Enter Values

Population or sample mean
Spread of the distribution (must be > 0)
Value to convert or find probability for
Start of the probability range
End of the probability range
Enter as decimal (e.g., 0.95 for 95th percentile)
Common Z-Values
Confidence Z-Score Tail Area
90% ±1.645 5% each
95% ±1.960 2.5% each
99% ±2.576 0.5% each
z = (x - μ) / σ

Calculation Result

Z-Score 1.96
Z-Score 1.96
Left Tail P(X≤x) 0.9750
Right Tail P(X>x) 0.0250
σ from Mean 1.96

Formula Breakdown

Z-Score: z = (x - μ) / σ

Interpretation

The 68-95-99.7 Rule

±1σ
68.27%
±2σ
95.45%
±3σ
99.73%
Ryan O'Connell, CFA
CALCULATOR BY
Ryan O'Connell, CFA
CFA Charterholder & Finance Educator

Finance professional building free tools for options pricing, valuation, and portfolio management.

About YouTube

Understanding the Normal Distribution

What is the Normal Distribution?

The normal distribution (also called Gaussian distribution) is a symmetric, bell-shaped probability distribution that is fundamental to statistics and finance. It is completely described by two parameters:

  • Mean (μ): The center of the distribution
  • Standard Deviation (σ): The spread or width of the distribution
Probability Density Function
f(x) = (1 / σ√2π) × e-(x-μ)²/(2σ²)

Z-Scores and Standardization

A z-score measures how many standard deviations a value is from the mean. Converting to z-scores allows comparison across different normal distributions:

  • z = 0: Value equals the mean (50th percentile)
  • z = 1: One standard deviation above (84th percentile)
  • z = -1: One standard deviation below (16th percentile)
  • z = 1.96: Approximately 97.5th percentile (used for 95% confidence intervals)

The Cumulative Distribution Function (CDF)

The CDF, denoted Φ(z), gives the probability that a random variable is less than or equal to a given value:

  • Left-tail probability: P(X ≤ x) = Φ(z)
  • Right-tail probability: P(X > x) = 1 - Φ(z)
  • Between probability: P(a ≤ X ≤ b) = Φ(zb) - Φ(za)

Note: For continuous distributions, P(X > x) = P(X ≥ x) since P(X = x) = 0.

Applications in Finance

Value at Risk (VaR)

Normal distribution is used to estimate potential portfolio losses at a given confidence level (e.g., 95% VaR uses z = 1.645).

Option Pricing

The Black-Scholes model assumes log-returns are normally distributed. The d1 and d2 terms use the standard normal CDF.

Important: Financial returns often exhibit fat tails (more extreme events than normal predicts) and skewness. The normal distribution may underestimate tail risks. Consider this limitation in risk management applications.
Download This Calculator as an Excel Template Interactive model with editable formulas — customize, save, and share.
Get Excel Template

Frequently Asked Questions

A normal distribution (also called Gaussian distribution) is a symmetric, bell-shaped probability distribution defined by two parameters: the mean (mu) which determines the center, and the standard deviation (sigma) which determines the spread. About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

A z-score measures how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean. Positive z-scores indicate values above the mean, negative z-scores indicate values below. For example, a z-score of 2 means the value is 2 standard deviations above the mean, occurring in roughly the top 2.5% of a normal distribution.

To find probability from a z-score, use the cumulative distribution function (CDF). The CDF gives P(X ≤ x), the probability that a random variable is less than or equal to x. For P(X > x), subtract from 1. For P(a ≤ X ≤ b), calculate CDF(b) minus CDF(a). This calculator handles all these calculations automatically.

The 68-95-99.7 rule (empirical rule) states that for a normal distribution: 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 helps quickly estimate probabilities without detailed calculations.

The normal distribution is widely used in finance for modeling stock returns, calculating Value at Risk (VaR), option pricing (Black-Scholes model), portfolio optimization, and hypothesis testing. However, be aware that financial returns often exhibit fat tails and skewness, so the normal assumption may underestimate extreme events.

The inverse normal function (probit function) finds the value corresponding to a given probability. It answers: "What value has X% of the distribution below it?" This is essential for calculating confidence intervals, critical values for hypothesis tests, and percentile-based risk measures like VaR.
Model Assumptions
  • Data follows a normal (Gaussian) distribution
  • Population parameters (μ, σ) are known or estimated accurately
  • Observations are independent and identically distributed
  • The normal approximation is appropriate for your use case

For educational purposes only. Not financial advice. Results should be verified for critical applications. The normal distribution may not accurately model financial data with fat tails or skewness.

Related Calculators

Professional finance illustration representing Hypothesis Testing Calculator

Hypothesis Testing Calculator

Perform statistical hypothesis tests with critical values.

Try Calculator →
Professional finance illustration representing Value at Risk (VaR) Calculator

Value at Risk (VaR) Calculator

Calculate parametric, historical, and Monte Carlo Value at Risk for any portfolio. Estimate potential losses...

Try Calculator →
Professional finance illustration representing Black Scholes Calculator

Black Scholes Calculator

Price European call and put options using the Black-Scholes model. Calculate option values, Greeks, and...

Try Calculator →