{ "version": "https:\/\/jsonfeed.org\/version\/1.1", "title": "Ivan Baldin, Data Analyst: posts tagged binomial", "_rss_description": "Data analytics, probability theory, and applied statistics — from the practitioner’s perspective. Ivan Baldin writes about sales funnels, call center performance, conversion analysis, confidence intervals, binomial distributions, Z-scores, and the math behind data-driven decisions. Years of hands-on experience turning raw CRM data into actionable insights for sales teams and business leaders.", "_rss_language": "en", "_itunes_email": "", "_itunes_categories_xml": "", "_itunes_image": "", "_itunes_explicit": "", "home_page_url": "https:\/\/ivanbaldin.blog\/tags\/binomial\/", "feed_url": "https:\/\/ivanbaldin.blog\/tags\/binomial\/json\/", "icon": "https:\/\/ivanbaldin.blog\/pictures\/userpic\/userpic@2x.jpg?1774085044", "authors": [ { "name": "Ivan Baldin", "url": "https:\/\/ivanbaldin.blog\/", "avatar": "https:\/\/ivanbaldin.blog\/pictures\/userpic\/userpic@2x.jpg?1774085044" } ], "items": [ { "id": "2", "url": "https:\/\/ivanbaldin.blog\/all\/wilson-score-confidence-interval-for-binomial-distribution\/", "title": "Wilson Score Confidence Interval for Binomial Distribution", "content_html": "

You listen to 10 random call center calls. In 4 of them, the operator never asks the client where they heard about you from. So now you have a number: 40% of calls are missing the advertising source. That’s an issue. You bring it to your manager, maybe flag it in a report.<\/p>\n

But here’s the thing. Is it really 40%?<\/p>\n

The trouble with small samples<\/h2>\n

10 calls is not a lot. You picked them at random, and randomness is fickle. The true<\/i> failure rate could be 20%. It could be 60%. You just don’t know — and the small sample is the reason.<\/p>\n

There is, of course, a standard way to express this uncertainty. You calculate the SD of the binomial distribution:<\/p>\n

\n\"\"\n<\/div>\n

You get SD = 15.49%. Then you build a 90% confidence interval: 40%±1.645 SD, which gives you a range from 14.52% to 65.48%.<\/p>\n

That’s a huge spread. But at least now you’re honest about it: with 90% probability, the true failure rate lies somewhere in that range.<\/p>\n

The 40% figure is just your best guess, not a fact.<\/p>\n

So far, so good.<\/p>\n

Except there’s a catch.<\/p>\n

When the standard method breaks down<\/h2>\n

The ±1.645 SD<\/i> approach works well enough — until it doesn’t. When your sample is small or your proportion is extreme (very close to 0% or 100%), the formula starts to wobble. There’s a simple rule of thumb: if np<\/i> < 10, you’re in trouble. In our case, np<\/i> = 4. We’re in trouble.<\/p>\n

Think of it this way. If “tails” came up fewer than 10 times, either you haven’t flipped the coin enough, or you’re dealing with a biased coin that rarely lands on tails. Either way, the standard formula isn’t built for this.<\/p>\n

\n

Many statisticians consider the Wilson method<\/b> to be the most reliable approach for estimating confidence intervals for proportions <...>. This method not only handles very small and very large proportions, but also works well with small sample sizes.<\/i><\/p>\n<\/blockquote>\n

The method was proposed in 1927. It has been around for nearly a century. And yet most analysts still reach for ±1.645 SD<\/i> by default.<\/p>\n

The Wilson method<\/h2>\n

The lower and upper bounds of the confidence interval p = 1–α\/2<\/i> are computed using the following formulas:<\/p>\n

\n
\n\"\"\n\"\"\n<\/div>\n
Formulas for the lower and upper bounds, respectively.<\/div>\n<\/div>\n

where p<\/b><\/i> is the observed proportion, N<\/b><\/i> is the number of trials, and z<\/b><\/i> is the z-<\/i>score (e. g., 1.960 for a 95% confidence interval, or 1.645 for 90%).<\/p>\n

