Normal Distribution Calculator

Find normal distribution probabilities and critical values for any mean and standard deviation, instantly.

📊 Normal Distribution Calculator
Mean (μ)
Standard Deviation (σ)
Probability Type
X Value
Lower Bound (a)
Upper Bound (b)
Mean (μ)
Standard Deviation (σ)
Probability (%)
Tail Direction
P(X < x)
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P(X > x)
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Z-Score
—
PDF f(x)
—
X Value
—
Z-Score
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Verification P(X < x)
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📊 What is the Normal Distribution?

The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes the spread of a continuous random variable that clusters symmetrically around a central mean, with the probability of values decreasing smoothly as you move away from the center. The shape is completely determined by two parameters: the mean (mu), which controls the center, and the standard deviation (sigma), which controls the spread. A wider bell means higher standard deviation; a narrower, taller bell means lower standard deviation.

The normal distribution appears throughout science, engineering, finance, and social science. Heights and weights in a population, measurement errors in scientific instruments, IQ scores (standardized to mean 100, SD 15), blood pressure readings, exam score distributions, and the daily returns of financial assets are all commonly modeled as approximately normal. More fundamentally, the Central Limit Theorem guarantees that the mean of any sufficiently large sample follows a normal distribution regardless of the underlying population distribution, which is why the normal distribution is central to statistical inference.

A common misconception is that the normal distribution is always the right model. Many real-world distributions have heavier tails (more extreme values) than the normal, including financial returns, income distributions, and many biological measurements. The normal distribution is also bounded by the symmetric bell shape, whereas data like heights, weights, and prices must be positive. Despite these limitations, the normal distribution provides an excellent approximation for many purposes, and its mathematical tractability makes it indispensable in statistics.

This calculator covers the two most common normal distribution computations. The Find Probability mode calculates the area under the normal curve to the left of x (left-tail CDF), to the right of x, between two values, or outside a range. The Find X (Inverse) mode solves the reverse problem: given a probability, find the x value at which the cumulative area equals that probability. This is used for computing percentiles, critical values, and confidence interval boundaries.

📐 Formula

z = (x − μ) ÷ σ
z = standard score (z-score)
x = observed value
μ = population mean
σ = population standard deviation

Cumulative distribution function (CDF) giving P(X < x):

P(X < x) = Φ(z) = ½ × [1 + erf(z ÷ √2)]
Φ(z) = standard normal CDF at z
erf = error function (numerical approximation)
P(X > x) = 1 − P(X < x)
P(a < X < b) = Φ(z⊂b;) − Φ(z⊂a;) where z⊂a; = (a−μ)/σ, z⊂b; = (b−μ)/σ
Example: μ = 100, σ = 15, x = 115. z = (115−100)/15 = 1.0. P(X < 115) = Φ(1.0) = 84.13%

Probability density function (PDF, the height of the bell curve at x):

f(x) = (1 ÷ (σ × √(2π))) × e−(x−μ)² ÷ (2σ²)
f(x) = probability density at point x (not a probability itself)
e = Euler's number (approximately 2.71828)
π = pi (approximately 3.14159)

📖 How to Use This Calculator

Steps

1
Enter mean and standard deviation - type the population mean and standard deviation (must be positive) for your normal distribution. For the standard normal distribution, use mean = 0 and SD = 1.
2
Choose probability type and enter X - in Probability mode, select Left Tail P(X less than x), Right Tail P(X greater than x), Between P(a less than X less than b), or Outside a range. Enter the x value or range bounds, then click Calculate.
3
Read results or switch to Inverse mode - results show the probability, complement, z-score, and PDF value. Switch to Find X (Inverse) mode to enter a probability percentage and find the x value at which the cumulative probability equals that amount.

💡 Example Calculations

Example 1 - IQ Score Probability (Left Tail)

IQ scores: mean = 100, SD = 15. What fraction of people score below 115?

