Conversion Rate Intervals

Turn a benchmark into a beta distribution, simulate funnel uncertainty with Monte Carlo, and update the forecast with pilot results using interactive calculators.

Key takeaways

A beta distribution defined by mean $\mu$ and concentration $k$ turns a benchmark into a range. At a 3% mean and $k=200$ , the 90% interval is 1.32–5.21%.
When quantiles are available, fit the mean and $k$ together. Littledata conversion data fit roughly a 1.8% mean and $k\approx71$ . Otherwise, choose concentration using mean × $k$ .
Updating after a pilot is simple: add conversions to $\alpha$ and nonconversions to $\beta$ .
The simulator's expected-count interval captures uncertainty in rates. When counts are small, also inspect the actual-count interval and the probability of zero conversions.
A small mean × $k$ lowers the median even with an unchanged mean. When transferring a benchmark between contexts, widen the interval and assess outcomes at the 5th percentile instead of cutting the mean.
We found no direct empirical evidence that turning benchmarks into priors improves forecast accuracy. Concentration $k$ is a judgment call; sensitivity analysis is essential.

The Problem with a Single Number

Multiplying benchmark values to predict “100 conversions” gives no indication of how far the actual outcome might differ. Budget approval needs a view of downside outcomes and which stage contributes the most uncertainty.

The Beta Distribution

A conversion rate lies between 0 and 1, making the beta distribution useful. Combining a beta prior with binomial observations produces a beta posterior (BDA3, Chapter 2).

p \sim \text{Beta}(\alpha, \beta), \quad \mu = \frac{\alpha}{\alpha + \beta}, \quad k = \alpha + \beta

$k$ controls the spread around the mean. For your own sample it can be interpreted as an equivalent number of observations, but that interpretation does not transfer directly to a benchmark. The intervals below are this book's calculations from 200,000 Monte Carlo draws.

90% intervals by concentration k at a 3% mean

Dots show medians; bars show the 5th–95th percentiles. Calculated for this book.

k = 502.40% (0.36% ~ 7.70%)
k = 2002.85% (1.32% ~ 5.21%)
k = 1,0002.97% (2.17% ~ 3.94%)

Estimate $k$ from Quantiles

With published 10th and 90th percentiles, estimate the standard deviation and use the beta-distribution identity to obtain $k$ . This normal approximation is rough for skewed distributions.

\sigma \approx \frac{P_{90} - P_{10}}{2.56}, \quad k = \frac{\mu(1-\mu)}{\sigma^2} - 1

Littledata's conversion figures (median 1.4%, 10th–90th percentiles 0.3–3.7%) give $k\approx77$ . This describes differences between stores, an appropriate uncertainty when you do not know where your store falls. However, substituting a median for the mean moves the distribution's center down to about 1.0%. For skewed data, increase the mean as part of the fit. Fitting all three quantiles gives a mean of approximately 1.8% and $k\approx71$ (calculated).

Choosing $k$ for Your Context

There is no standard rule. This book chooses mean × $k$ , rather than $k$ directly, because it determines the distribution's skew. Divide mean × $k$ by the mean to obtain $k$ .

Context	Mean × $k$	$k$ at a 2% mean
Quantiles available for the same category and market	Fit the mean and $k$ to the quantiles. Between-company distributions yield 0.6–2.5	Depends on the data
Only one value, or a different context, such as US data applied to Korea	1–2	50–100
Same industry and company size	Around 4	200
Your own historical data	At least 16, or a fraction of the historical sample size	At least 800

For rates above 50%, use (100% − mean) instead of the mean. When mean × $k$ is below 1, mass piles up near zero; avoid this unless fitting published quantiles. Do not compare confidence using $k$ alone: a low rate such as 0.016% can require a $k$ in the thousands for the same uncertainty.

Mean × $k$ and the Median

Even with a fixed mean, skew makes the displayed median lower than the point estimate. More stages amplify this effect. The table gives median ÷ point estimate (this book's calculations). It resembles a discount factor but arises from the distribution's shape.

Mean × $k$	1 stage	2 stages	3 stages
0.5	0.48	0.15	0.04
1	0.72	0.41	0.24
2	0.85	0.66	0.51
4	0.93	0.82	0.73
8	0.96	0.91	0.85

If the source gives a median, increase the mean so the distribution's median matches it. With $a$ = mean × $k$ , the mean is approximately median × $a$ ÷ ( $a$ − 1/3). The approximation error is about 5% at $a=1$ and 1% at $a=2$ .

Do not reduce the mean simply because the context differs. No published discount factor supports transferring a benchmark to a new store or another country. Lower mean × $k$ to widen the interval, then make budget decisions at the 5th percentile rather than the median.

Funnel Simulation

Repeatedly draw a value for each stage and multiply the draws.

