Quick answer: pick the result type first
- Given a score x, find probability: use CDF (left/right/two-tailed).
- Given two cutoffs a and b, find the area between them: use Interval probability.
- Given percentile p, find cutoff x: use Inverse CDF (quantile).
- Given x, compare across tests: use Z-score mode to standardize.
For cleaner results, set mu and sigma from your dataset first, then run this calculator for probability or percentile interpretation.
How to use (3 steps)
- Enter the mean and standard deviation, then choose what you want to find (PDF/CDF, interval, percentile, or z-score). The default values use a test with mean 100 and σ = 15.
- Fill in x, the interval [a, b], the percentile p, or the z-score z, and pick left, right, or two-tailed if you are in CDF mode.
- Click “Calculate” to see the numeric result, working steps, and a shaded normal curve. You can copy the result text or a shareable URL for homework or lesson notes.
Common tasks (examples)
- Tail probability: find P(X ≤ x), P(X ≥ x), or a two-tailed probability for exam scores, manufacturing tolerances, or measurement error → use CDF and switch Left/Right/Two-tailed.
- Interval probability: find P(a < X < b) for ranges such as “between 85 and 115” → use Interval probability.
- Percentile / cutoff score: find x for a percentile p (e.g., 0.95) when you need a benchmark, passing threshold, or screening cutoff → use Inverse CDF (quantile).
- Z-score: convert between x and z for a given mean μ and σ when you want to compare results across different tests or scales.
- If you only have raw data: compute mean and standard deviation first with the Mean and Standard Deviation Calculator or Descriptive Statistics.
Interpretation (and when to use a normal model)
- CDF gives the probability P(X ≤ x). For a right tail, use P(X ≥ x) = 1 − CDF. Two-tailed means “as extreme or more extreme than x” on both sides of the mean.
- PDF is a density, not a probability. Probabilities come from areas under the curve (intervals), not from the height at a single point.
- Z-score standardises your value: z = (x − μ)/σ. This converts any normal distribution to the standard normal, so you can compare results across different μ and σ.
- When it fits: the normal distribution is a good model for many measurement errors and approximately symmetric, unimodal data. Averages of many small effects often look more normal than the raw components.
- Limits: if your variable is discrete, strongly skewed, bounded (e.g., 0–1), or has heavy tails/outliers, a normal approximation can be misleading.
Worked examples
- Test scores: μ = 100, σ = 15, x = 120 → z ≈ 1.33, so P(X ≤ 120) ≈ 0.909 (90.9%).
- Middle-range probability: μ = 100, σ = 15, a = 85, b = 115 → the interval is one standard deviation around the mean, so the probability is about 0.683.
- 95th percentile cutoff: μ = 100, σ = 15 → x ≈ 100 + 1.645×15 ≈ 124.7 for p = 0.95.
References
Results
Value: —
Steps
Probabilities are shown both as a 0–1 value and as a percentage. Use these as an educational guide; final decisions should rely on appropriate professional or academic judgement.
Normal curve preview
We plot the range from −4σ to +4σ and highlight the chosen tail or interval in blue, so you can see at a glance which part of the distribution the probability refers to.
How to choose the right normal distribution mode
Use this page when you already know the mean and standard deviation and need a probability, z-score, percentile, or inverse normal cutoff.
Start with the question you are trying to answer
- What is the probability below or above x? Use CDF and choose left-tail or right-tail.
- What is the probability between a and b? Use Interval probability.
- What score corresponds to percentile p? Use Inverse CDF (quantile).
- How unusual is score x? Use Z-score to standardize the value.
Common interpretation mistakes
- Treating the PDF height as a probability for a single point.
- Mixing up a percentile with a tail probability.
- Using a normal model for strongly skewed, bounded, or clearly discrete data without checking whether the approximation is sensible.
- Switching between sample and population assumptions in upstream mean and standard-deviation calculations.
Worked examples
For test scores with μ = 100 and σ = 15, enter x = 120 in CDF mode to get the percentile below 120. If you need the 95th-percentile cutoff instead, switch to Inverse CDF and enter p = 0.95 to recover the score threshold directly.
See also
FAQ
What is a normal distribution?
A normal distribution models a continuous variable that clusters around a mean μ with spread measured by the standard deviation σ. The familiar bell-shaped curve is symmetric: values close to μ are common, while very large or very small values are rare.
What do the mean μ and standard deviation σ represent?
The mean μ is the central or average value, and the standard deviation σ describes how spread out the data are around μ. A small σ means most observations lie close to μ; a large σ means observations are more dispersed.
What is a z-score?
A z-score shows how many standard deviations a value x is above or below the mean: z = (x − μ)/σ. For example, z = 1 means “one σ above the mean”, and z = −2 means “two σ below the mean”. This makes different normal distributions comparable.
How do I find the probability between two values?
Use Interval probability and enter the lower and upper bounds a and b. The calculator converts both endpoints to z-scores and subtracts the corresponding CDF values to get the shaded area between them.
When should I use left-tail, right-tail, or two-tailed probabilities?
Left-tail covers P(X ≤ x), right-tail covers P(X ≥ x), and two-tailed combines both ends where |X−μ| is at least |x−μ|. Use the tail that matches your question, test setup, or classroom example before interpreting the result.
Can I use a normal model for percentages or counts?
Sometimes, but not always. A normal model works best for continuous, roughly symmetric data without hard bounds. If your variable is strongly skewed, discrete, bounded, or dominated by outliers, check whether another model fits better before relying on the result.
How accurate is the inverse CDF (quantile) result?
We rely on Peter Acklam’s rational approximation with a single Newton step. The typical absolute error stays near 10−6, which is more than adequate for teaching, homework, and exam preparation.
How it’s calculated