1
Compute z-score: z = (115 - 100) / 15 = 15 / 15 = 1.000. This is 1 standard deviation above the mean.
2
Look up P(Z less than 1.000): CDF at z = 1.0 is 84.13%. So approximately 84% of people score below 115.
3
Complement: P(IQ greater than 115) = 1 - 84.13% = 15.87%. About 1 in 6 people score above 115.
P(X < 115) = 84.13% | P(X > 115) = 15.87%
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Example 2 - Empirical Rule Verification (Between)

Standard normal: what percentage of values fall within 2 standard deviations of the mean?

1
For the standard normal N(0, 1), "within 2 SDs" means the range a = -2 to b = 2. Both z-scores are already -2 and +2.
2
P(X less than 2) = Phi(2.0) = 97.725%. P(X less than -2) = Phi(-2.0) = 2.275%.
3
P(-2 less than X less than 2) = 97.725% - 2.275% = 95.45%. This confirms the "95%" part of the 68-95-99.7 empirical rule.
P(-2 < X < 2) = 95.45% for N(0,1)
Try this example →

Example 3 - Finding the 90th Percentile (Inverse)

Exam scores: mean = 70, SD = 10. What score is at the 90th percentile?

1
Use Inverse mode. Set mean = 70, SD = 10, probability = 90%, Left Tail direction.
2
The standard normal quantile for 90% is z = 1.2816. Convert to the exam scale: x = mean + z * SD = 70 + 1.2816 * 10 = 70 + 12.816 = 82.82.
3
Verification: P(X less than 82.82) = Phi((82.82 - 70) / 10) = Phi(1.2816) = 90.0%. Correct.
90th percentile score = 82.82
Try this example →

❓ Frequently Asked Questions

What is the normal distribution formula?+
The normal distribution PDF is f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-(x-mu)^2 / (2*sigma^2)). The CDF, which gives P(X less than x), is expressed as Phi((x-mu)/sigma) where Phi is the standard normal CDF. The PDF tells you the relative likelihood (density) at a specific x; the CDF gives the probability of being at or below x. This calculator uses the Abramowitz and Stegun error function approximation, accurate to within 1.5e-7.
How do I find P(X less than x) for any normal distribution?+
Compute z = (x - mean) / SD, then look up the standard normal CDF at z. For example, mean = 100, SD = 15, x = 115: z = (115-100)/15 = 1.0. Phi(1.0) = 0.8413, so P(X less than 115) = 84.13%. This calculator performs all steps automatically. Enter mean, SD, and x in Probability mode with Left Tail selected, then click Calculate.
What is the difference between left tail and right tail probability?+
Left tail P(X less than x) is the area under the bell curve to the left of x, representing the probability that a value falls below x. Right tail P(X greater than x) is the area to the right, representing the probability of exceeding x. They always sum to 1. For x equal to the mean, both are exactly 50%. Left-tail probability is what standard z-tables report; right-tail probability is used for upper-tailed hypothesis tests.
How is the inverse normal distribution calculated?+
The inverse normal (quantile function) finds x given P(X less than x) = p. Step 1: find the standard normal quantile z such that Phi(z) = p using the inverse error function. Step 2: convert to the original scale: x = mean + z * SD. For example, 95th percentile of N(100, 15): z for 95% is 1.6449, so x = 100 + 1.6449 * 15 = 124.7. This calculator uses the Beasley-Springer-Moro rational approximation for high accuracy.
What is the 68-95-99.7 rule for normal distributions?+
The empirical rule states that for any normal distribution: 68.27% of values fall within 1 standard deviation of the mean, 95.45% fall within 2 standard deviations, and 99.73% fall within 3 standard deviations. This applies regardless of the specific mean and SD values. You can verify this using the Between mode: set mean = 0, SD = 1, and enter range -1 to 1 to get 68.27%, or -2 to 2 for 95.45%.
What z-score corresponds to 95% confidence in a two-tailed test?+
For a 95% two-tailed confidence interval or hypothesis test, you need 2.5% in each tail, so the critical z-score is 1.96 (more precisely, 1.9599...). Common critical values: z = 1.645 for 90% two-tailed (5% in each tail); z = 1.96 for 95% two-tailed; z = 2.576 for 99% two-tailed. Use the Inverse mode: enter probability = 97.5%, Left Tail to get z = 1.96 for the standard normal (mean = 0, SD = 1).
What is the standard normal distribution and how is it different?+
The standard normal distribution is the special case N(0, 1): mean = 0, standard deviation = 1. Any normal distribution N(mu, sigma^2) can be converted to standard normal by computing z = (x - mu) / sigma. Standard normal tables (z-tables) give CDF values for the standard normal. This calculator works with any normal distribution by converting to z internally, so you do not need to standardize manually.
How accurate is this normal distribution calculator?+
This calculator uses the Abramowitz and Stegun (1964) polynomial approximation to the error function, which has a maximum absolute error of 1.5 x 10^-7. For the inverse CDF, it uses the Beasley-Springer-Moro rational approximation. Both are accurate to approximately 6 to 7 significant figures, which is sufficient for all practical statistics and scientific applications. For comparison, standard 4-decimal z-tables have precision to 0.0001, while this calculator gives 4 to 6 significant figures.
Can I calculate P(a less than X less than b) for any normal distribution?+
Yes. The probability between two values a and b is P(a less than X less than b) = Phi((b-mu)/sigma) - Phi((a-mu)/sigma), where Phi is the standard normal CDF. Select the Between option in Probability mode, enter the mean and standard deviation, then input the lower bound a and upper bound b. The calculator shows the probability in the range and the outside probability (1 minus the between probability).
What does the PDF value (probability density) mean?+
The PDF (probability density function) value f(x) is the height of the bell curve at point x. Unlike probabilities, PDF values can exceed 1. The PDF value itself is not a probability but a density; the probability of being in a small interval [x, x+dx] is approximately f(x) * dx. The PDF peaks at the mean (where f(mu) = 1/(sigma*sqrt(2*pi))) and is useful for understanding the relative likelihood of values. The area under the entire PDF equals exactly 1.
How is the normal distribution used in hypothesis testing?+
In hypothesis testing, you compute a test statistic (such as a z-score) and compare it to the normal distribution to obtain a p-value. The p-value is the probability of observing a test statistic at least as extreme as the one obtained, assuming the null hypothesis is true. For a right-tailed test, p-value = P(Z greater than z_observed). For a two-tailed test, p-value = 2 * P(Z greater than |z_observed|). Use the Right Tail or appropriate probability type in this calculator to compute p-values for z-tests.
Tags: Normal Distribution Calculator Bell Curve Calculator Z-Score Probability CDF Normal Distribution Inverse Normal Distribution Calculator Formula Free Online

