Normal Distribution Calculator

The normal distribution, also called the Gaussian distribution, is the most fundamental continuous probability distribution in statistics. It models random variables that cluster around a central mean value μ (mu) with spread determined by the standard deviation σ (sigma). The distribution is perfectly symmetric and bell-shaped, with the familiar curve that appears everywhere from test scores to measurement errors to natural phenomena like height and weight distributions.

A normal distribution is fully characterized by just two parameters: the mean μ (which centers the distribution) and the standard deviation σ (which controls its width). The mathematical formula for the probability density function (PDF) is f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²)), but you don't need to memorize this—our calculator handles all the complex math automatically.

Key Components of the Normal Distribution

PDF (Probability Density Function): The PDF gives the height of the bell curve at any point x. The peak occurs at the mean μ, and the curve is symmetric on both sides. The total area under the curve equals 1, representing 100% probability. While the PDF height itself is not a probability, the area under the curve between two points represents the probability of observing a value in that range.

CDF (Cumulative Distribution Function): The CDF at a point x gives the probability that a random variable X is less than or equal to x, written as P(X ≤ x). It represents the area under the PDF curve from negative infinity up to x. The CDF starts at 0 (for x far below the mean) and approaches 1 (for x far above the mean). The CDF is the integral of the PDF and is what you use to answer "what's the probability of getting a value below x?"

z-Score (Standard Score): The z-score is a standardized measure that tells you how many standard deviations a value x is away from the mean. The formula is z = (x - μ) / σ. A z-score of 0 means the value equals the mean, z = 1 means one standard deviation above the mean, z = -2 means two standard deviations below the mean. Converting to z-scores allows you to compare values from different normal distributions on a common scale—the standard normal distribution with μ = 0 and σ = 1.

Quantile / Inverse CDF: This is the reverse operation of the CDF. Given a probability p (often called a percentile or tail area), the quantile function returns the value x (or z) where the cumulative probability equals p. For example, the 95th percentile returns the value below which 95% of the data falls. This is essential for finding critical values in hypothesis testing and constructing confidence intervals.

One-Tailed vs Two-Tailed Probabilities

Left-Tail (Lower-Tail): The probability that X is less than or equal to a specific value x. This is simply the CDF value: P(X ≤ x). Use this when you care about values below a threshold, such as "What's the probability a student scores below 70?"

Right-Tail (Upper-Tail): The probability that X is greater than a specific value x, written as P(X > x). This equals 1 - CDF(x). Use this when you care about values above a threshold, such as "What's the probability a measurement exceeds the upper specification limit?"

Two-Tailed: The probability that X falls outside a symmetric interval around the mean, either below -|z| or above +|z|. For a given z-score, the two-tailed probability is p = 2 × (1 - Φ(|z|)), where Φ is the standard normal CDF. This is commonly used in hypothesis testing when you care about deviations in either direction, such as "Is this result significantly different from the expected mean?"

Between-Bounds (Interval Probability): The probability that X falls between two values a and b, calculated as P(a ≤ X ≤ b) = CDF(b) - CDF(a). Our calculator supports this directly—just enter lower and upper bounds to see the shaded area and numeric probability without manual subtraction.

The 68-95-99.7 Rule (Empirical Rule)

For any normal distribution, approximately 68% of values fall within one standard deviation of the mean (μ ± σ), 95% within two standard deviations (μ ± 2σ), and 99.7% within three standard deviations (μ ± 3σ). This rule provides quick mental estimates: if you know μ and σ, you can immediately gauge where most of the data will lie. Values beyond ±3σ are rare (less than 0.3% probability) and often considered outliers in quality control or scientific measurements.

