Normal Distribution Calculator

Calculate probabilities, percentiles, and values for the standard normal distribution. Find P(X < x), P(a < X < b), and z-scores for any normal distribution using the bell curve.

📊 Statistics📐 P(X < x) = Φ((x-μ)/σ)🔢 Math
Calculate
Value x (or a for between)
Mean μ
Standard deviation σ
Upper value b (for between only)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Normal Distribution Calculator—P(X < x) = Φ((x-μ)/σ)probability 0-1

Step-by-Step Examples

Example 1
Test Score Probability

Scores: Normal(70, 10). What fraction score above 85?

  • z = (85-70)/10 = 1.50
  • P(X > 85) = 1 - Φ(1.50) = 1 - 0.9332 = 0.0668
  • 6.68% of students score above 85
✓ P(X > 85) = 6.68%
Example 2
Between Two Values

Heights: Normal(170, 8) cm. Fraction between 162 and 178?

  • z1 = (162-170)/8 = -1.00 → Φ(-1) = 0.1587
  • z2 = (178-170)/8 = 1.00 → Φ(1) = 0.8413
  • P(162 < X < 178) = 0.8413 - 0.1587 = 0.6827
  • The famous 68% rule: μ ± 1σ contains 68.27%
✓ P(within 1σ) = 68.27%
Example 3
Percentile Finding

What score is at the 95th percentile? Normal(70,10).

  • 95th percentile: z = 1.645
  • x = μ + z×σ = 70 + 1.645×10 = 86.45
  • 95% of students score below 86.45
✓ 95th percentile = 86.45

Real-World Applications

📊
Quality Control
Normal distribution models manufacturing tolerances. Z-scores identify defective parts.
💊
Medical Diagnosis
Normal ranges for blood tests, height, weight based on normal distribution percentiles.
💼
Finance
Returns of diversified portfolios approximate normal distribution (Central Limit Theorem).
🏫
Standardized Testing
SAT, IQ scores deliberately designed as normal distributions with known parameters.

Common Mistakes to Avoid

⚠️
Using normal distribution for non-normal data

Many phenomena are NOT normally distributed (income is right-skewed, wait times are exponential). Check normality with histograms or normality tests before assuming normal.

⚠️
Confusing X and Z

Always standardize: z = (x - μ)/σ before looking up probabilities. Look up z, not raw x values, in the standard normal table.

⚠️
Off-by-one in tails

P(X > x) = 1 - P(X < x). P(X = x) = 0 for continuous distributions (exact probability of any single value is zero).

Frequently Asked Questions

What is the normal distribution? ▾
A symmetric, bell-shaped probability distribution defined by mean μ (center) and standard deviation σ (spread). About 68% of data falls within μ±σ, 95% within μ±2σ, 99.7% within μ±3σ.
What is the standard normal distribution? ▾
Normal distribution with μ=0 and σ=1. Any normal distribution can be converted to standard normal via z = (x-μ)/σ. Standard normal tables (z-tables) give Φ(z) = P(Z
What is the Central Limit Theorem? ▾
The distribution of sample means approaches normal as n increases, regardless of the original population distribution. Key reason normal distribution is so widely applicable.
What are the 68-95-99.7 rules? ▾
μ±1σ: 68.27% of data. μ±2σ: 95.45%. μ±3σ: 99.73%. These are exact results for normal distributions, not approximations.
When is normal distribution not appropriate? ▾
Skewed data (income, prices), count data (Poisson), proportions (binomial), time-to-event (exponential/Weibull). Use appropriate distributions for these data types.

Related Math Calculators

📊 Z-score📊 P-Value📊 Percentile📊 Binomial Distribution📊 Confidence Interval

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.