Back to the call center<\/h2>\n

Let’s revisit our 10 calls. The standard method gave us a confidence interval from 14.52% to 65.48%. Now let’s apply Wilson.<\/p>\n

The bounds tighten: with 90% probability, the true failure rate lies between 19.42% and 64.84%. The effective SD drops to 13.80%.<\/p>\n

The difference may look subtle, but notice what happened at the lower end: it shifted from 14.52% to 19.42%. That’s not a rounding error. In a real business decision — say, whether to retrain your call center team — that shift matters.<\/p>\n

And this is the point. When your data is scarce, every percentage point of precision counts. The Wilson score interval doesn’t give you more data. It just makes better use of the data you already have.<\/p>\n

Calculator<\/b><\/a> in Google Sheets (menu “File” → “Make a copy”).<\/p>\n

See also:<\/h2>\n

Online calculator<\/a> for a 95% confidence interval
\nCalculator<\/a> on WolframAlpha.com
\nBinomial confidence intervals and contingency tests<\/a> (pp. 4–5)
\nhttps:\/\/influentialpoints.com\/Training\/confidence_intervals_of_proportions.htm#wils<\/a>
\nWilson score interval<\/a> on Wikipedia<\/p>\n", "summary": "The standard ±1.645 SD approach to binomial distribution confidence intervals breaks down with small samples or extreme proportions. The Wilson score interval, proposed in 1927, handles both cases gracefully. This post walks through a practical call center QA example with formulas and a ready-made calculator.", "date_published": "2020-04-01T17:54:00+03:00", "date_modified": "2026-04-12T22:51:34+03:00", "tags": [ "analytics", "binomial", "statistics" ], "image": "https:\/\/ivanbaldin.blog\/pictures\/CodeCogsEqn.png", "_date_published_rfc2822": "Wed, 01 Apr 2020 17:54:00 +0300", "_rss_guid_is_permalink": "false", "_rss_guid": "2", "_e2_data": { "is_favourite": false, "links_required": [ "jquery\/jquery.js", "fotorama\/fotorama.css", "fotorama\/fotorama.js" ], "og_images": [ "https:\/\/ivanbaldin.blog\/pictures\/CodeCogsEqn.png", "https:\/\/ivanbaldin.blog\/pictures\/wilson01.png", "https:\/\/ivanbaldin.blog\/pictures\/wilson02.png" ] } }, { "id": "29", "url": "https:\/\/ivanbaldin.blog\/all\/4-for-4-against-confidence-interval\/", "title": "4 for, 4 against: the confidence interval hiding behind a 50\/50 split", "content_html": "

I was recently listening to Vesti FM (a Russian national talk-radio station), where the hosts were going over the results of Russia’s nationwide election day on September 8<\/a>.<\/p>\n

\n\"\"\n<\/div>\n

One remark from the host caught my attention (at 01:45:05<\/a>):<\/p>\n

\n

By the way, listeners from Khabarovsk Krai are texting in, and going by the number of texts, they’re splitting about 50\/50. Half of them think they made a positive choice, half think things got worse and it was a negative choice. This isn’t a public opinion survey, obviously. I’m just seeing eight or ten texts come in, and they split roughly down the middle. Still, it’s interesting.<\/i><\/p>\n<\/blockquote>\n

Credit where it’s due: the disclaimer is exactly right, this is not a survey. But the question is still worth asking. What can you actually say about how opinion splits in reality<\/i> when all you have in hand is 4 texts in favor and 4 against? How much does a 50\/50 result on a sample of 8 tell you about the same split in the population?<\/i><\/p>\n

Working it out in Google Sheets<\/h2>\n

Let’s reproduce the setup in a spreadsheet:<\/p>\n

\n\"\"\n<\/div>\n

That day, 4 listeners texted in favor and 4 texted against. But no two days are the same. Today it’s one set of listeners, tomorrow it’s another, and the score could just as easily come out 3 to 5, 5 to 3, 2 to 6, or 7 to 1 — any combination is possible in principle. If we assume the audience really is evenly divided, then 4-to-4, 3-to-5 and 5-to-3 are somewhat more likely, while 8-to-0 or 1-to-7 are much less likely.<\/p>\n

