Normal Distribution Curve Calculator
Quick Presets
Enter a raw value, mean, and standard deviation to calculate the z-score and corresponding percentile.
Enter a percentile to find the corresponding raw value. For the 95th percentile of IQ scores (μ=100, σ=15), enter 95.
What Is a Normal Distribution Curve?
The normal distribution curve is a continuous probability distribution shaped like a symmetric bell. The center of the bell sits at the mean, and the curve falls away on both sides in a predictable pattern governed by the standard deviation. It appears throughout statistics, measurement science, and natural phenomena because many real-world quantities vary around a central value in exactly this way.
Two numbers fully define any normal distribution: the mean (μ), which controls where the center sits, and the standard deviation (σ), which controls how wide or narrow the bell is. A large σ produces a low, wide curve. A small σ produces a tall, narrow one. The total area under every normal curve equals 1, so area directly represents probability.
Properties of the Normal Distribution
- Symmetric: the left half is a mirror image of the right half around the mean.
- Unimodal: there is exactly one peak, at the mean.
- Mean = median = mode: all three measures of center coincide at the peak.
- Total area = 1: probability is represented by area under the curve, and all probabilities must sum to 1.
- Asymptotic tails: the curve approaches but never touches the x-axis, meaning no value is technically impossible.
- Fully determined by μ and σ: no other parameters are needed.
Normal Distribution Formula
The probability density function (PDF) gives the height of the curve at any value x:
Note that the height f(x) is the probability density, not the probability of exactly one value. For a continuous distribution, P(X = x) = 0 for any single point. Probability is always an area, found by integrating the PDF over an interval.
What Is the Standard Normal Distribution?
The standard normal distribution is the special case where μ = 0 and σ = 1. Any normal distribution can be converted to standard normal by transforming each value to a z-score. This matters because standard normal probabilities are extensively tabulated, and the interactive tool above uses the standard normal CDF internally to calculate every probability regardless of what μ and σ you enter.
What Is a Z-Score?
A z-score measures how far a value sits from the mean in units of standard deviations. The formula is:
If x = 85, μ = 70, and σ = 10, then z = (85 - 70) / 10 = 1.5. This means 85 lies 1.5 standard deviations above the mean. A negative z-score means the value is below the mean. The z-score also points directly to the cumulative probability: P(X < 85) = P(Z < 1.5) ≈ 0.9332, so 85 is at the 93.32nd percentile of that distribution.
Area Under the Normal Curve
Because the height of the PDF at any single point is not a probability, probability is always the area under the curve over an interval. The cumulative distribution function (CDF) gives the area from negative infinity up to a value x:
The erf term is the error function, approximated numerically in the calculator above using the Abramowitz and Stegun rational approximation, which is accurate to better than 7.5 × 10⁻⁸ across all z-values.
The 68-95-99.7 Rule
Three intervals cover well-defined and memorable shares of the total area under any normal curve. These percentages come from integrating the PDF, not from rounding:
| Interval | Probability | Percentage | 1 in how many fall outside? |
|---|---|---|---|
| μ ± 1σ | 0.6827 | 68.27% | About 1 in 3 |
| μ ± 2σ | 0.9545 | 95.45% | About 1 in 22 |
| μ ± 3σ | 0.9973 | 99.73% | About 1 in 370 |
| μ ± 1.96σ | 0.9500 | 95.00% | 1 in 20 (used in hypothesis testing) |
| μ ± 2.576σ | 0.9900 | 99.00% | 1 in 100 |
How to Find a Probability From a Normal Distribution
How to Find a Percentile
To find the value at the pth percentile, you need the inverse CDF (also called the quantile function or probit). First find the z-score that corresponds to cumulative probability p/100, then convert back to raw values using x = μ + zσ. The Percentile tab above handles this automatically using a rational approximation of the inverse standard normal CDF, validated against known quantiles:
- 50th percentile: z = 0 (any μ)
- 90th percentile: z ≈ 1.2816
- 95th percentile: z ≈ 1.6449
- 97.5th percentile: z ≈ 1.9600
- 99th percentile: z ≈ 2.3263
Normal Distribution Examples
Real datasets should not automatically be assumed to follow a normal distribution. Many physical and social measurements are approximately normal in large samples, but actual normality should be tested using methods such as a Q-Q plot or a Shapiro-Wilk test before relying on normal probability calculations for critical decisions.
Normal Distribution vs Bell Curve
Bell curve is an informal visual description of the shape. Normal distribution is a precise mathematical model with specific properties, a defined formula, and exact probability calculations. Every normal distribution has a bell shape, but not every bell-shaped distribution is technically normal. The terms are used interchangeably in casual settings, but in statistics the normal distribution refers specifically to the family of distributions defined by the PDF above.
Normal Distribution vs Standard Normal Distribution
Any normal distribution with any μ and σ belongs to the same family. The standard normal distribution is the specific member with μ = 0 and σ = 1. Converting to z-scores standardizes any normal distribution to the standard normal, which is why a single CDF table covers all normal distributions. This tool performs that conversion internally so you can work directly with your own mean and standard deviation.
