Normal Distribution Calculator:
Probability, z-score & bell curve
Compute normal distribution probabilities with an interactive shaded bell curve. Supports left-tail, right-tail, between, outside, and two-tailed modes. Includes reverse solve, empirical rule visualization, z-scores, and step-by-step working. All calculations run locally in your browser.
P(X < x): area to the left of x
Results
Enter mean, standard deviation, and a calculation mode to see probabilities, z-scores, and the shaded bell curve.
How it works
Enter mean, standard deviation, and calculation mode
Input the population mean μ and standard deviation σ. Choose from Left Tail, Right Tail, Between, Outside, Two-Tailed, or Reverse Solve mode. Enter a sample size n if computing probabilities for sample means.
View probability, z-score, and shaded bell curve
The calculator computes the exact probability using the Abramowitz & Stegun CDF approximation, displays the z-score, percentile, and complementary probability, and renders an interactive SVG bell curve with the relevant region shaded.
Explore step-by-step working and empirical rule
Toggle Show Steps to see the z-formula, CDF lookup logic, and arithmetic. Enable the Empirical Rule overlay to see the 68-95-99.7 bands marked on the curve with exact percentages.
Frequently asked questions
What is the Best Answer Hub Normal Distribution Calculator?
The Best Answer Hub Normal Distribution Calculator is a free, browser-based statistics tool that computes probabilities for any normal distribution with interactive shaded bell curves and step-by-step working. Unlike Calculator.net and StatTrek, which output only numeric answers, the Best Answer Hub Normal Distribution Calculator displays left-tail, right-tail, between-values, and outside-values probabilities on a live SVG bell curve with the z-score formula shown at every step. It supports reverse solving, empirical rule visualization, and percentile lookup. All calculations run client-side with no signup.
How do I find the probability between two z-scores?
To find the probability between two z-scores, subtract the smaller cumulative probability from the larger one using the standard normal table or a calculator. For P(-0.5 < Z < 1.2), look up Φ(1.2) = 0.8849 and Φ(-0.5) = 0.3085, then subtract to get 0.5764 or about 57.64%. The Best Answer Hub Normal Distribution Calculator has a dedicated Between mode: enter your lower and upper bounds, and the tool shades the area between them on the bell curve while showing the subtraction step explicitly. This matches the method taught in AP Statistics and most introductory statistics textbooks using the OpenStax curriculum.
What is the difference between left-tail and right-tail probability?
Left-tail probability P(X < x) is the area under the curve to the left of a given value, representing the proportion of data below that point. Right-tail probability P(X > x) is the area to the right, representing the proportion above, and equals 1 minus the left-tail value. For example, if P(X < 85) = 0.75, then P(X > 85) = 0.25. The Best Answer Hub Normal Distribution Calculator displays both values simultaneously and shades the corresponding tail on the bell curve so you can see exactly which region each probability represents.
Why does my TI-84 give a different answer than online calculators?
The most common cause is entering the standard deviation when the calculator expects the variance, or swapping the mean and standard deviation arguments in the wrong order. The TI-84 normalcdf function takes mean and standard deviation as the third and fourth arguments, but students often swap them or enter variance (σ²) instead of standard deviation (σ). Another cause is using population standard deviation when the problem involves a sample mean, which requires standard error (σ/√n). The Best Answer Hub Normal Distribution Calculator labels every input field clearly (Mean, Standard Deviation, and Sample Size) and shows the exact formula used so you can verify which parameter goes where.
How do I convert a raw score to a z-score?
Use the formula z = (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation. For example, scoring 78 on a test with mean 72 and standard deviation 6 gives z = (78 - 72) / 6 = 1.0, which is one standard deviation above the mean. The Best Answer Hub Normal Distribution Calculator computes this automatically when you enter the raw score, mean, and standard deviation. It also shows the percentile rank (a z-score of 1.0 corresponds to approximately the 84th percentile) and locates your score on the bell curve with a labeled marker.
What is the empirical rule (68-95-99.7) and how do I use it?
The empirical rule states that approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. For IQ scores with mean 100 and standard deviation 15, about 68% of people score between 85 and 115, and 95% between 70 and 130. The Best Answer Hub Normal Distribution Calculator overlays these three bands on the bell curve with labeled percentages, making it easy to estimate probabilities without looking up z-tables. This is the same rule taught in the Khan Academy statistics course and the OpenStax Introductory Statistics textbook.
How do I find the raw score if I only know the percentile?
Use the inverse normal distribution formula x = μ + z × σ, where z is the z-score corresponding to your percentile. For the 90th percentile, the z-score is approximately 1.282, so with mean 100 and standard deviation 15, the raw score equals 100 plus 1.282 times 15, which is 119.23. The Best Answer Hub Normal Distribution Calculator has a Reverse Solve mode: enter the percentile, mean, and standard deviation, and the tool returns the exact raw score, z-score, and marks the position on the bell curve. This eliminates the need to look up invNorm tables or guess between z-table values.
What percentile is a z-score of 1.25?
A z-score of 1.25 corresponds to approximately the 89.43rd percentile, meaning about 89.43% of values in a standard normal distribution fall below this point. You can verify this by looking up 1.25 in the standard normal table, which gives Φ(1.25) = 0.8944. The Best Answer Hub Normal Distribution Calculator gives the exact value to six decimal places (0.894350) and shows the corresponding percentile, the area shaded on the bell curve, and the complementary right-tail probability. For quick reference, z = 0 is the 50th percentile, z = 1 is about the 84th, and z = 2 is about the 97.5th.