This uses only the standard library (Python 3.12 or later). It reproduces approximately the B2B SaaS example's expected-count quantiles of 0.07, 3.7, and 26 customers, an actual-count interval of 0–27, and a roughly 22% probability of zero. The simulator's currency is illustrative; calculations work the same with your currency.

Estimate your funnel's interval

Run the calculation above in your browser. Changing inputs updates distributions and sensitivity, and results can be shared by link. Total cost affects only cost per conversion, not incoming volume.

Fixed inputs

Incoming volume(visits)

Total cost($)

Stage rates

k sets the spread around the mean. For the same mean, a larger k narrows the interval. A small mean×k skews the distribution and pulls the median below the mean.

Stage 1 conversion

Mean(%)

Strength k

Stage 2 conversion

Mean(%)

Strength k

Final conversions

3.7conversions

90% interval 0.1conversions ~ 27.2conversions

Mean 7.3conversions · Actual count 0conversions ~ 28conversions · Chance of zero 21%

Cost per conversion

$2,720

90% interval $368 ~ $149,000

Distribution of Final conversions5% 0.1conversions · Median 3.7conversions · 95% 27.2conversions

0.0conversions23.0conversions46.0conversions

Inputs that move the result most

Each input alone moves to its 5th and 95th percentile with others at the mean. Measure the top one first.

Stage 2 conversion0.1conversions ~ 23.7conversions
Stage 1 conversion2.8conversions ~ 13.0conversions

8,000 runs. Stages are independent. If they move together the real interval is wider; if they move in opposite directions it is narrower. The 90% interval holds only rate uncertainty around the expected value; the actual count adds chance variation in counts. Vertical lines mark the 5th percentile, median (darker), and 95th. The last bin collects values above the 99th percentile.

Limits of the independence assumption

Low ad quality can depress both click-to-signup and signup-to-paid conversion. When stages move together, the real interval is wider. Conversely, a credit-card-required trial can reduce entry while increasing downstream conversion, narrowing the interval. Either way, read the 5th percentile as the start of an unfavorable scenario, not an absolute worst case.

Expected Counts and Actual Counts

The simulator's 90% interval for expected counts captures only uncertainty in rates. Actual counts also include random variation: an expected count of 1 can produce 0 or 2. Count outputs therefore also show the mean, actual-count interval, and probability of zero.

The difference matters most at low volumes. For 10,000 cold emails, the expected number of wins spans 0.01–0.90, but actual counts span 0–2 and the probability of zero is approximately 78%.

Use quantiles rather than means for ratio outputs such as CAC. Near-zero counts can inflate the mean dramatically. To estimate an average unit cost across campaigns, divide the budget by the mean conversion count.

Update with Pilot Results

p \mid y \sim \text{Beta}(\alpha + y,\ \beta + n - y)

The posterior mean lies between the sample rate and the prior mean.

Illustrative example. A 3% benchmark with $k=200$ gives Beta(6, 194). Observing 40 conversions from 2,000 visits yields Beta(46, 2154), with a 2.09% mean and a 90% interval of 1.62–2.62%. The mean moves substantially from the 3% benchmark toward the pilot's 2.0%.

Sensitivity Analysis

Halve and double $k$ to see how the interval changes.
Move each stage individually to its own 5th and 95th percentiles to find which stage moves the result most. This is the simulator's input-sensitivity calculation. Moving every mean by the same relative amount, such as ±20%, gives identical effects in a multiplicative funnel and cannot rank stages.
Spend pilot budget first on the stage with the largest effect on uncertainty.

When Not to Use This Approach

The benchmark population differs greatly from yours, but you set a high mean × $k$ .
A new product changes the market itself. Also consider models such as Bass.
A few customers dominate revenue. The distribution of monetary outcomes matters more than the count distribution.

Key takeaways

A beta distribution defined by mean $\mu$ and concentration $k$ turns a benchmark into a range. At a 3% mean and $k=200$ , the 90% interval is 1.32–5.21%.
When quantiles are available, fit the mean and $k$ together. Littledata conversion data fit roughly a 1.8% mean and $k\approx71$ . Otherwise, choose concentration using mean × $k$ .
Updating after a pilot is simple: add conversions to $\alpha$ and nonconversions to $\beta$ .
The simulator's expected-count interval captures uncertainty in rates. When counts are small, also inspect the actual-count interval and the probability of zero conversions.
A small mean × $k$ lowers the median even with an unchanged mean. When transferring a benchmark between contexts, widen the interval and assess outcomes at the 5th percentile instead of cutting the mean.
We found no direct empirical evidence that turning benchmarks into priors improves forecast accuracy. Concentration $k$ is a judgment call; sensitivity analysis is essential.