Technically, this is a binomial<\/i> setup: out of 8 texts we expect<\/i> 4 in favor, but we don’t know in advance how many we’ll actually get. If the probability that any given text is “for” is 50% (let’s assume exactly half the audience is in favor), then the standard error (SE, often written σ)<\/i><\/b> of the observed share is:<\/p>\n

\n\"\"\n<\/div>\n

where p<\/i><\/b> = 50% and n<\/i><\/b> = 8.<\/p>\n

Let’s compute it:<\/p>\n

\n\"\"\n<\/div>\n

So if the probability of a “for” text is 50%, then on a sample of 8 texts the standard error comes out to 17.68%<\/b>.<\/p>\n

What does that mean in practice?<\/p>\n

It means that because the sample (8 texts) is tiny, the amount of randomness baked into the “4 for, 4 against” result is enormous, and we cannot confidently claim a strict 50\/50 split across the entire Vesti FM audience. The only thing we can state with any confidence is that the true share of “for” votes lies somewhere within an interval<\/i> around 50%. And the wider we want that interval’s reliability to be, the wider the interval itself gets.<\/p>\n

Say we want to be 90% confident in our interval — in other words, we accept being wrong 1 time in 10. Since the normal distribution is a decent approximation of the binomial once n<\/i> is large enough, the interval works out to 50% ± 1.645 SE.<\/b><\/p>\n

That’s an easy calculation in Google Sheets:<\/p>\n

\n\"\"\n<\/div>\n

The true share of “for” votes lies in the interval 50% ± 29.08%, that is, anywhere from 20.92% to 79.08%.<\/b> Roughly like this:<\/p>\n

\n\"\"\n<\/div>\n

So we’re nowhere near being able to say that “listeners are splitting about 50\/50.” At best — and even then we’re allowing a 10% chance of being wrong — all we can say is somewhere between 21% and 79%.<\/b><\/p>\n

Sharpening the estimate<\/h2>\n

That said, the p<\/i> ± 1.645SE<\/i><\/b> interval is itself a crude approximation, and it’s at its worst exactly where we’re using it: small samples. There are more sophisticated and somewhat more accurate ways to set the bounds.<\/p>\n

Running the numbers through the Wolfram Alpha calculator<\/a> gives these bounds:<\/p>\n\n\n\n\n\n\n\n
Clopper-Pearson confidence interval for a binomial parameter<\/td>\n0.1929<\/td>\n0.8071<\/td>\n<\/tr>\n
Wilson score confidence interval for a binomial parameter with continuity correction<\/td>\n0.2034<\/td>\n0.7966<\/td>\n<\/tr>\n
standard confidence interval for a binomial parameter<\/td>\n0.2092<\/td>\n0.7908<\/td>\n<\/tr>\n
Jeffreys confidence interval for a binomial parameter<\/td>\n0.2393<\/td>\n0.7607<\/td>\n<\/tr>\n
Wilson score confidence interval for a binomial parameter<\/td>\n0.2486<\/td>\n0.7514<\/td>\n<\/tr>\n
Agresti-Coull confidence interval for a binomial parameter<\/td>\n0.2486<\/td>\n0.7514<\/td>\n<\/tr>\n<\/table>\n

Every one of them is still enormously wide. The method you pick barely matters when the sample is this small.<\/p>\n

So how many texts would it take to get to 45–55%?<\/h2>\n

Here’s the flip side of the question: how large would the sample have to be — still splitting exactly 50\/50 — before we could narrow the interval to at least 45–55%?<\/p>\n

That’s easy enough to solve. For an interval of 50% ± 5% (and sticking with our 90% confidence level), those 5 percentage points have to equal 1.645 standard errors. So SE = 3.04%. Plugging that back into the standard error formula:<\/p>\n

\n\"\"\n<\/div>\n

gives n<\/i> = 270.6. In other words, you’d need somewhere around 271 texts, split exactly down the middle, before you could talk about a 45%–55% range at 90% confidence.<\/p>\n