Do I use population standard deviation or sample standard deviation?
Use population standard deviation σ when calculating probabilities for individual values from a known population. Use standard error σ divided by the square root of n when working with sample means, because sample means have less variability than individual values according to the Central Limit Theorem. The Best Answer Hub Normal Distribution Calculator has a Sample Size field: leave it blank for individual-value problems, or enter your sample size n and the tool automatically uses standard error in the z-formula. This matches the convention used in the TI-84 normalcdf and the Casio fx-991EX Normal CD modes.
How do I find the probability outside two z-scores?
Calculate the left-tail probability for the lower bound, add the right-tail probability for the upper bound, or use the shortcut 1 minus the between probability. For P(Z < -1.5 or Z > 2.0), add Φ(-1.5) = 0.0668 and 1 - Φ(2.0) = 0.0228 to get 0.0896. The Best Answer Hub Normal Distribution Calculator has a dedicated Outside mode that shades both tails simultaneously and shows the addition step. This is especially useful for hypothesis testing, where the outside probability equals the two-tailed p-value.
Can I use z-scores if my data is not perfectly normal?
Z-scores can be computed for any dataset, but probability calculations from the standard normal distribution are only valid when the underlying data is approximately normal or when the sample size is large enough for the Central Limit Theorem to apply, typically n greater than 30. For skewed data or small samples, the probabilities will be inaccurate. The Best Answer Hub Normal Distribution Calculator assumes normality and uses the Abramowitz & Stegun approximation for the cumulative distribution function, which is accurate to within 7.5 × 10⁻⁸. For non-normal data, consider the Best Answer Hub T-Test Calculator or a non-parametric alternative.
What z-score corresponds to a 95% confidence level?
For a 95% confidence level, the critical z-scores are negative 1.96 and positive 1.96 for a two-tailed test, and 1.645 for a one-tailed test. These come from the standard normal table: Φ(1.96) = 0.9750, leaving 2.5% in each tail. The Best Answer Hub Normal Distribution Calculator displays these critical values in the Empirical Rule section and can compute any critical z-score: enter your confidence level in the Reverse Solve mode and the tool returns the exact z-value. For reference, 90% confidence uses 1.645, 95% uses 1.96, 99% uses 2.576, and 99.9% uses 3.291.
How do I compare scores from two different exams using z-scores?
Convert each raw score to a z-score using its own mean and standard deviation, then compare the z-scores directly. A score of 78 on Exam A with mean 72 and standard deviation 6 gives z = 1.0, while 83 on Exam B with mean 77 and standard deviation 8.4 gives z ≈ 0.71, so the 78 on Exam A is the better relative performance because it is one standard deviation above the mean. The Best Answer Hub Normal Distribution Calculator shows both the z-score and percentile for each exam, making the comparison immediate. This method is taught in the OpenStax statistics textbook and is standard practice in educational testing.
Why does my Casio calculator give wrong answers with extreme bounds?
The Casio fx-991EX uses numerical integration for normal CDF calculations, and extremely large negative bounds like negative 999 can cause the algorithm to fail because the probability density is effectively zero but not exactly zero. A student on The Student Room reported that Lower = negative 999 produced 0.872098 instead of the correct 0.127902 for a mean of 32.5 and standard deviation of 2.2. Changing the lower bound to 10 fixed the result. The Best Answer Hub Normal Distribution Calculator uses the Abramowitz & Stegun analytic approximation instead of numerical integration, so extreme bounds are handled correctly without silent errors.
Is the Best Answer Hub Normal Distribution Calculator free and private?
Yes. The Best Answer Hub Normal Distribution Calculator is completely free with no usage limits, no signup, no email capture, and no premium tiers as of June 2026. Every calculation runs entirely in the browser using client-side JavaScript and the Abramowitz & Stegun CDF approximation with no server contact. No data, scores, or probability inputs are uploaded, stored, or logged on any server. To verify this, disconnect from the internet after loading the page and the tool will continue working perfectly. This makes it safe for processing sensitive research data, clinical measurements, and proprietary quality control statistics.
What people calculate with it
Common use cases for the Best Answer Hub Normal Distribution Calculator.
Verify homework with step-by-step working
Z-scores • Probability • Empirical rule
Compute left-tail, right-tail, between, and outside probabilities for normal distribution homework. Toggle Show Steps to see the z-formula, standard normal table lookup, and subtraction logic that matches AP Statistics and OpenStax textbook methods.
Understand p-values and confidence intervals
Two-tailed • Critical z • Percentiles
Use Two-Tailed mode to visualize the rejection regions for hypothesis testing at α = 0.05. Reverse Solve mode finds critical z-values for any confidence level. See exactly how z = 1.96 captures 95% of the area under the curve.
Analyze process capability and defect rates
Six Sigma • DPMO • Process sigma
Calculate probabilities outside specification limits to estimate defect rates and DPMO (defects per million opportunities). The shaded bell curve shows exactly what percentage of output falls outside tolerance bands for Six Sigma process improvement.
Convert test scores to percentiles
IQ • Standardized tests • Percentile rank
Enter mean 100 and standard deviation 15 for Wechsler IQ scores, or mean 100 and SD 16 for Stanford-Binet. Reverse Solve mode converts any percentile to the exact raw score, z-score, and position on the bell curve for clinical reporting.
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Built & maintained by Shahbaz Ali Malik Last updated: