Normal Distribution Calculator (Bell Curve)

Work with the normal distribution easily. Find probabilities, percentiles, and critical values for any mean and standard deviation, visualized on a classic bell curve.

Normal Distribution Calculator

Calculate probabilities and percentiles for normal distributions

Mean (μ)

Standard Deviation (σ)

Calculation Type

Lower Value (X₁)

Upper Value (X₂)

About Normal Distribution

The normal (Gaussian) distribution is a continuous probability distribution characterized by its bell-shaped curve. It's defined by two parameters: mean (μ) which determines the center, and standard deviation (σ) which determines the spread.

The empirical rule states that approximately 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

How the Normal Distribution Calculator Works

Enter the mean (μ) and standard deviation (σ) for your normal distribution. These parameters define the bell curve's center and spread. Use sample statistics or population parameters depending on your data.

Choose what to calculate: probability below a value (left tail), above a value (right tail), between two values, or find the value for a given percentile. Input your x-value(s) or target probability, and the calculator computes the result using the cumulative distribution function.

The interactive bell curve displays your distribution with the relevant area shaded. Z-scores show how many standard deviations values are from the mean. Results include both the probability and its interpretation in context.

When You'd Actually Use This

Quality control specifications

Find what percentage of products fall within tolerance limits. If widget diameters are normally distributed, calculate the defect rate outside specs.

Test score percentile ranking

Determine what percentile a score represents. If SAT scores have mean 1050 and SD 200, find what percentage scored below 1300.

Setting cutoff thresholds

Find the score that separates top 10% from the rest. Useful for grading curves, scholarship eligibility, or identifying outliers.

Confidence interval calculations

Find critical values for confidence intervals. The middle 95% of a normal distribution falls within ±1.96 standard deviations of the mean.

Risk assessment modeling

Model financial returns or measurement errors. Calculate probability of losses exceeding a threshold or errors beyond acceptable limits.

Biological measurements analysis

Analyze heights, blood pressures, or lab values. Determine what percentage of a population falls in normal ranges or outside clinical thresholds.

What to Know Before Using

The 68-95-99.7 rule applies.About 68% of values fall within 1 SD of mean, 95% within 2 SD, 99.7% within 3 SD. This quick rule helps estimate probabilities without calculation.

Z-scores standardize any normal distribution.Z = (x - μ) / σ converts any normal to standard normal (mean 0, SD 1). Z-tables and calculators work with these standardized values.

Normal distribution is symmetric.Mean, median, and mode are all equal. Left and right halves are mirror images. This symmetry simplifies many probability calculations.

Many phenomena approximate normality.Heights, test scores, measurement errors often follow normal distributions due to the Central Limit Theorem. But not everything is normal.

Pro tip: Always check if your data is approximately normal before using these calculations. Use histograms, Q-Q plots, or normality tests. Skewed or heavy-tailed data need different approaches.

Common Questions

What if I don't know the standard deviation?

Use the sample standard deviation as an estimate. For small samples, use t-distribution instead of normal. The t-distribution accounts for SD uncertainty.

How do I find the z-score?

Z = (x - mean) / standard deviation. A z-score of 1.5 means the value is 1.5 standard deviations above the mean. Negative z-scores are below the mean.

What's the standard normal distribution?

Normal distribution with mean 0 and standard deviation 1. Any normal distribution can be converted to standard normal using z-scores.

Can I use this for sample means?

Yes, by the Central Limit Theorem. Sample means are normally distributed with SD = σ/√n. This is the basis for confidence intervals and hypothesis tests.

What percentile is the mean?

The mean is the 50th percentile (median) in a normal distribution. Half the values fall below the mean, half above, due to symmetry.

How accurate are the calculations?

Very accurate - using numerical integration of the normal CDF. Results are precise to many decimal places. Real-world data uncertainty usually dominates.

What if my data isn't normal?

Consider data transformation (like log), use non-parametric methods, or apply the Central Limit Theorem for sample means with large n.

Other Free Tools

Search tools

Normal Distribution Calculator (Bell Curve)