1. Set the Mean (μ) and Standard Deviation (σ): Enter the mean and standard deviation that define your normal distribution. For example, if you're analyzing test scores with a mean of 75 and standard deviation of 10, enter μ = 75 and σ = 10. If you're working with the standard normal distribution (z-scores directly), set μ = 0 and σ = 1, or use the z-mode toggle if available to skip this step.
2. Choose Your Task: The calculator supports multiple modes depending on what you need:
   • PDF/CDF at a specific x: Enter the value x to see the probability density (curve height) and the cumulative probability P(X ≤ x). This is useful for finding percentiles or probabilities below a threshold.
   • Between-bounds probability: Enter a lower and upper value to compute P(lower ≤ X ≤ upper) = CDF(upper) - CDF(lower). The chart shades the interval and displays the numeric probability.
   • z ↔ x conversion: Toggle between standardized z-scores and original x values. The calculator converts automatically using x = μ + zσ or z = (x - μ) / σ. Use z-mode when working with published critical values (e.g., z = 1.96 for 95% confidence) or when comparing across different distributions.
   • Inverse CDF (quantile): Enter a target probability (percentile) to find the corresponding x or z value. For example, "What score is at the 90th percentile?" Input 0.90 to get the x value where 90% of scores fall below.
3. Select Tail Type: If your calculator interface offers tail-type selection, choose the appropriate option:
   • Left-tail: P(X ≤ x) — probability below the value
   • Right-tail: P(X > x) — probability above the value
   • Two-tailed: Symmetric probability outside ±|z| — used in two-sided hypothesis tests
   The chart will shade the corresponding area and display the numeric result.
4. Click Calculate: Once you've entered your parameters and values, click the Calculate button. The interactive bell curve will render with the shaded region corresponding to your query (left-tail, right-tail, between-bounds, etc.). Numeric outputs display PDF(x), CDF(x), z-score, and the requested probability. The chart is fully responsive and updates dynamically as you change inputs.
5. Interpret and Share Results: Review the shaded area on the curve to visually confirm the probability region. Check the numeric outputs for precise values. If available, use the "Copy Link" or "Share" feature to save a URL with your inputs and results encoded, making it easy to share calculations with colleagues, students, or for documentation. You can also screenshot the chart for reports or presentations.
6. Iterate for Comparisons: Change μ, σ, or x values to see how the distribution and probabilities shift. For example, compare how increasing σ (more spread) reduces the peak and widens the tails, or how moving x further from μ decreases the PDF and increases tail probabilities. This is a powerful way to build intuition about normal distributions and sensitivity analysis for statistical models.

• Quality Control and Six Sigma: In manufacturing, product specifications often have upper and lower tolerance limits. Use the normal distribution to calculate the probability a measurement exceeds a tolerance (defect rate). For example, if a part dimension must be 50 ± 0.5 mm and the process has μ = 50, σ = 0.2 mm, compute P(X < 49.5 or X > 50.5) to estimate the proportion of out-of-spec parts. Six Sigma methodology aims for processes where the specification limits are ±6σ from the mean, yielding defect rates below 3.4 per million.
• Test Scores and Percentile Ranks: Standardized tests often report scores along with percentiles. If you know the test's mean and standard deviation (e.g., SAT Math: μ = 520, σ = 115), you can compute your percentile rank by finding CDF(your score). Conversely, if you want to know what score is needed to reach the 75th percentile, use the inverse CDF (quantile) at p = 0.75. This helps students, educators, and admissions offices interpret performance relative to the population.
• Error Modeling in Analytics and Machine Learning: Many statistical models assume measurement errors or residuals are normally distributed with mean 0. Use the normal distribution to compute prediction intervals, assess outlier probabilities, or validate model assumptions. For example, if residuals have σ = 5, the probability of an error exceeding ±10 (2σ) is about 5%, while errors beyond ±15 (3σ) are rare (<1%). Understanding these probabilities helps set alert thresholds and interpret model diagnostics.
• Confidence Intervals and Hypothesis Testing: Critical z-values from the normal distribution underpin many statistical tests and confidence intervals. For a 95% confidence interval, the two-tailed critical value is z ≈ 1.96 (leaving 2.5% in each tail). For 99% confidence, z ≈ 2.576. Use the inverse CDF at p = 0.025 (left-tail) or p = 0.975 (right-tail) to find these values. In hypothesis testing, compare your test statistic z to these critical values (or compute the p-value directly) to decide whether to reject the null hypothesis.
• Between-Bounds Probability for Range Questions: Instead of computing CDF(upper) - CDF(lower) manually, use the between-bounds mode. For example, "What's the probability a randomly selected adult male (μ = 70 inches, σ = 3 inches) is between 68 and 74 inches tall?" Enter lower = 68, upper = 74 to get P(68 ≤ X ≤ 74) directly. The chart shades the interval, making it easy to visualize and communicate the result.
• Standardize First for Comparisons: If you only have z-scores (e.g., from a published table or research paper), set μ = 0 and σ = 1 to work directly with the standard normal distribution. Alternatively, if the calculator offers a z-mode toggle, enable it to skip entering μ and σ. Standardizing allows you to compare values from different populations on a common scale—for instance, comparing a student's z-score in Math (z = 1.5) to their z-score in English (z = 0.8) to see which subject is a relative strength.
• Estimating Sample Means and Central Limit Theorem: The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as sample size increases, even if the underlying population is not normal. If you're working with sample means (X̄) from a population with mean μ and standard deviation σ, the sampling distribution of X̄ is approximately normal with mean μ and standard deviation σ/√n (the standard error). Use this calculator with μ (population mean) and σ/√n (standard error) to compute probabilities for sample means, such as "What's the probability the sample mean exceeds a certain value?"
• Financial Risk and Value at Risk (VaR): In finance, asset returns are often modeled as normally distributed (though real returns have "fat tails"). To estimate Value at Risk, compute the quantile at a given confidence level. For example, for a portfolio with daily return μ = 0.05%, σ = 1.2%, the 5th percentile (p = 0.05) gives the return level below which you expect to fall 5% of the time—representing a loss threshold for risk management. Use the inverse CDF at p = 0.05 to find this critical return level.

