Understanding your normal-distribution result
The table below lists the standard normal cumulative probability Φ(z) at whole-number z-scores, consistent with standard normal distribution tables.
| Z-score | Φ(z) — cumulative probability below z |
|---|---|
| −3 | 0.13% |
| −2 | 2.28% |
| −1 | 15.87% |
| 0 | 50.00% |
| +1 | 84.13% |
| +2 | 97.72% |
| +3 | 99.87% |
- Empirical rule: about 68.27% of a normal distribution lies within ±1 standard deviation of the mean, about 95.45% within ±2 standard deviations, and about 99.73% within ±3 standard deviations.
- In "between" mode, the upper bound b must be greater than or equal to x; the calculator returns no result if b is smaller than x.
- Because the normal distribution is continuous, the probability of any single exact value is 0 — only ranges (below, above, between) have nonzero probability.
- These calculations assume the variable is genuinely normally distributed. Skewed, bounded, or heavy-tailed data can depart substantially from these probabilities even though a z-score can still be computed.
What is the normal distribution?
The normal distribution (Gaussian distribution) is a continuous, symmetric, bell-shaped probability distribution fully described by two parameters: its mean (μ), which sets the center, and its standard deviation (σ), which sets the spread. Many natural and measured quantities — heights, measurement errors, standardized test scores — are approximately normal, which is why the distribution is central to statistical theory and practice.
Because a continuous distribution assigns probability to ranges rather than single points, this calculator answers questions of the form 'what proportion of values falls below x', 'above x', or 'between x and b' — never 'what is the probability of exactly x', which is always 0 for a continuous distribution.
Every normal distribution can be converted to the standard normal distribution (mean 0, standard deviation 1) by computing a z-score, z = (x − μ) / σ. The probability associated with a range of x-values is then read from the standard normal cumulative distribution function (CDF), often written Φ(z), which gives the proportion of the distribution lying at or below z.
The empirical rule (also called the 68-95-99.7 rule) summarizes the standard normal distribution's spread: about 68.27% of values lie within 1 standard deviation of the mean, about 95.45% lie within 2 standard deviations, and about 99.73% lie within 3 standard deviations.
How to use this normal distribution calculator
- Enter the mean (μ) and standard deviation (σ) of the distribution. The standard deviation must be greater than zero.
- Choose a calculation mode: below a value, above a value, or between two values.
- Enter the value x. In "between" mode, also enter the upper bound b, which must be greater than or equal to x.
- Read the probability (as a percentage and as a proportion between 0 and 1), along with the corresponding z-score for x.
The formula behind normal distribution probability
Every calculation starts by standardizing x into a z-score: z = (x − μ) / σ. The probability is then read from the standard normal CDF, Φ(z), which has no simple closed-form expression and is evaluated numerically.
Below mode: P(X < x) = Φ(z). Above mode: P(X > x) = 1 − Φ(z). Between mode: P(x < X < b) = Φ(zᵦ) − Φ(zₓ), where zᵦ is the z-score of the upper bound b.
Worked example (below mode): mean = 100, standard deviation = 15, x = 120. z = (120 − 100) / 15 = 1.3333. Looking up Φ(1.3333) on the standard normal distribution gives approximately 0.9088, so P(X < 120) ≈ 90.88%.
Worked example (between mode, standard normal): mean = 0, standard deviation = 1, between −1 and +1. z lower = −1, z upper = +1. Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827, so P(−1 < X < 1) ≈ 68.27% — the first step of the empirical rule.
Common mistakes
- Selecting the wrong mode — "below", "above" and "between" give different probabilities for the same x, and mixing them up is a frequent source of error.
- Entering an upper bound b that is smaller than x in "between" mode, which is not a valid range for this calculator.
- Treating the probability of a single exact value as meaningful — for a continuous distribution such as the normal, P(X = x) is always 0.
- Assuming a dataset is normal without checking — the probabilities returned here are only accurate when the underlying variable is approximately normally distributed.
- Confusing standard deviation with variance — this calculator asks for the standard deviation (σ), not its square.
Câu hỏi thường gặp
How do I calculate a normal distribution probability?
Convert the value to a z-score using z = (x − mean) / standard deviation, then look up the corresponding cumulative probability on the standard normal distribution. For a value below x, use Φ(z) directly; for above x, use 1 − Φ(z); for a range between x and b, subtract the two cumulative probabilities.
What is the empirical rule?
The empirical rule (68-95-99.7 rule) states that for a normal distribution, about 68.27% of values fall within 1 standard deviation of the mean, about 95.45% fall within 2 standard deviations, and about 99.73% fall within 3 standard deviations.
What is a z-score?
A z-score expresses how many standard deviations a value lies from the mean, calculated as z = (x − mean) / standard deviation. It is the intermediate step this calculator uses to look up a probability on the standard normal distribution.
Why is the probability of an exact value always 0?
The normal distribution is continuous, so probability is defined only over ranges, not single points. Asking for P(X = 120) exactly is mathematically 0; the meaningful questions are P(X < 120), P(X > 120), or P(a < X < b).
Does this calculator work for any dataset?
Only if the underlying variable is approximately normally distributed. Many real variables are approximately normal, but skewed, bounded (such as counts or proportions), or heavy-tailed data can give misleading probabilities if forced into this framework.
What is the difference between the 'below' and 'between' modes?
"Below" reports the cumulative probability up to a single value x, P(X < x). "Between" reports the probability of falling in a range, P(x < X < b), which is the difference between two cumulative probabilities and requires both a lower value x and an upper bound b.
Tài liệu tham khảo
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.6.1: Normal distribution. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (the normal distribution and standardized values).
- Abramowitz M, Stegun IA (eds). Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series 55, 1964 (standard normal distribution tables).
- Casella G, Berger RL. Statistical Inference (2nd ed). Duxbury, 2002 (the normal distribution and its properties).