How to use the exponential distribution calculator
- Choose whether you know the average time between events or the rate of events, and type that number. Use the same time unit, such as minutes, for everything.
- Pick what to Find: the chance the wait is within a time, beyond a time, between two times, or the time by which a chance of your choice is reached.
- Enter the time or times, or the chance wanted.
- Read the headline, then the mean, median and spread in the stats, and the percentile table for how long waits usually take.
Formula
rate = 1 / mean
P(wait <= x) = 1 - e^(-rate x) P(wait > x) = e^(-rate x)
time for chance p: x = -ln(1 - p) / rate
median = mean x ln 2 standard deviation = mean
The exponential distribution describes the wait between independent events that happen at a steady average rate. Its standard deviation equals its mean, and it is memoryless: the time already waited does not change how long is left.
A bus that comes every 10 minutes on average
With a mean wait of 10 minutes, the chance the wait is 15 minutes or less is 1 - e^(-1.5) = 77.6870%, and the chance it is longer than 15 minutes is 22.3130%. The chance it falls between 5 and 20 minutes is 47.1195%. The median wait is 6.9314718 minutes, shorter than the mean, because a few long waits pull the mean up.
Nine waits in ten are over by 23.025851 minutes. If instead you enter a rate of 0.1 events per minute, the answers are identical, since a rate of 0.1 is a mean of 10. The result of the first question was also confirmed by integrating the density numerically from 0 to 15.
What memoryless means
If you have already waited 10 minutes for something modeled this way, the chance you wait at least 5 more is the same as it was at the start. That is unusual and it is the reason the model suits random arrivals, such as calls to a help line or radioactive decays, but not schedules, where waiting a while does mean the bus is nearly due.
The average wait is the reciprocal of the rate, and the number of events in a fixed window under this model follows the Poisson distribution. The two descriptions of the same process are linked, and you can find counts with the Poisson Distribution Calculator.
Where the model fits poorly
Many real waits are not exponential: service times often cluster around a usual length, machine parts wear out and so fail more often with age, and arrivals can cluster. If your data show that kind of pattern, the exponential model will mislead. The same formulas appear in the decay of a half-life, and the Half-Life Calculator works with them in that setting.
More statistics tools
For score scales, sampling, runs and agreement, try the Geometric and Negative Binomial Calculator, Cohen's Kappa Calculator and Robust Center Calculator.
Assumptions and limits
- Events happen independently at a constant average rate.
- You use one time unit for the mean or rate and for every time you enter.
- The mean or rate must be greater than zero, and times are zero or more.
- The result is a probability model, not a prediction for any one wait.
- The example mean of 10 is an illustration. Replace it with your own.
Frequently asked questions
Why is the median smaller than the mean?
The distribution is skewed to the right: many short waits and a few very long ones. The long ones raise the average but not the middle value, so the median is the mean times the natural log of 2, about 0.693.
Can the rate be a fraction of an event?
Yes. A rate of 0.25 per hour means one event every 4 hours on average. Enter the rate directly or enter the mean wait of 4.
How is this different from the Poisson distribution?
The Poisson distribution counts how many events fall in a window of time. The exponential distribution measures the time between events. They describe the same random process from two sides.
What does a chance of 100% mean for the time?
There is always some chance of a longer wait, so no finite time gives exactly 100%. The page asks for a chance below 100% for that reason, and 99.9% is a useful practical upper value.