PDF(x) — Probability Density

The PDF value is the height of the bell curve at a specific point x. It is not a probability itself, but rather a density. The units are "probability per unit of x." For example, if x is measured in inches and PDF(x) = 0.12, that means the density is 0.12 per inch at that point. To get an actual probability, you integrate (sum) the PDF over an interval—this is what the CDF and between-bounds calculations do. The PDF is highest at the mean μ and decreases symmetrically as you move away from the mean in either direction.

CDF(x) — Cumulative Probability

CDF(x) gives the probability that a random variable X is less than or equal to x: P(X ≤ x). This is the area under the PDF curve from negative infinity up to x. CDF values range from 0 to 1. A CDF of 0.5 at x means 50% of values fall below x (this occurs at x = μ for a normal distribution). A CDF of 0.975 means 97.5% of values fall below x, leaving 2.5% in the right tail. The right-tail probability is simply 1 - CDF(x).

Between-Bounds Probability

The probability that X falls between two values a and b is P(a ≤ X ≤ b) = CDF(b) - CDF(a). Our calculator computes this directly when you enter lower and upper bounds. The chart shades the interval from a to b, and the numeric output displays the probability. This is the most intuitive way to answer range questions like "What's the chance of scoring between 80 and 90 on the test?" or "What fraction of parts will fall within the tolerance zone?"

z ↔ x Conversion

The z-score standardizes any value x to the number of standard deviations it is from the mean: z = (x - μ) / σ. The reverse conversion is x = μ + zσ. Our calculator handles both directions automatically. Use z when you want to compare values from different distributions (e.g., comparing scores from two different exams) or when working with published critical values. Use x when you need results in the original units (e.g., dollars, inches, test points). The probabilities are identical for corresponding z and x values—only the scale changes.

Two-Tailed Probability

The two-tailed probability for a z-score represents the chance of observing a value at least as extreme in either direction from the mean. Mathematically, for a given |z|, the two-tailed p-value is p = 2 × (1 - Φ(|z|)), where Φ is the standard normal CDF. For example, z = 1.96 has a two-tailed probability of approximately 0.05 (5%), meaning there's a 5% chance of observing a z-score below -1.96 or above +1.96 under the null hypothesis. This is the basis for the classic 95% confidence interval and two-sided hypothesis tests at α = 0.05.

Quantile / Critical Value

The quantile (inverse CDF) function solves the reverse problem: given a probability p and a tail type, find the value x (or z) where the cumulative probability equals p. For example, the 95th percentile (p = 0.95, left-tail) returns the x value below which 95% of the distribution falls. In hypothesis testing, critical values are quantiles that define rejection regions. For a one-tailed test at α = 0.05, the critical z is approximately 1.645 (right-tail at p = 0.95). For a two-tailed test at α = 0.05, the critical z is ±1.96 (each tail has 2.5%).

Accuracy and Numerical Precision

Our calculator uses high-precision numerical approximations for the normal CDF (Φ) and its inverse, based on well-established statistical algorithms. These methods are accurate to many decimal places and suitable for all typical statistical work, including research, quality control, and finance. Extremely far into the tails (e.g., probabilities below 10⁻¹⁰), small numerical errors may appear, but for practical purposes (p-values down to 0.0001 or beyond), the results are reliable and match published statistical tables and software like R, Python SciPy, or Excel.

Limitations and Assumptions

The normal distribution is appropriate when data are continuous, symmetric, and unimodal. Real-world data often deviate from perfect normality—skewed distributions, outliers, or heavy tails may require transformations or alternative distributions (e.g., log-normal, t-distribution for small samples, or non-parametric methods). Always verify the normality assumption using histograms, Q-Q plots, or formal tests (Shapiro-Wilk, Anderson-Darling) before relying on normal-based calculations for critical decisions. For small sample sizes (n < 30) or unknown population standard deviation, use the t-distribution instead of the normal distribution for inference.