Updated Probability & Statistics Tool

Normal Distribution Calculator

Compute PDF, CDF, Z-scores, tail probabilities and inverse normal values for any normal distribution. Enter a mean μ and standard deviation σ, then evaluate probabilities, convert between X and Z and find critical values.

PDF & CDF Z ↔ X Conversion Tail Probabilities Inverse Normal

Work with Any Normal Distribution N(μ, σ²)

Use this Normal Distribution Calculator to evaluate the probability density function (PDF), cumulative distribution function (CDF), tail probabilities and Z-score conversions for a normal random variable X with mean μ and standard deviation σ. You can also find inverse normal values (quantiles) for common confidence levels and probabilities.

By default, the calculator uses the standard normal distribution N(0, 1). To work with a general normal distribution, change the mean μ and standard deviation σ in any tab. All results use high-quality numerical approximations for the normal CDF and its inverse.

This tab evaluates the PDF f(x) and CDF F(x) for X ∼ N(μ, σ²). It also reports the corresponding Z-score z = (x − μ) / σ.

Enter either x or z and click convert. If you fill both, the calculator prioritizes converting from x to z.

For left or right tail probabilities, only a is used. For between and outside probabilities, both a and b are used and the calculator automatically enforces a ≤ b when interpreting the interval.

Use this tab to find critical values. For example, with tail type "left" and p = 0.975 you obtain the 97.5th percentile; with tail type "central" and p = 0.95 you obtain symmetric bounds around μ that capture 95 percent of the distribution.

This numeric table summarizes standard normal CDF values Φ(z) = P(Z ≤ z) for Z ∼ N(0, 1) at common z-values. You can use it to quickly approximate probabilities or verify results from the other tabs.

Normal Distribution Calculator – Complete Guide to PDF, CDF, Z-Scores and Probabilities

The Normal Distribution Calculator on MyTimeCalculator is built for students, teachers, analysts and anyone who works with bell-shaped curves. It lets you evaluate the probability density function (PDF), cumulative distribution function (CDF), Z-scores, tail probabilities and inverse normal values for any normal distribution with mean μ and standard deviation σ.

Instead of flipping between different tables and formulas, you can explore all the key normal distribution operations in one place. The calculator supports the standard normal distribution N(0, 1) as well as custom distributions such as heights, test scores and measurement errors that follow a bell-shaped pattern.

1. The Normal Distribution and N(μ, σ²)

A continuous random variable X is normally distributed with mean μ and variance σ² if its PDF has the form:

f(x) = \(\frac{1}{\sigma \sqrt{2\pi}}\) exp\(\,-\frac{(x - \mu)^2}{2\sigma^2}\).

The curve is symmetric about x = μ, and the spread is controlled by the standard deviation σ. Larger values of σ stretch the distribution, creating a wider, flatter bell; smaller values of σ create a narrower, sharper peak.

The CDF of X, denoted F(x) = P(X ≤ x), is the area under the PDF curve from minus infinity up to x. There is no simple closed-form expression for F(x) in elementary functions, so numerical methods and tables are used in practice. The Normal Distribution Calculator uses stable approximations to evaluate F(x) to high precision.

2. Standardization and the Z-Score

Any normal random variable X ∼ N(μ, σ²) can be standardized to the standard normal distribution Z ∼ N(0, 1) using the transformation:

Z = \(\frac{X - \mu}{\sigma}\).

This Z-score tells you how many standard deviations above or below the mean a particular value x lies. For example, z = 2 means that x is two standard deviations above the mean, and z = −1.5 means one and a half standard deviations below the mean.

The Z ↔ X converter tab in the calculator automates this transformation. You can enter x, μ and σ to find the corresponding Z-score, or enter a Z-score and convert back to the original scale, making it easy to interpret results in everyday units.

3. PDF and CDF for Any Normal Distribution

The PDF tab of the calculator reports three core quantities:

  • Z-score: z = (x − μ) / σ.
  • PDF f(x): the height of the normal curve at x.
  • CDF F(x): the probability P(X ≤ x).

While the PDF is useful for understanding the shape of the distribution and comparing relative likelihoods, it is the CDF that gives probabilities. For example, P(X ≤ 70) is calculated as F(70), and P(X ≥ 70) is 1 − F(70).

Because the calculator works directly with μ and σ, you can move quickly between different scenarios such as exam scores with mean 75 and standard deviation 8, or manufacturing measurements with mean 10 and standard deviation 0.5, without having to rescale everything manually.

4. Tail and Interval Probabilities

Many real questions involve tail or interval probabilities rather than a single point. Typical examples include:

  • P(X ≤ a): left-tail probability.
  • P(X ≥ a): right-tail probability.
  • P(a ≤ X ≤ b): probability that X lies in a range.
  • P(X ≤ a or X ≥ b): probability that X falls outside a central band.