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What is the normal distribution formula?

The normal distribution PDF is f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-(x-mu)^2 / (2*sigma^2)). The CDF, which gives P(X < x), has no closed form but is expressed as Phi((x-mu)/sigma) where Phi is the standard normal CDF. In practice, CDF values are obtained from tables or numerical approximations. This calculator uses the Abramowitz and Stegun error function approximation accurate to within 1.5e-7.

How do I find P(X < x) for a normal distribution?

To find P(X < x): (1) Calculate the z-score: z = (x - mean) / standard_deviation. (2) Look up the standard normal CDF at z, written as Phi(z). For example, mean = 100, SD = 15, x = 115. z = (115-100)/15 = 1.0. Phi(1.0) = 0.8413, so P(X < 115) = 84.13%. This calculator does all steps automatically.

What is the difference between a z-score and a normal distribution probability?

A z-score is a standardized value: z = (x - mean) / SD. It tells you how many standard deviations x is from the mean. A probability (from the CDF) is the area under the normal curve to the left of that z-score. They are related: a z-score of 1.645 corresponds to a left-tail probability of 95%. One is a location on the x-axis; the other is an area (probability).

What does P(X > x) mean for a normal distribution?

P(X > x) is the right-tail probability, or the probability that a randomly selected value from the distribution exceeds x. Because the total probability is 1, P(X > x) = 1 - P(X < x). For example, if P(X < 115) = 84.13%, then P(X > 115) = 15.87%. In hypothesis testing, P(X > x) is the one-tailed p-value for testing whether an observed value is significantly above the mean.