Eight texts, it turns out, is not a small sample. It’s barely a sample at all.<\/p>\n

See also<\/h2>\n

https:\/\/en.wikipedia.org\/wiki\/Binomial_proportion_confidence_interval<\/a>
\nCalculator at WolframAlpha.com<\/a>
\nhttps:\/\/cyberleninka.ru\/article\/n\/doveritelnye-intervaly-dlya-chastot-i-doley.pdf<\/a>
\nStatistical significance for real-estate developers<\/a><\/p>\n", "summary": "A radio host saw 4 listener texts for and 4 against, and called the audience evenly split. The math says otherwise: on a sample of 8, the true share is anywhere from 21% to 79%. Here's how to compute that – and why you'd need 271 texts.", "date_published": "2019-09-16T20:03:00+03:00", "date_modified": "2026-08-17T00:26:20+03:00", "tags": [ "binomial", "Google Sheets", "statistics" ], "image": "https:\/\/ivanbaldin.blog\/pictures\/4smski05[1].JPG", "_date_published_rfc2822": "Mon, 16 Sep 2019 20:03:00 +0300", "_rss_guid_is_permalink": "false", "_rss_guid": "29", "_e2_data": { "is_favourite": false, "links_required": [], "og_images": [ "https:\/\/ivanbaldin.blog\/pictures\/4smski05[1].JPG", "https:\/\/ivanbaldin.blog\/pictures\/4smski00.PNG", "https:\/\/ivanbaldin.blog\/pictures\/CodeCogsEqn[1].png", "https:\/\/ivanbaldin.blog\/pictures\/4smski01.PNG", "https:\/\/ivanbaldin.blog\/pictures\/4smski03.PNG", "https:\/\/ivanbaldin.blog\/pictures\/4smski02[1].PNG", "https:\/\/ivanbaldin.blog\/pictures\/4smski04[1].png" ] } }, { "id": "8", "url": "https:\/\/ivanbaldin.blog\/all\/one-in-five-testing-a-connection-complaint\/", "title": "One in Five? Testing a Connection Complaint", "content_html": "

A contractor swears the phones drop out on every fifth call.<\/b> You make one — it’s fine. Another — also fine. How many do you need before you can confidently tell him he’s wrong?<\/p>\n

Our job is to check whether the claim that 20% of calls have connection problems holds up.<\/b> And as always, we’re not going to just make 100 test calls — we don’t have the resources for that. Instead, we’ll state a null hypothesis, an alternative hypothesis, and test it at a chosen confidence level.<\/p>\n

Stating the hypothesis and choosing a confidence level<\/h2>\n

The null hypothesis (H0<\/sub>)<\/i><\/b> is the contractor’s claim: at least 20% of calls have connection problems. This is the claim we’re putting on trial.<\/p>\n

The alternative hypothesis (H1<\/sub>)<\/i><\/b> is that the connection is fine — problems happen in fewer than 20% of calls. This is the position we’ll lean toward if the data supports it.<\/p>\n

(That framing matters. In hypothesis testing, you never “prove” H0<\/sub>; you either reject it or you fail to reject it. So when we say we want to side with the contractor or not, what we really mean is: can we gather enough evidence to reject<\/i> his claim?)<\/p>\n

The confidence level<\/b> is how sure we want to be about the outcome. The higher it is, the more test calls we’ll need to make, so we’ll budget 1% for error and go with 99% confidence<\/b> — leaving a 1% chance that we reject the contractor’s claim when problems really are there.<\/p>\n

Building the formula for the sample size<\/h2>\n

To reject H0<\/sub><\/i>, all we need is to pull off N<\/i> successful calls in a row with no signs of trouble — while keeping the probability at or below 1% that we simply got lucky and that real equipment problems just happened not to show up in any of the N<\/i> calls.<\/p>\n

If the contractor is right, 80% of calls go through cleanly. The probability of no issues across N<\/i> calls in a row is 0.80N<\/sup>. We need the smallest N<\/i> for which that probability drops to 1%:
\n0.80N<\/sup> = 0.01<\/p>\n

