Normal Distribution Calculator
Calculate probabilities, z-scores, and percentiles for the normal (Gaussian) distribution. Find P(X ≤ x), probability density, or inverse normal values. See also our Z-Score Calculator and Standard Deviation Calculator.
How to Use the Normal Distribution Calculator
The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes how data clusters around a mean value with a characteristic bell-shaped curve. Approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations — this is the empirical rule (68-95-99.7 rule).
This calculator supports three modes: (1) CDF mode calculates the cumulative probability P(X ≤ x) — the area under the curve to the left of x; (2) PDF mode calculates the probability density function value at a specific point; (3) Inverse mode finds the x-value corresponding to a given cumulative probability (percentile). Enter the mean and standard deviation of your distribution, then the x-value or probability depending on the mode.
The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Any normal distribution can be converted to the standard normal using the z-score transformation: Z = (X - μ)/σ. This allows you to use standard normal tables or this calculator for any normal distribution regardless of its parameters. The z-score tells you how many standard deviations a value is from the mean.
Normal Distribution Formulas
Probability Density Function (PDF):
f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
Cumulative Distribution Function (CDF):
F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt
Z-Score Transformation:
Z = (X - μ) / σ
Empirical Rule:
P(μ-σ ≤ X ≤ μ+σ) ≈ 0.6827 (68.27%)
P(μ-2σ ≤ X ≤ μ+2σ) ≈ 0.9545 (95.45%)
P(μ-3σ ≤ X ≤ μ+3σ) ≈ 0.9973 (99.73%)
Properties:
Mean = Median = Mode = μ
Variance = σ², Skewness = 0, Kurtosis = 3
Example Calculation
IQ scores follow a normal distribution with μ = 100 and σ = 15. What percentage of people have an IQ above 130?
Given: μ = 100, σ = 15, x = 130
Z = (130 - 100)/15 = 30/15 = 2.0
P(X ≤ 130) = Φ(2.0) = 0.9772
P(X > 130) = 1 - 0.9772 = 0.0228 = 2.28%
About 2.28% of people have IQ above 130
Inverse: What IQ is the 95th percentile?
Z₀.₉₅ = 1.645
X = μ + Zσ = 100 + 1.645×15 = 124.67
Standard Normal Z-Table
| Z-Score | P(Z ≤ z) | P(Z > z) | Common Use |
|---|---|---|---|
| 0.0 | 0.5000 | 0.5000 | — |
| 0.5 | 0.6915 | 0.3085 | — |
| 1.0 | 0.8413 | 0.1587 | — |
| 1.28 | 0.8997 | 0.1003 | — |
| 1.5 | 0.9332 | 0.0668 | — |
| 1.645 | 0.9500 | 0.0500 | 90% CI (one-tail 5%) |
| 1.96 | 0.9750 | 0.0250 | 95% CI (two-tail 2.5%) |
| 2.0 | 0.9772 | 0.0228 | — |
| 2.326 | 0.9900 | 0.0100 | 99% CI (one-tail 1%) |
| 2.5 | 0.9938 | 0.0062 | — |
| 2.576 | 0.9950 | 0.0050 | 99% CI (two-tail 0.5%) |
| 3.0 | 0.9987 | 0.0013 | — |
Applications of the Normal Distribution
The normal distribution appears throughout science and engineering: in physics (thermal noise, measurement uncertainty, Brownian motion), biology (height, weight, enzyme activity), psychology (IQ scores, reaction times), finance (stock returns under certain models, risk assessment), manufacturing (quality control, tolerance analysis), and education (standardized test scores). Understanding the normal distribution is fundamental to hypothesis testing, confidence intervals, and statistical inference.
When to Use the Normal Distribution
Use when:
- Data is continuous and approximately symmetric around the mean (bell-shaped histogram)
- The variable results from many small additive random effects (Central Limit Theorem)
- Working with sample means from large samples (n > 30), regardless of population shape
- Modeling measurement errors, biological measurements (height, weight), or IQ scores
- Performing z-tests or constructing confidence intervals for means
Do NOT use when:
- Data is bounded (e.g., percentages 0-100, counts that cannot be negative)
- Distribution is heavily skewed (use log-normal, gamma, or Weibull instead)
- Data is discrete counts (use Poisson or binomial instead)
- Data has heavy tails with frequent extreme values (use t-distribution or Cauchy)
- Sample size is small and you cannot verify normality (use non-parametric methods)
Additional Solved Examples
Example: Manufacturing Tolerance
A machine fills cereal boxes with mean 500g and standard deviation 8g (normally distributed). What percentage of boxes weigh between 488g and 516g?
