Calculator
Example Data Table
| Trials | p | Query | Exact Result | Normal Estimate | Use Case |
|---|---|---|---|---|---|
| 50 | 0.40 | P(X ≤ 20) | Calculated by tool | With correction | Quality checks |
| 100 | 0.55 | P(X ≥ 60) | Calculated by tool | With correction | Survey results |
| 200 | 0.25 | P(40 ≤ X ≤ 60) | Calculated by tool | With correction | Defect counts |
Formula Used
The exact binomial probability for exactly x successes is:
P(X = x) = C(n, x) × px × (1 - p)n - x
The normal approximation uses these values:
Mean = μ = n × p
Variance = σ² = n × p × (1 - p)
Standard Deviation = σ = √(n × p × (1 - p))
The z score is:
z = (x - μ) / σ
When continuity correction is enabled, the calculator adjusts discrete limits by 0.5. For example, P(X ≤ x) becomes P(Y ≤ x + 0.5) under the normal curve.
How To Use This Calculator
- Enter the number of trials.
- Enter the success probability as a decimal or percent.
- Select the probability type.
- Enter x, or enter lower and upper values for a range.
- Choose decimal places.
- Enable continuity correction for better normal approximation.
- Press Calculate to show results above the form.
- Use CSV or PDF buttons to download the report.
Binomial Distribution With Normal Distribution Approximation
Purpose Of The Calculator
A binomial distribution models repeated trials. Each trial has only two outcomes. The usual names are success and failure. This calculator compares the exact binomial result with a normal curve estimate. It helps when trial counts are large and manual summation becomes slow.
Why Approximation Matters
Exact binomial probability is precise. Yet it can involve many terms. A normal distribution can estimate the same area more quickly. This works best when expected successes and expected failures are both large. The tool checks these values automatically.
Understanding Inputs
The number of trials is n. The success probability is p. The value x is the number of successes being tested. For a range calculation, lower and upper values define the included success counts. The calculator supports exact, cumulative, tail, and interval questions.
Continuity Correction
The binomial model is discrete. The normal model is continuous. Continuity correction helps connect these two shapes. It adds or subtracts 0.5 from boundaries before z scores are found. This often improves estimates, especially near the center of the distribution.
Reading The Output
The exact result is the benchmark value. The normal estimate shows the approximation. The absolute difference measures direct gap size. The relative difference shows the gap as a percent of the exact probability. Mean, variance, standard deviation, skewness, and kurtosis describe distribution shape.
Best Practical Use
Use this calculator for statistics homework, sampling work, production checks, survey planning, and probability reviews. For small samples, trust the exact value more. For large balanced samples, the normal estimate is often useful. Always review the condition message before using an approximation in reports.
FAQs
What does this calculator compare?
It compares exact binomial probability with a normal approximation. It also shows mean, variance, standard deviation, z scores, and error difference.
When is normal approximation useful?
It is useful when trials are large. It works better when expected successes and expected failures are both at least five, and preferably ten.
What is continuity correction?
Continuity correction adjusts a discrete binomial boundary by 0.5. It helps a continuous normal curve estimate a discrete probability more accurately.
Should I trust exact or normal results?
The exact binomial result is more precise. The normal result is an estimate. Use the approximation when conditions are acceptable.
Can I enter probability as a percent?
Yes. Select percent as the input type. Then enter values like 40 for forty percent or 12.5 for twelve point five percent.
What does P(X ≤ x) mean?
It means the probability of getting x or fewer successes. The calculator sums all exact probabilities from zero through x.
What does P(X ≥ x) mean?
It means the probability of getting x or more successes. The calculator sums all exact probabilities from x through n.
Why are CSV and PDF exports included?
CSV is useful for spreadsheets. PDF is useful for saving reports, sharing results, or attaching calculations to assignments and documentation.