Which is exactly the logarithm of 1% to base 80%.<\/p>\n

\n\"\"\n<\/div>\n

Into Google Sheets it goes:<\/p>\n

\n\"\"\n<\/div>\n

The formula for cell C5 looks like<\/p>\n

=LOG(1-C2,1-C3)<\/code><\/pre>
We need 20.64 calls. (Check: 0.8020.64<\/sup> = 0.9995%, right on target.)
\nAll that’s left is rounding up:<\/p>\n
=ROUNDUP(C5)<\/code><\/pre>
or, in one shot,<\/p>\n
=ROUNDUP(LOG(1-C2,1-C3))<\/code><\/pre>
\n\"\"\n<\/div>\n
Testing the hypothesis<\/h2>\n
If the contractor’s claim H0<\/sub><\/i> is correct and every fifth call really does have trouble, then the probability of seeing zero problems across 21 test calls in a row is about 1%. In other words: either we just hit a one-in-a-hundred coincidence, or the claim that 20% of calls have problems doesn’t hold. At the 99% confidence level, we reject H0<\/sub><\/i> — in everyday terms, the connection is fine.<\/p>\n",
            "summary": "A contractor claims every fifth call has a bad connection. How many good calls in a row do you need to disprove him? This post uses a binomial hypothesis test to calculate the exact sample size for 99% confidence — with the Google Sheets formula included.",
            "date_published": "2019-06-09T16:45:58+03:00",
            "date_modified": "2026-04-24T21:06:41+03:00",
            "tags": [
                "binomial",
                "call center",
                "Google Sheets",
                "statistics"
            ],
            "image": "https:\/\/ivanbaldin.blog\/pictures\/every5th01.png",
            "_date_published_rfc2822": "Sun, 09 Jun 2019 16:45:58 +0300",
            "_rss_guid_is_permalink": "false",
            "_rss_guid": "8",
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        },
        {
            "id": "6",
            "url": "https:\/\/ivanbaldin.blog\/all\/the-cognitive-fallacy-of-conversion-rates\/",
            "title": "The Cognitive Fallacy of Conversion Rates",
            "content_html": "
A curious thing about working with the concept of conversion rate<\/i> is that, strictly speaking, a conversion rate is almost never precisely defined.<\/b><\/p>\n
All those “call-to-sale conversion of 18.4%,” “CTR of 3.1%,” “deal conversion of 30%” — it’s always a somewhat simplified approach, as if the conversion rate were reliably measured and, should it change, we’d explain it with objective factors, never entertaining the thought that there never was any “18.4%” to begin with — there were just 38 contracts we closed on 206 calls, and that doesn’t at all mean there couldn’t have been more or fewer.<\/p>\n
Much like the position of an electron around the nucleus of an atom isn’t defined by exact coordinates, but is merely described by a region where it’s most likely to be found, our conversion rate is not a specific number either. In reality, it is an interval within which it lies.<\/b><\/p>\n
Calculating Conversion Rates and Cognitive Bias<\/h2>\n
Consider a fictional sales department that started selling a new product this year. Let’s say, hmm, trucks.<\/b> The product isn’t in high demand, so, for now, there isn’t much data for analysis — or, better said, very little at all:<\/p>\n\n\n\n\n\n\n\n\n
Month<\/b><\/td>\nLeads<\/b><\/td>\nSales<\/b><\/td>\n<\/tr>\n
August<\/td>\n48<\/td>\n1<\/td>\n<\/tr>\n
September<\/td>\n35<\/td>\n1<\/td>\n<\/tr>\n
October<\/td>\n24<\/td>\n0<\/td>\n<\/tr>\n
November<\/td>\n61<\/td>\n2<\/td>\n<\/tr>\n
December<\/td>\n32<\/td>\n0<\/td>\n<\/tr>\n
TOTAL:<\/b><\/td>\n200<\/b><\/td>\n4<\/b><\/td>\n<\/tr>\n<\/table>\n
As the sales data shows, over several months we’ve got 4 deals out of 200 leads, i.e. our conversion rate is 4 \/ 200 = 2.0%.<\/p>\n