The Problem with a Single Number

The Beta Distribution

A conversion rate lies between 0 and 1, making the beta distribution useful. Combining a beta prior with binomial observations produces a beta posterior (BDA3, Chapter 2).

p \sim \text{Beta}(\alpha, \beta), \quad \mu = \frac{\alpha}{\alpha + \beta}, \quad k = \alpha + \beta

90% intervals by concentration k at a 3% mean

Dots show medians; bars show the 5th–95th percentiles. Calculated for this book.

k = 502.40% (0.36% ~ 7.70%)
k = 2002.85% (1.32% ~ 5.21%)
k = 1,0002.97% (2.17% ~ 3.94%)

Estimate $k$ from Quantiles

With published 10th and 90th percentiles, estimate the standard deviation and use the beta-distribution identity to obtain $k$ . This normal approximation is rough for skewed distributions.

\sigma \approx \frac{P_{90} - P_{10}}{2.56}, \quad k = \frac{\mu(1-\mu)}{\sigma^2} - 1

Choosing $k$ for Your Context

There is no standard rule. This book chooses mean × $k$ , rather than $k$ directly, because it determines the distribution's skew. Divide mean × $k$ by the mean to obtain $k$ .

Context	Mean × $k$	$k$ at a 2% mean
Quantiles available for the same category and market	Fit the mean and $k$ to the quantiles. Between-company distributions yield 0.6–2.5	Depends on the data
Only one value, or a different context, such as US data applied to Korea	1–2	50–100
Same industry and company size	Around 4	200
Your own historical data	At least 16, or a fraction of the historical sample size	At least 800

Mean × $k$ and the Median

Mean × $k$	1 stage	2 stages	3 stages
0.5	0.48	0.15	0.04
1	0.72	0.41	0.24
2	0.85	0.66	0.51
4	0.93	0.82	0.73
8	0.96	0.91	0.85

Funnel Simulation

Repeatedly draw a value for each stage and multiply the draws.

Estimate your funnel's interval

Run the calculation above in your browser. Changing inputs updates distributions and sensitivity, and results can be shared by link. Total cost affects only cost per conversion, not incoming volume.

Fixed inputs

Incoming volume(visits)

Total cost($)

Stage rates

k sets the spread around the mean. For the same mean, a larger k narrows the interval. A small mean×k skews the distribution and pulls the median below the mean.

Stage 1 conversion

Mean(%)

Strength k

Stage 2 conversion

Mean(%)

Strength k

Final conversions

3.7conversions

90% interval 0.1conversions ~ 27.2conversions

Mean 7.3conversions · Actual count 0conversions ~ 28conversions · Chance of zero 21%

Cost per conversion

$2,720

90% interval $368 ~ $149,000

Distribution of Final conversions5% 0.1conversions · Median 3.7conversions · 95% 27.2conversions

0.0conversions23.0conversions46.0conversions

Inputs that move the result most

Each input alone moves to its 5th and 95th percentile with others at the mean. Measure the top one first.

Stage 2 conversion0.1conversions ~ 23.7conversions
Stage 1 conversion2.8conversions ~ 13.0conversions

Limits of the independence assumption

Expected Counts and Actual Counts

The difference matters most at low volumes. For 10,000 cold emails, the expected number of wins spans 0.01–0.90, but actual counts span 0–2 and the probability of zero is approximately 78%.

Update with Pilot Results

p \mid y \sim \text{Beta}(\alpha + y,\ \beta + n - y)

The posterior mean lies between the sample rate and the prior mean.

Sensitivity Analysis

Halve and double $k$ to see how the interval changes.
Move each stage individually to its own 5th and 95th percentiles to find which stage moves the result most. This is the simulator's input-sensitivity calculation. Moving every mean by the same relative amount, such as ±20%, gives identical effects in a multiplicative funnel and cannot rank stages.
Spend pilot budget first on the stage with the largest effect on uncertainty.

When Not to Use This Approach

The benchmark population differs greatly from yours, but you set a high mean × $k$ .
A new product changes the market itself. Also consider models such as Bass.
A few customers dominate revenue. The distribution of monetary outcomes matters more than the count distribution.

The Problem with a Single Number

The Beta Distribution

Estimate $k$ from Quantiles

Choosing $k$ for Your Context

Mean × $k$ and the Median

Funnel Simulation

Estimate your funnel's interval

Expected Counts and Actual Counts

Update with Pilot Results

Sensitivity Analysis

When Not to Use This Approach

On this page

Conversion Rate Intervals

The Problem with a Single Number

The Beta Distribution

Estimate $k$ from Quantiles

Choosing $k$ for Your Context

Mean × $k$ and the Median

Funnel Simulation

Estimate your funnel's interval

Expected Counts and Actual Counts

Update with Pilot Results

Sensitivity Analysis

When Not to Use This Approach

On this page

Conversion Rate Intervals

Python code to run yourself

Estimate your funnel's interval

On this page

Conversion Rate Intervals

Python code to run yourself

Estimate your funnel's interval

On this page