Normal Distribution Calculator
Results
Results
The 68–95–99.7 Rule (Empirical Rule)
For any normal distribution N(μ, σ²), the following proportions hold:
Text diagram of bell curve regions:
|----0.13%----|----2.15%----|----13.59%----|--34.13%--|--34.13%--|----13.59%----|----2.15%----|----0.13%----|
μ-4σ μ-3σ μ-2σ μ-σ μ μ+σ μ+2σ μ+3σ μ+4σ
Standard Normal Table — Key Z-Values
| z | Φ(z) = P(Z < z) | P(Z > z) | Context |
|---|---|---|---|
| -3.00 | 0.0013 | 0.9987 | 3σ below mean |
| -2.58 | 0.0049 | 0.9951 | 1% two-tail critical |
| -2.33 | 0.0099 | 0.9901 | 1% one-tail critical |
| -2.00 | 0.0228 | 0.9772 | 2σ below mean |
| -1.96 | 0.0250 | 0.9750 | 5% two-tail critical |
| -1.645 | 0.0500 | 0.9500 | 5% one-tail critical |
| -1.00 | 0.1587 | 0.8413 | 1σ below mean |
| 0.00 | 0.5000 | 0.5000 | At the mean |
| 1.00 | 0.8413 | 0.1587 | 1σ above mean |
| 1.645 | 0.9500 | 0.0500 | 5% one-tail critical |
| 1.96 | 0.9750 | 0.0250 | 5% two-tail critical |
| 2.00 | 0.9772 | 0.0228 | 2σ above mean |
| 2.33 | 0.9901 | 0.0099 | 1% one-tail critical |
| 2.58 | 0.9951 | 0.0049 | 1% two-tail critical |
| 3.00 | 0.9987 | 0.0013 | 3σ above mean |
Understanding the Normal Distribution
The normal distribution is the most important probability distribution in statistics. It describes the pattern of variation in countless natural measurements — from human heights and weights to exam scores and manufacturing tolerances. Its characteristic bell-shaped curve is perfectly symmetric about the mean, meaning that exactly half the probability lies on each side.
In UK A-Level Statistics (both AQA, Edexcel, and OCR exam boards), the normal distribution is a central topic. Students must be able to calculate probabilities, find z-scores, and apply the distribution to real-world contexts. This calculator covers all the core techniques required at A-Level and beyond.
The Normal Distribution Formula
The probability density function (PDF) of a normal distribution is:
where μ is the mean and σ is the standard deviation.
The cumulative distribution function (CDF), denoted Φ(z), gives the area under the curve from −∞ to z. There is no closed-form expression for Φ(z), so it is computed using numerical approximation. This calculator uses the Hart approximation algorithm, which achieves accuracy to better than 7 significant figures for all practical purposes.
Properties of the Normal Distribution
- Symmetry: The distribution is perfectly symmetric about the mean μ.
- Mean = Median = Mode: All three measures of central tendency coincide at μ.
- Total area = 1: The total probability under the bell curve equals 1 (or 100%).
- Asymptotic tails: The curve never touches zero — it extends to ±∞.
- Inflection points: The curve inflects at μ − σ and μ + σ.
- Scale parameter: Larger σ makes the curve flatter and wider; smaller σ makes it taller and narrower.
Standardisation and Z-Scores
Any normal random variable X ~ N(μ, σ²) can be standardised to a standard normal variable Z ~ N(0, 1) using the formula:
Standardisation is essential because it allows us to use a single standard normal table (z-table) for all normal distributions. Once you have the z-score, you can look up or compute the CDF value Φ(z). In UK exam papers, you will typically use either statistical tables or a calculator — both give the same answer.
The Normal Approximation to the Binomial
When the sample size n is large and the probability p is not too extreme, the binomial distribution B(n, p) can be approximated by a normal distribution N(μ, σ²) where μ = np and σ² = np(1−p). A continuity correction is applied: P(X ≤ k) becomes P(X < k + 0.5) in the normal approximation. The approximation works well when np > 5 and n(1−p) > 5.
Real-World Applications
The normal distribution models many real phenomena. Heights of adult men and women in the UK follow approximately normal distributions. IQ scores are designed to follow N(100, 15²). Manufacturing processes use the normal distribution in quality control, with the capability index measuring how many standard deviations the process mean is from the specification limits. Blood pressure, exam marks, and environmental measurements also commonly follow normal distributions, making this one of the most practically useful tools in statistics.
How to Use This Calculator
Select the type of probability you want to find using the tabs above. For P(X < a) or P(X > a), enter the mean, standard deviation, and the value x. The calculator will output P(X < x), P(X > x), and the z-score. For P(a < X < b), enter both boundary values a and b to find the probability that X lies between them. Results are displayed to 6 decimal places for maximum precision.
Frequently Asked Questions
What is the normal distribution?
The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about the mean. It is defined by two parameters: the mean (μ) and the standard deviation (σ). It is fundamental to statistics, appearing naturally in many real-world phenomena such as heights, test scores, and measurement errors.
What is a z-score and how is it calculated?
A z-score measures how many standard deviations a value X is from the mean μ. The formula is z = (X − μ) / σ. A positive z-score means the value is above the mean; a negative z-score means it is below the mean. Z = 0 means the value equals the mean exactly.
What is the 68-95-99.7 rule?
The empirical rule states that approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule is a quick way to estimate probabilities in a normal distribution and is frequently tested in UK A-Level Statistics.
How do I find P(a < X < b)?
Convert both values to z-scores: z₁ = (a − μ)/σ and z₂ = (b − μ)/σ. Then P(a < X < b) = Φ(z₂) − Φ(z₁), where Φ is the standard normal CDF. This calculator handles all the computation automatically — just enter your values in the P(a < X < b) tab.
What is the standard normal distribution?
The standard normal distribution has mean μ = 0 and standard deviation σ = 1, denoted Z ~ N(0, 1). Any normal variable can be standardised to Z using the z-score formula, allowing a single table to serve all normal distributions.
How is the normal distribution used in UK A-Level Statistics?
A-Level Statistics students must calculate z-scores, look up probabilities in tables, apply the inverse normal, and model real-world scenarios with normal distributions. Exam questions from AQA, Edexcel, and OCR all test these skills regularly. This calculator supports all standard A-Level probability types.
What is the difference between P(X < a) and P(X ≤ a)?
For continuous distributions like the normal distribution, P(X < a) = P(X ≤ a) because the probability of any single exact value is zero. This differs from discrete distributions where individual probability masses exist at each value.