(Additionally, based on five months of data, we can roughly project 480 leads for the next year and, accordingly, 480 × 0.02 = 9.6 deals.)<\/p>\n
Given such scarce data, it’s impossible to go wrong, so, of course, such a forecast won’t be incorrect. However, it contains an important cognitive bias: 2.0% is not the exact value, but rather the most probable so far<\/i> value of our lead-to-sale conversion rate for trucks.<\/p>\n
In reality, the conversion rate cannot be determined precisely. It lies within a confidence interval from 0.4% to 3.6%.<\/b> And for the coming year, we should be projecting not 9.6 deals, but somewhere between 5 and 15 trucks sold. Unfortunately, narrowing this range any further would be overly optimistic.<\/p>\n
The Conversion Bell Curve<\/h2>\n
Assuming that our true<\/i> conversion rate is stable and exactly<\/i> equals 2.0%, we can estimate the possible range of deals out of 480 leads expected next year. Since we can only vouch for the consistency of our own<\/i> work but can’t account for randomness (customer moods, force majeure, a lucky sale to the CEO’s buddy), there’s always a chance the number of deals will differ slightly from the projected 480 × 0.02 = 9.6 deals — much like the number of tails in 480 coin flips may differ slightly from 240, coming out to 235, 248, or possibly even 223.<\/p>\n
Graphically, this looks like a normal distribution bell curve,<\/i> where the further we move from the expected value of 9 deals at the center of the bell, the lower the probability of closing significantly fewer or significantly more deals:<\/p>\n
\n\"\"\n<\/div>\n
Looking at the resulting graph, we have to admit that seeing fewer than 2 or more than 19 sales is practically impossible.<\/p>\n
But can we narrow our confidence interval?<\/i><\/p>\n
Confidence Interval of a Conversion Rate<\/h2>\n
The standard error (SD) for a binomial distribution is calculated using the formula:<\/p>\n
\n\"\"\n<\/div>\n
where n<\/i><\/b> is the number of trials, and p<\/i><\/b> is the probability of success.<\/p>\n
For our 200 leads this year, we get:<\/p>\n
\n\"\"\n<\/div>\n
SD = 1.98 deals.<\/b> In other words, according to the laws of normal distribution (and binomial distribution is a special case of normal distribution), in roughly 68% of cases, while operating with a true conversion rate of 2.0%, we’d land within the confidence interval of 2.02 to 5.98 deals, i.e. ±1SD.<\/p>\n
For the projected 480 leads next year, we get:<\/p>\n
\n\"\"\n<\/div>\n
SD = 3.07 deals.<\/b> By the laws of binomial (normal) distribution, it is known that in 68% of cases, next year’s sales will fall within ±1SD of the expected value of 9.6 deals, and in 90% of cases — within ±1.645SD of the expected value. 3.07 deals × 1.645 = 5.05 deals, meaning that in 90% of cases, if we work the entire next year at a 2% conversion rate, we won’t fall outside the confidence interval of “4.55 to 14.65 deals.” (Notably, conversely, in 1 out of 10 cases we will<\/i> fall outside these bounds, all while still operating at a “true” conversion rate of 2%.)<\/p>\n
One can only wonder what kind of uproar the sales manager would raise if, by year’s end, we sold just 4 trucks on 480 leads, formally showing a conversion rate of 0.83%... and what’s even more curious is that, statistically, this happens in 1 out of 27 sales departments.<\/p>\n
In one out of 27 cases, you get fired for missing your sales target, even though you were still operating at a “true” conversion rate of 2%.<\/p>\n
Three Conversion Rates at the Confidence Interval Boundary<\/h2>\n
So how should we interpret this year’s results, where we got 4 sales out of 200 leads?<\/p>\n
