Normal Distribution Calculator
Find z-scores, probabilities and percentiles of any normal distribution. Compute P(X ≤ x) and ranges between bounds instantly.
Optional — for P(lower < X < upper).
Z-score
1.5
Probability density
0.013
P(X ≤ x)
93.3193%
P(X > x)
6.6807%
P(lower < X < upper)
81.8595%
Interval
60 – 90
Given lower and upper bounds
Assumes a normal (bell-curve) distribution. The z-score is (x − mean) ÷ standard deviation.
Last updated: January 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Use the normal distribution calculator whenever your data clusters symmetrically around a centre — exam marks, body measurements, blood pressure readings, packaging weights or machine tolerances. Students preparing for entrance tests model their mock-score distributions to estimate where a cutoff lands, and coaching institutes in cities like Kota and Delhi use it to convert raw marks into percentile estimates. Quality engineers check whether a batch's measurements stray beyond spec limits by computing tail probabilities, and HR teams compare salary bands by standardizing values to z-scores. Because the calculator accepts any mean and standard deviation, it works for any approximately normal dataset without tables — you get the same P(X ≤ x), between-range and tail probabilities that printed z-tables provide, computed instantly for your exact numbers.
How to Use This Calculator
- Step 1: Enter the value you are interested in as x, such as a cutoff mark or measurement.
- Step 2: Enter the mean and standard deviation of the distribution — every normal curve needs these two numbers.
- Step 3: Optionally enter lower and upper bounds to find the probability that a value falls inside that range.
- Step 4: Read the z-score, probability density, P(X ≤ x) and P(X > x) from the result cards.
Worked Example
A coaching centre in Kota records mock-test marks that are roughly normal with a mean of 70 and a standard deviation of 10. A student scored 85. Enter x = 85, mean = 70 and sd = 10: the z-score is 1.5, P(X ≤ 85) is about 0.9332, so roughly 93% of students scored 85 or below. Adding bounds of 60 and 90 shows that about 81.9% of scores fall between 60 and 90 — the middle of the distribution where most students land.
Tips and Common Mistakes
- •Tip 1: Set the bounds to mean ± 1, 2 or 3 standard deviations to confirm the 68–95–99.7 rule on your own data.
- •Tip 2: Use P(X > x) when ranking cutoffs — it is the share of candidates scoring above the mark, which decides merit lists.
- •Tip 3: Check the standard deviation is in the same units as the mean, otherwise the z-score will be meaningless.
- ✗Mistake 1: Entering variance in place of standard deviation — the calculator expects the square root of variance.
- ✗Mistake 2: Forgetting the distribution must be approximately normal; skewed data like salaries gives misleading probabilities.
Frequently Asked Questions
What is the normal distribution used for?
The normal (Gaussian) distribution describes how many natural measurements are spread — exam scores, heights, blood pressure and manufacturing tolerances. Most values cluster around the mean and taper off symmetrically on both sides, forming the familiar bell curve. If your data is roughly bell-shaped, this calculator gives the probability of values in any range.
How do I find the probability that a value is below a cutoff?
Enter the value as x, the mean and the standard deviation, and read P(X ≤ x). For example, with a mean of 70 and a standard deviation of 10, a score of 85 has a z-score of 1.5 and a probability of about 0.9332 — meaning roughly 93% of values fall at or below 85.
What does the z-score tell me?
The z-score tells you how many standard deviations a value sits above or below the mean. A z-score of 1.5 means the value is 1.5 standard deviations above average. Positive z-scores are above the mean, negative ones below, and the z-score is the standard way to look up normal probabilities in tables.
What is the 68-95-99.7 rule?
For any normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. You can verify this with the calculator by setting the lower and upper bounds to mean ± 1, 2 or 3 standard deviations. This rule is widely used for quick quality-control and outlier checks.
Does this work for scores in competitive Indian exams?
Yes, when the raw scores are converted to a standard scale the distribution is often approximately normal, which is how percentile calculations in exams like JEE and CAT work. This calculator takes any mean and standard deviation, so you can model your exam's score distribution directly and estimate the percentage of candidates above or below a cutoff.
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