Normal Distribution Calculator
Calculate probabilities for the normal distribution using z-scores.
About Normal Distribution Calculator
The normal distribution is the bell curve. About 68% of data falls within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.
$$P(X \leq x) = \Phi\left(\frac{x-\mu}{\sigma}\right)$$
How to use this calculator
- Enter the mean \(\mu\), which is the center of the distribution.
- Enter the standard deviation \(\sigma\), which measures how spread out the data are.
- Type the value \(x\) you want to evaluate.
- Read the probability \(P(X \le x)\), which is the area under the curve to the left of \(x\).
The formula explained
The formula \(P(X \le x) = \Phi\left(\frac{x-\mu}{\sigma}\right)\) computes the cumulative probability that a normal random variable is at or below a given value. The expression inside \(\Phi\) is the z-score, which tells how many standard deviations \(x\) is from the mean.
- x = the value you are finding the probability for
- \(\mu\) = the mean, or average, of the normal distribution
- \(\sigma\) = the standard deviation, which measures spread
- \(\Phi\) = the standard normal cumulative distribution function
- \(P(X \le x)\) = the probability that the random variable is less than or equal to \(x\)
Step by step method
- Find the z-score with \(z = \frac{x-\mu}{\sigma}\).
- Use the z-score to locate the corresponding cumulative probability in the standard normal distribution.
- Interpret that probability as the area to the left of \(x\) under the bell curve.
Worked example
Problem. A test score is normally distributed with mean \(\mu = 70\) and standard deviation \(\sigma = 8\). What is the probability that a randomly chosen score is at most \(78\)?
- Compute the z-score: \(z = \frac{78-70}{8} = 1\).
- Find the standard normal cumulative probability: \(\Phi(1) \approx 0.8413\).
- So \(P(X \le 78) \approx 0.8413\), which means about 84.13% of scores are at or below 78.
Answer. \(P(X \le 78) \approx 0.8413\)
Tips and common mistakes
- Make sure \(\sigma\) is positive, because a standard deviation of 0 is not valid for a normal distribution.
- This calculator gives the probability to the left of \(x\). If you need the probability to the right, subtract the result from 1.
Frequently asked questions
How do I use the normal distribution calculator with a mean and standard deviation?+
Enter the mean \(\mu\), the standard deviation \(\sigma\), and the value \(x\) you want to evaluate. The calculator converts \(x\) to a z-score using \((x-\mu)/\sigma\), then finds the probability \(P(X \le x)\).
What does the formula P(X ≤ x) = Φ((x - μ)/σ) mean?+
It means you first standardize the value \(x\) by measuring how many standard deviations it is from the mean. Then \(\Phi\) gives the cumulative probability to the left of that z-score, so the result is the area under the normal curve up to \(x\).
Can this calculator find probabilities for values greater than x or between two values?+
This tool is set up for left-tail probabilities, \(P(X \le x)\). For \(P(X > x)\), use \(1 - P(X \le x)\), and for an interval like \(P(a \le X \le b)\), subtract the left-tail probabilities at \(b\) and \(a\).
What happens if the standard deviation is zero or negative?+
A standard deviation must be positive, because it measures spread. If \(\sigma = 0\), the normal distribution formula does not apply, and if \(\sigma < 0\), the input is invalid.
How should I interpret the result from a normal distribution example?+
If the calculator returns, for example, 0.8413, that means about 84.13% of values are at or below the given \(x\). In a normal model, this is the proportion of the total area under the curve to the left of that value.
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