The Standardized Precipitation Index (SPI) is normally computed from a long-term precipitation record by fitting a probability distribution to accumulated precipitation (commonly a gamma distribution) and then transforming the cumulative probability to a standard normal value.

Standardized Precipitation Index (SPI) Calculator

Z-score Approximation
From Monthly Series (Gamma SPI)

Z‑score approximation: enter any 3 values to calculate the missing variable (z = (P − mean)/SD)

Related Calculators

Standardized Precipitation Index Formula

The Standardized Precipitation Index (SPI) is computed by converting the cumulative probability of an observed accumulated precipitation amount to a standard normal value (dimensionless). A common approach fits a gamma distribution to accumulated precipitation totals and then applies a normal (Gaussian) inverse CDF transform.

SPI = \Phi^{-1}(H(P)), \quad H(P)=q + (1-q)\,G(P;\alpha,\beta)

Variables:

  • SPI is the Standardized Precipitation Index (dimensionless; standard normal units)
  • P is the accumulated precipitation over the chosen timescale (any consistent unit, e.g., mm)
  • G(P; α, β) is the gamma cumulative distribution function (CDF) evaluated at P with shape α and scale β (fit from a long-term record for the same timescale)
  • q is the probability of zero precipitation for that timescale (fraction of accumulated totals equal to 0 in the record), used because the gamma distribution is defined for P > 0
  • Φ−1(·) is the inverse CDF of the standard normal distribution

Note: If you only standardize using a mean and standard deviation, z = (P − mean)/SD, you are computing a z‑score (standardized precipitation anomaly). This can be a rough proxy in some contexts, but it is not the formal SPI distribution-based method.

What is a Standardized Precipitation Index?

The Standardized Precipitation Index (SPI) is a meteorological index used to quantify precipitation deficits (drought) and surpluses (wetness) over a specified accumulation period (e.g., 1-, 3-, 6-, 12-month). SPI is derived by fitting a probability distribution to long-term precipitation totals for that accumulation period, converting an observed total to a cumulative probability, and transforming that probability to a standard normal value. Negative SPI values indicate drier-than-typical conditions, while positive values indicate wetter-than-typical conditions.

How to Calculate Standardized Precipitation Index?

The following steps outline how SPI is typically calculated.

  1. Choose a timescale (e.g., 3-month SPI) and compute accumulated precipitation totals for that timescale from a long-term monthly precipitation record (commonly 30+ years).
  2. Fit an appropriate probability distribution to the long-term accumulated totals (commonly a gamma distribution), and estimate the probability of zero precipitation (q) if zeros occur.
  3. For the observed accumulated precipitation amount P, compute the cumulative probability H(P) (including q for zero precipitation, if applicable).
  4. Transform H(P) to the standard normal scale using the inverse normal CDF: SPI = Φ−1(H(P)).
  5. Interpret SPI values (negative = drier than typical; positive = wetter than typical). Many references classify SPI using thresholds such as ±1, ±1.5, and ±2.
  6. If you only have a long-term mean and standard deviation (not the full record), you can compute a standardized precipitation anomaly (z‑score) as an approximation, but it is not the formal SPI computation.

Example Problem (Z-score approximation):

Use the following variables as an example problem to test your knowledge of the z‑score approximation.

Accumulated precipitation for the chosen period (P) = 120 mm

Long-term mean for the same period = 80 mm

Long-term standard deviation for the same period (SD) = 20 mm

Standardized value (z) = (120 − 80) / 20 = 2.00