How do I calculate the inverse normal distribution?

The inverse normal (quantile function or probit) answers: for what x does P(X < x) = p? To find it: (1) Find the standard normal quantile z such that Phi(z) = p using a table or this calculator's Inverse mode. (2) Convert back to x: x = mean + z * SD. For example, for the 90th percentile of a distribution with mean 50 and SD 10: z for 90% is 1.2816, so x = 50 + 1.2816 * 10 = 62.8.

What is the 68-95-99.7 rule for the normal distribution?

The empirical rule states that for a normal distribution: approximately 68.27% of values fall within 1 standard deviation of the mean (mu minus sigma to mu plus sigma); approximately 95.45% fall within 2 standard deviations; and approximately 99.73% fall within 3 standard deviations. These percentages apply to any normal distribution regardless of the specific mean and SD values. Use the Between mode in this calculator to verify: for mean = 0, SD = 1, range -1 to 1 gives 68.27%.

How is the normal distribution used in real life?

The normal distribution appears in many natural and social phenomena: IQ scores are standardized to mean 100, SD 15; heights in a population approximate a normal distribution; measurement errors in scientific instruments follow a normal distribution; financial returns are often approximated as normal (though fat tails are common in practice); quality control uses the normal distribution to set tolerance limits. The central limit theorem guarantees that sample means are approximately normally distributed for large samples, making the normal distribution central to statistics.

What is the standard normal distribution?

The standard normal distribution is the special case of the normal distribution with mean 0 and standard deviation 1. It is written N(0, 1). Any normal distribution N(mu, sigma^2) can be converted to standard normal by the z-score transform: z = (x - mu) / sigma. Standard normal tables (z-tables) give P(Z < z) for a range of z values. This calculator works with any normal distribution by converting to z internally.

What z-score corresponds to a 95% confidence level?

For a 95% confidence interval (two-tailed), you need the z-score that leaves 2.5% in each tail. This is z = 1.96 (more precisely, 1.9599...). For a one-tailed 95% test, the critical z-score is 1.645 (P(Z < 1.645) = 95%). Common critical values: 90% two-tailed: z = 1.645; 95% two-tailed: z = 1.96; 99% two-tailed: z = 2.576. Use the Inverse mode in this calculator: enter probability 97.5% and left-tail direction to get 1.96.

Can I use this calculator for a t-distribution instead of normal?

No. This calculator is for the normal distribution only. The t-distribution has heavier tails than the normal and is characterized by degrees of freedom (df). When the sample size is large (df above 30), the t-distribution approximates the normal distribution closely. For small samples, use the t-Test Calculator instead. The critical t-value at 95% confidence converges to 1.96 as df increases to infinity.

Why is the normal distribution called the bell curve?

The probability density function (PDF) of the normal distribution produces a symmetric, bell-shaped curve when plotted. The peak of the bell is at the mean (mu), and the curve tapers off toward zero in both directions. The width of the bell is controlled by the standard deviation (sigma): a larger sigma produces a wider, flatter bell; a smaller sigma gives a narrower, taller bell. Despite this shape, the area under the entire bell always equals exactly 1, since it represents total probability.

📌 Quick Tips

💡The standard normal distribution has mean 0 and standard deviation 1. For any other normal distribution, this calculator converts to a z-score first using z = (x - mean) / SD, then looks up the probability from the standard normal table.
💡The 68-95-99.7 rule (empirical rule) states that 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. You can verify this: set mean = 0, SD = 1, and use the Between mode with a = -1, b = 1 to see 68.27%.
💡For the inverse calculation, entering a probability of 97.5% with left-tail gives 1.96 standard deviations above the mean, which is the critical value commonly used in 95% two-tailed hypothesis tests.
💡When comparing two normal distributions, standardize both to z-scores first. The z-score tells you how many standard deviations an observation is from its distribution's mean, making values from different distributions directly comparable.
💡The normal distribution is symmetric around the mean, so P(X > x) always equals P(X < 2*mean - x). If you know the left-tail probability, the right-tail probability is always 1 minus that value.