Z_lower = (488 - 500)/8 = -1.5
Z_upper = (516 - 500)/8 = 2.0
P(Z < 2.0) = 0.9772
P(Z < -1.5) = 0.0668
P(-1.5 < Z < 2.0) = 0.9772 - 0.0668 = 0.9104
Answer: Approximately 91.04% of boxes weigh between 488g and 516g. This information helps quality control determine whether the filling process meets specifications.
Example: Finding a Cutoff Score
A company wants to select the top 5% of applicants on an aptitude test. If scores are normally distributed with mean 70 and SD 12, what is the minimum passing score?
Need the 95th percentile (top 5% cutoff)
Z for 95th percentile = 1.645
X = 70 + 1.645(12) = 70 + 19.74 = 89.74
Answer: The minimum passing score should be set at approximately 90 (rounding up). Only 5% of applicants would be expected to score at or above this threshold.
Practice Questions
Question 1
Light bulb lifespans are normally distributed with mean 1200 hours and SD 100 hours. What proportion of bulbs last more than 1400 hours?
Answer: Z = (1400-1200)/100 = 2.0. P(Z > 2.0) = 1 - 0.9772 = 0.0228. About 2.28% of bulbs last longer than 1400 hours.
Question 2
Newborn weights are normal with mean 3.4 kg and SD 0.5 kg. What weight range contains the middle 90% of newborns?
Answer: Middle 90% leaves 5% in each tail. Z = +/-1.645. Lower = 3.4 - 1.645(0.5) = 2.58 kg. Upper = 3.4 + 1.645(0.5) = 4.22 kg. The middle 90% of newborns weigh between 2.58 and 4.22 kg.
Question 3
If X is normal with mean 100 and SD 15, find P(85 < X < 115).
Answer: Z_lower = (85-100)/15 = -1.0. Z_upper = (115-100)/15 = 1.0. P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6827. About 68.27% of values fall within one standard deviation, consistent with the empirical rule.
Key Takeaways
- The normal distribution is fully described by two parameters: mean (center) and standard deviation (spread).
- The empirical rule (68-95-99.7) provides quick probability estimates for values within 1, 2, or 3 SDs of the mean.
- Any normal distribution can be standardized using Z = (X-mean)/SD to use standard normal tables.
- The Central Limit Theorem makes the normal distribution applicable to sample means regardless of population shape.
- Always verify approximate normality before using normal-based methods on raw data (histograms, Q-Q plots, Shapiro-Wilk test).
Frequently Asked Questions
What is the normal distribution?
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (μ) which determines the center, and the standard deviation (σ) which determines the spread. Many natural phenomena follow a normal distribution due to the Central Limit Theorem — the sum of many independent random variables tends toward a normal distribution regardless of the original distributions.
What is the difference between PDF and CDF?
The PDF (probability density function) gives the relative likelihood of a continuous random variable taking a specific value — it is the height of the bell curve at that point. The CDF (cumulative distribution function) gives the probability that the variable is less than or equal to a value — it is the area under the curve to the left of that point. For continuous distributions, P(X = x) = 0; only intervals have non-zero probability, which is why we use the CDF.
How do I interpret a z-score?
A z-score tells you how many standard deviations a value is from the mean. Z = 0 means the value equals the mean. Z = 1 means one standard deviation above the mean (84th percentile). Z = -2 means two standard deviations below the mean (2.3rd percentile). Z-scores allow comparison across different normal distributions — a z-score of 2 is equally unusual whether the original data is heights, test scores, or temperatures.
When should I use the normal distribution?
Use the normal distribution when: (1) data is continuous and symmetric around the mean; (2) the Central Limit Theorem applies (sample means of large samples); (3) measurement errors; (4) natural biological measurements (height, weight, blood pressure); (5) quality control (process variation). Do NOT use it for: skewed data, count data (use Poisson), binary outcomes (use binomial), or data with hard boundaries (like income, which is right-skewed).
What is the 68-95-99.7 rule?
The empirical rule states that for a normal distribution: approximately 68.27% of data falls within ±1σ of the mean, 95.45% within ±2σ, and 99.73% within ±3σ. This means values beyond 3σ are extremely rare (0.27% or about 1 in 370). In quality control, the "six sigma" standard means defects beyond 6σ — only 3.4 per million opportunities. This rule provides quick probability estimates without calculation.
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population distribution. For most distributions, n ≥ 30 is sufficient. The sampling distribution has mean = population mean and standard error = σ/√n. This is why the normal distribution is so important — it applies to sample means even when individual data is not normally distributed.