Case one, “2.00%.”<\/b> We covered this one right away. 4 \/ 200 = 0.02, meaning our conversion rate is 2%. At the same time, by the laws of binomial distribution, there’s still a chance of fluctuating within the 90% confidence interval of “±1.645SD,” i.e. within the range of 0.74 to 7.26 deals per 200 leads.<\/p>\n
It looks roughly like this:<\/p>\n
\n\"\"\n<\/div>\n
Our result of 4 deals matched the expected value for a 2.0% conversion rate, although, in general, it didn’t have to — it could have landed anywhere within the 90% confidence interval of 1 to 7 sales.<\/p>\n
Case two, “1.22%.”<\/b> In this scenario, our “true” conversion rate is actually lower, equaling, say, 1.22%.<\/b> Then the expected number of trucks sold is approximately 2, and we were lucky<\/b> to close 4 sales. Our luck was such that we could have made more<\/i> than 4 sales only in 10% of cases. That is, we remain within the 90% probability field, though we’re sitting at its boundary. A little more, and our luck would be too good — but for now, we’re lucky “within reason”:<\/p>\n
\n\"\"\n<\/div>\n
Case three, “3.31%.”<\/b> Now let’s suppose that this year we were unlucky,<\/b> even though our true conversion rate was above 2.0% all year and actually equaled 3.31%.<\/b> The expected value for 200 leads would then be approximately 6 trucks sold, and making fewer<\/i> than 4 sales would only be possible in 10% of cases. So we’d still be within the 90% probability field, but sitting at the left boundary with our unlucky 4 deals.<\/p>\n
\n\"\"\n<\/div>\n
Thus, the takeaway is clear: we don’t know for sure which of the 3 conversion rates is the true one.<\/b> We’re used to thinking it’s Case 1 — dividing 4 sales by 200 leads and getting a conversion rate of 2.00%. But nobody knows for certain whether the current year was typical or atypical, whether we were lucky or unlucky. In 90% of cases, it could have been a matter of luck,<\/b> and we were actually operating at a conversion rate of 1.22%, or a matter of bad luck,<\/b> where we were operating at a conversion rate of 3.31%. In all 3 cases, the probability of closing 4 sales out of 200 leads did not exceed the 90% threshold.<\/b><\/p>\n
Unfortunately, we simply don’t have enough data yet to make more precise claims.<\/p>\n
We Need More Data<\/h2>\n
A logical question to ask is — how much data do we need to accumulate before we can speak of a 2.0% conversion rate with any reliability? Let’s try gradually increasing the sample size (number of leads, and consequently, sales) until we see the 90% confidence interval converge around the 2.00% conversion rate value:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                    Leads<\/b><\/td>\n             Deals<\/b><\/td>\nLower bound of 90% confidence interval (-1.645SD)<\/b><\/td>\nUpper bound of 90% confidence interval (+1.645SD)<\/b><\/td>\nLower conversion bound<\/b><\/td>\nUpper conversion bound<\/b><\/td>\n<\/tr>\n
200<\/td>\n4<\/td>\n0.7<\/td>\n7.3<\/td>\n0.37%<\/td>\n3.63%<\/td>\n<\/tr>\n
500<\/td>\n10<\/td>\n4.9<\/td>\n15.1<\/td>\n0.97%<\/td>\n3.03%<\/td>\n<\/tr>\n
1,000<\/td>\n20<\/td>\n12.7<\/td>\n27.3<\/td>\n1.27%<\/td>\n2.73%<\/td>\n<\/tr>\n
5,000<\/td>\n100<\/td>\n83.7<\/td>\n116.3<\/td>\n1.67%<\/td>\n2.33%<\/td>\n<\/tr>\n
10,000<\/td>\n200<\/td>\n177.0<\/td>\n223.0<\/td>\n1.77%<\/td>\n2.23%<\/td>\n<\/tr>\n
50,000<\/td>\n1,000<\/td>\n948.5<\/td>\n1,051.5<\/td>\n1.90%<\/td>\n2.10%<\/td>\n<\/tr>\n
100,000<\/td>\n2,000<\/td>\n1,927.2<\/td>\n2,072.8<\/td>\n1.93%<\/td>\n2.07%<\/td>\n<\/tr>\n
500,000<\/td>\n10,000<\/td>\n9,837.2<\/td>\n10,162.8<\/td>\n1.97%<\/td>\n2.03%<\/td>\n<\/tr>\n
1,000,000<\/td>\n20,000<\/td>\n19,769.7<\/td>\n20,230.3<\/td>\n1.98%<\/td>\n2.02%<\/td>\n<\/tr>\n
10,000,000<\/td>\n200,000<\/td>\n199,271.7<\/td>\n200,728.3<\/td>\n1.99%<\/td>\n2.01%<\/td>\n<\/tr>\n
