n: 20, p: 0.1, x: 5

Inequality: lt

Other Titles
Understanding the Continuity Correction Calculator: A Comprehensive Guide
An in-depth look at how to use the normal distribution to approximate binomial probabilities accurately.

What is Continuity Correction?

  • Bridging Discrete and Continuous Distributions
  • The Role of the 0.5 Adjustment
  • When is Normal Approximation Appropriate?
Continuity correction is a statistical technique used to improve the approximation of a discrete probability distribution (like the binomial distribution) with a continuous probability distribution (like the normal distribution). Since a discrete variable can only take specific integer values (e.g., 7 heads), while a continuous variable can take any value within a range (e.g., 7.125), a direct conversion is not perfectly accurate. The continuity correction 'fills in the gaps' between the discrete values.
The 0.5 Adjustment
The core of the correction involves adding or subtracting 0.5 from the discrete value of interest (x). This converts a specific point into a range that can be measured on a continuous scale. For example, the probability of getting exactly 10 successes, P(X = 10), is approximated by finding the area under the normal curve between 9.5 and 10.5, P(9.5 < Y < 10.5).

Step-by-Step Guide to Using the Continuity Correction Calculator

  • Inputting Your Data
  • Selecting the Correct Inequality
  • Interpreting the Results
Using this calculator is straightforward. Here's how to do it:
Number of Trials (n): Enter the total number of events or trials.
Probability of Success (p): Input the probability of a single success, as a decimal.
Number of Successes (x): Enter the specific number of successes you're interested in.
Inequality: Choose the relationship to 'x' from the dropdown menu (e.g., less than or equal to, equal to, etc.). This is the most critical step for applying the correct adjustment.
Reading the Output
The calculator provides a full breakdown: the mean and standard deviation of the distribution, the exact range used after correction, the calculated Z-score representing how many standard deviations your corrected value is from the mean, and the final approximated probability.

Real-World Applications of Continuity Correction

  • Polling and Election Predictions
  • Quality Control in Manufacturing
  • Medical and Biological Research
While modern software can compute exact binomial probabilities, the normal approximation with continuity correction is a fundamental concept with many applications:
Manufacturing
A factory manager can approximate the probability of finding more than a certain number of defective products in a large batch without needing to compute complex binomial sums. This helps in making quick decisions about quality control.
Polling
A political analyst can estimate the probability that at least a certain number of people in a sample will vote for a candidate, providing a quick way to assess polling results.

Common Misconceptions and Correct Methods

  • Forgetting the Correction Factor
  • Applying to Inappropriate Distributions
  • Ignoring the Validity Check
The Validity Check: np and n(1-p)
A very common mistake is to use the normal approximation when it's not appropriate. The rule of thumb is that both np and n(1-p) should be at least 5. If this condition is not met, the shape of the binomial distribution is too skewed to be accurately represented by the symmetrical normal curve. Our calculator includes a warning for this.
Applying the Wrong Adjustment
The adjustment depends entirely on the inequality. P(X < 10) is different from P(X ≤ 10). The former excludes 10 (so we adjust down to 9.5), while the latter includes 10 (so we adjust up to 10.5). It is crucial to select the correct operator.

Mathematical Derivation and Examples

  • The Binomial to Normal Formula
  • Calculating the Z-Score
  • Worked Example
The process relies on standardizing the binomial distribution into a standard normal distribution (where mean=0, std.dev=1).
Formulas Used
Mean (μ): μ = n * p
Standard Deviation (σ): σ = sqrt(n p (1 - p))
Z-Score: Z = (x_corrected - μ) / σ
Example: To find P(X ≤ 45) for n=100, p=0.5. First, μ = 50 and σ = 5. We correct 45 to 45.5. The Z-score is (45.5 - 50) / 5 = -0.9. Using a Z-table or standard normal calculator, P(Z ≤ -0.9) is approximately 0.1841.