Normal Distribution vs Empirical Rule
The empirical rule (68-95-99.7 rule) is a consequence of the normal distribution's mathematics, not a separate concept. It describes three particular areas under the normal curve. It applies only to data that follows a normal distribution. The Empirical Rule Calculator applies these intervals directly to any mean and standard deviation you provide.
Common Mistakes With Normal Distribution Calculations
When Is the Normal Distribution Used?
The normal distribution appears across statistics and applied science because of the central limit theorem: the distribution of sample means approaches normal as sample size grows, regardless of the underlying population shape. This makes the normal distribution the foundation of much of classical inference, including confidence intervals, hypothesis testing, and regression residual analysis. Measurement errors, psychometric scores, and biological measurements in large populations also tend to approximate normal distributions, making the normal curve one of the most practically useful models in data analysis.
Related Tools and Pages
Frequently Asked Questions
A normal distribution curve is a symmetric, bell-shaped graph of a continuous probability distribution defined by its mean (μ) and standard deviation (σ). The peak sits at the mean, and the curve falls symmetrically on both sides. The total area under the curve equals 1, so any region's area gives the probability that a randomly drawn observation falls in that range.
The name comes from the shape: the graph rises to a rounded peak at the mean and falls away smoothly on both sides, producing an outline that resembles a bell. It is an informal visual description. The mathematically precise term is normal distribution (or Gaussian distribution, after Carl Friedrich Gauss), and bell curve simply refers to that characteristic shape.
The probability density function is f(x) = (1 / σ√(2π)) × e^(-(x-μ)² / (2σ²)). Here μ is the mean, σ is the standard deviation, e is Euler's number (~2.71828), and π is ~3.14159. The term 1/(σ√2π) normalizes the curve so its total area equals 1. The exponential term creates the bell shape and controls how quickly the tails fall off based on how far x is from μ.
The standard normal distribution is a special case of the normal distribution with mean μ = 0 and standard deviation σ = 1. Any normal distribution value can be converted to a standard normal z-score using z = (x - μ) / σ. This matters because standard normal probabilities are tabulated in z-tables and built into statistical software, making it the reference form for all normal distribution calculations.
A z-score measures how many standard deviations a value lies from the mean. The formula is z = (x - μ) / σ. For example, if a student scored 90 on an exam where μ = 75 and σ = 10, their z-score is (90 - 75) / 10 = 1.5. This means they scored 1.5 standard deviations above the mean. You can then find the corresponding percentile using the standard normal CDF.
The area under the normal curve between two values represents the probability that a randomly selected observation from that distribution falls in that interval. The total area under the entire curve is always 1 (or 100%). For example, the area between μ - σ and μ + σ is approximately 0.6827, meaning there is a 68.27% chance any observation falls within one standard deviation of the mean.
According to the 68-95-99.7 rule: approximately 68.27% of data falls within 1 standard deviation (μ ± 1σ), approximately 95.45% falls within 2 standard deviations (μ ± 2σ), and approximately 99.73% falls within 3 standard deviations (μ ± 3σ). These figures come from integrating the normal PDF and are exact properties of the distribution, not estimates.
Convert your x value to a z-score using z = (x - μ) / σ. Then apply the cumulative distribution function: P(X < x) = Φ(z). For a right-tail probability, P(X > x) = 1 - Φ(z). For a range, P(a < X < b) = Φ(z_b) - Φ(z_a). The Probability tab in the tool above performs all these calculations automatically, shading the corresponding area on the curve.
The percentile equals Φ(z) × 100, where Φ(z) is the standard normal CDF. For z = 1.5, Φ(1.5) ≈ 0.9332, so the 93.32nd percentile. To go in reverse, use the inverse CDF: given a percentile p, find the z-score and then compute x = μ + z × σ. The Percentile tab above handles both directions.
Yes. Every tab accepts your own μ and σ. The curve redraws dynamically around your mean, and all probability calculations use your specified parameters. Changing the mean shifts the curve left or right without altering its shape. Changing the standard deviation makes the curve wider and lower (larger σ) or narrower and taller (smaller σ). The only constraint is that σ must be greater than 0.
No. Many datasets are skewed, bimodal, bounded, or follow entirely different distributions. Income tends to be right-skewed. Counts of rare events often follow a Poisson distribution. The normal distribution is a model that fits many situations well, and the central limit theorem ensures sample means approach normality in large samples, but it should not be assumed without checking. Use Q-Q plots or formal normality tests to assess your data.
A bell curve is an informal name for any graph that looks like a bell. A normal distribution is a specific mathematical distribution with a precise formula, defined properties (symmetry, mean equals median equals mode, etc.), and exact probability calculations. Every normal distribution has a bell shape, but the term bell curve can loosely describe other unimodal, roughly symmetric distributions that are not technically normal.