Binomial Distribution
Your major gifts officer made 40 personalized asks last quarter. Based on three years of data, about 25% of these asks result in a gift. She brought in 14 this time. Your executive director calls it "an incredible quarter." Your board chair suspects the team got lucky. Who's right?
The binomial distribution answers questions like this. Whenever you have a fixed number of independent yes-or-no situations, each with the same probability of success, the binomial distribution tells you how likely each possible count of "yeses" is. Forty asks, each with a 25% chance of converting, produces a specific shape. The single most likely outcome is 10, because 40 times 0.25 equals 10. But the distribution fans out around that number, showing you the full range of what's plausible through chance alone.
Think of it as a close relative of the normal distribution, but built for counting discrete outcomes. Where the normal distribution describes continuous measurements like test scores, the binomial distribution describes counts of successes out of a fixed number of tries. How many emails get opened out of 500 sent. How many event registrants actually show up out of 120 registered. How many petition signatures are valid out of 3,000 collected. The answer is never a single number. It's a spread of possibilities, each with its own probability. And when the number of tries is large enough, that spread starts to look remarkably like a bell curve.
The entire distribution is defined by just two inputs. n is the number of trials, and p is the probability of success on each one. From those two numbers you can calculate the expected value (n times p) and the standard deviation (the square root of n times p times one minus p). For the major gifts example, n is 40 and p is 0.25. The expected value is 10, and the standard deviation is about 2.7. Roughly 95% of the time, you'd expect between 5 and 15 gifts. Fourteen is at the high end, but it's within normal range. It's a strong quarter, probably not a miracle.
This matters because nonprofits constantly deal with yes-or-no outcomes at scale and rarely know what variation to expect. If your email open rate is usually 22% and you send 200 emails, you'd expect about 44 opens. But the binomial distribution tells you that anything between roughly 33 and 55 opens is ordinary random variation. If you ran an A/B test and one version got 48 opens while the other got 44, that gap tells you almost nothing. In grant applications, if your historical success rate is 30% and you submit 10 proposals, getting just 1 funded in a given cycle isn't a sign that your writing declined. The binomial distribution puts the probability of one or fewer successes at about 15%, which is bad luck but not rare bad luck. In event planning, if 60% of registrants typically attend and you have 80 registrations, attendance could range from about 40 to 56 without anything unusual happening.
The binomial distribution turns gut feelings about "good" and "bad" results into actual probabilities. Before you celebrate or worry, check whether the number you got was likely all along.
See It
Drag the sliders to change the number of asks and the success rate. The highlighted bars show the range where 95% of outcomes fall.
Reflect
Think about a yes-or-no outcome your organization tracks, whether that's email opens, donation conversions, event attendance, or grant success rates. Do you know the typical probability of success? If the latest result came in higher or lower than usual, was it actually outside the range that chance alone would produce?
When your team celebrates a "great month" or worries about a "bad quarter," are they comparing against a fixed target, or against the full range of outcomes that random variation allows?