The Tail & Interval Probabilities tab handles all of these cases. You provide μ, σ and the bounds a and b. The calculator computes the requested probability and also reports:

  • The standardized bounds in terms of Z.
  • A plain-language description of the event.
  • The complement probability for the opposite event.

Internally, the probability P(a ≤ X ≤ b) is computed as F(b) − F(a), and outside probabilities use 1 − P(a ≤ X ≤ b). This makes it straightforward to check popular rules such as the empirical 68–95–99.7 rule for normal distributions.

5. Inverse Normal and Critical Values

Inverse normal calculations reverse the usual direction: instead of asking for the probability of a given value, you ask which value corresponds to a given probability. Typical examples are:

  • The 95th percentile of a test score distribution.
  • The cutoff above which only the top 2.5 percent of values lie.
  • Symmetric bounds around the mean that capture 90 percent of the distribution.

The Inverse Normal tab supports three tail types:

  • Left tail: find x such that P(X ≤ x) = p.
  • Right tail: find x such that P(X ≥ x) = p.
  • Central: find symmetric bounds −x and x around μ such that P(−x ≤ X ≤ x) = p.

You can enter p either as a decimal (0 to 1) or as a percentage (0 to 100). The calculator converts to the underlying Z-score using a high-quality approximation to the inverse standard normal CDF, then rescales to the original units using X = μ + zσ.

6. Normal Distribution Table and Standard Values

Standard normal tables list values of the CDF Φ(z) for a grid of z-values. They are commonly used in introductory statistics courses and for hand calculations. While the calculator can compute Φ(z) directly for any z, the Normal Distribution Table tab provides a quick numeric reference for selected values from negative to positive z.

You can use this table to check results from other sources, verify textbook exercises or build intuition about how probabilities change as z moves away from zero. For example, you can see that P(Z ≤ 1.96) is close to 0.975, a widely used value in confidence interval calculations.

7. Practical Steps for Using the Normal Distribution Calculator

  1. Identify the context and determine the mean μ and standard deviation σ of your normal model.
  2. Choose whether you need a point probability (through CDF), a tail or interval probability, a Z-score or an inverse normal value.
  3. Open the corresponding tab: PDF & CDF, Z ↔ X, Tail & Interval, or Inverse Normal.
  4. Enter μ, σ and the relevant value(s) such as x, a, b or p.
  5. Click calculate to obtain numeric results along with standardized Z-scores and helpful interpretations.
  6. Optionally, compare your results with the standard normal table tab to deepen your understanding.
  7. Use the outcomes to answer questions about proportions, cutoffs, pass/fail thresholds or confidence regions.

8. Common Applications of the Normal Distribution

  • Education: Modeling exam scores, determining grade cutoffs, estimating pass rates.
  • Quality control: Monitoring product dimensions, tolerances and defect rates.
  • Finance: Approximating returns, risk measures and error terms in regression models.
  • Health and biology: Modeling physiological measurements such as blood pressure or heights.
  • Engineering: Aggregating many small independent effects into an overall performance metric.

In many of these settings, the central limit theorem provides theoretical justification: sums and averages of many independent contributions tend to follow a normal distribution, even if the underlying components do not.

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Normal Distribution Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about Z-scores, probabilities, percentiles and how to interpret the outputs of this Normal Distribution Calculator.

The PDF f(x) describes the relative likelihood density at a single point x and gives the shape of the bell curve. The CDF F(x) is the probability that X is less than or equal to x, equal to the area under the curve from minus infinity to x. For probability questions, the CDF and differences of CDF values are the key quantities, while the PDF helps visualize how probability mass is distributed along the axis.

Z-scores are helpful when you want to compare values from different normal distributions or use standard tables that are based on N(0, 1). By converting to Z, you reduce every problem to the same standard normal scale. The calculator handles both perspectives: you can work directly with μ and σ or convert to Z and back as needed without manual algebra.

The Normal Distribution Calculator uses widely accepted approximation formulas for the normal CDF and its inverse, which are accurate to several decimal places for most practical purposes. For extreme tail probabilities or very large absolute values of z, small numerical differences can appear compared with different software packages, but these are typically beyond the precision needed in introductory and intermediate applications.

A valid normal distribution must have σ > 0. If you enter zero or a negative value for σ, the calculator will show an error message and skip the computation. Always check that the standard deviation you use is strictly positive and correctly reflects the spread of your data or model.

Yes. The Inverse Normal tab is designed for exactly these tasks. For a two-sided confidence interval with confidence level 1 − α, you can use the central tail type with p = 1 − α to obtain symmetric bounds. For one-sided tests, choose left or right tail and set p to the appropriate cumulative probability cut-off, such as 0.95 or 0.975. The tool then outputs the corresponding Z-score and X value for your specified μ and σ.

Internally, the calculator carries more precision than is shown in the final results. The displayed values are rounded to a fixed number of decimal places for readability, which can introduce small rounding differences when you add them by hand. The underlying calculations are consistent; the minor discrepancies come from rounding rather than from the probability model itself.

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