25,000,000<\/td>\n500,000<\/td>\n498,848.5<\/td>\n501,151.5<\/td>\n2.00%<\/td>\n2.00%<\/td>\n<\/tr>\n<\/table>\n
Needless to say, very few sales departments can generate more than a few tens of thousands of leads. Therefore, the truth is, setting sales targets and making promoting and firing decisions based on 1–5% conversion rates is madness — at these small numbers, math in sales simply doesn’t work.<\/b><\/p>\n",
            "summary": "Your conversion rate isn't a fixed number — it's a confidence interval. Using binomial distribution, this post shows why a 2% conversion rate on 200 leads could really be anywhere from 0.4% to 3.6%, and why setting sales targets on small samples is madness.",
            "date_published": "2019-05-07T12:39:07+03:00",
            "date_modified": "2026-04-12T11:39:08+03:00",
            "tags": [
                "binomial",
                "conversion",
                "sales funnel"
            ],
            "image": "https:\/\/ivanbaldin.blog\/pictures\/norm_dist.png",
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        {
            "id": "4",
            "url": "https:\/\/ivanbaldin.blog\/all\/staff-decisions-or-lets-fire-grandma\/",
            "title": "Staff Decisions, or Let’s Fire Grandma",
            "content_html": "
It’s a Thursday morning, and the VP of Sales is staring at a spreadsheet. One number — 10.1% — is about to end someone’s career. He doesn’t know it yet, but he’s about to make a decision that mathematics would call premature.<\/p>\n
I would call this problem “Promote or Fire”, or “Let’s Fire Grandma”.<\/p>\n
Here we’re looking at our sales department’s results, and questions are arising about our new manager, Drennan:<\/p>\n
\n\"\"\n<\/div>\n
Matt Drennan joined the team two and a half months ago. He came from a smaller company, interviewed well, and his first few weeks were promising — he learned the product fast, clients liked him on the phone. His probation target was set at “16 deals per 100 calls”. Reason? Because historically, our VP of Sales observed a call-to-deal conversion rate of 16.1%.<\/p>\n
Now, with 89 calls behind him and only 9 deals to show for it, his conversion sits at 10.1%.<\/p>\n
“Fire him,” the VP of Sales decides.<\/p>\n
Here’s the question: is this decision justified?<\/b> Has enough time passed (enough data accumulated) to make this call? Has the “bad luck” factor been accounted for? Could it be that Drennan is simply performing no worse<\/b> than the other managers, actually converting at around the required 16%, but consistently running into clients’ force majeure events (5 clients “dropped off”), a “rough patch” in his personal life, and an unfortunately positioned Moon in the third house of Taurus?<\/p>\n
Rolling the Dice<\/h2>\n
Before we get to the spreadsheet formula that will settle this, let me take you on a brief detour.<\/p>\n
In 1913, at the Monte Carlo Casino, the roulette ball landed on black twenty-six times in a row<\/a>. Gamblers lost millions betting against the streak, convinced that red was “due.” They were wrong — not because the wheel was rigged, but because they could not intuitively grasp what randomness actually looks like. Streaks happen. Cold spells happen. And sometimes a perfectly good sales manager just has a bad quarter.<\/p>\n
This is the same trap the VP of Sales is walking into.<\/p>\n
Recalling the coin flip example, let’s switch things up this time and roll a 6-sided die. The probability of rolling a “1” is 1\/6, or roughly 16.7%.<\/p>\n
The expected value for 89 rolls of a die is 89 × 1\/6 = 14.8 “ones” (and the same number of “twos,” “threes,” and so on), but obviously you could get not just 14–15, but also 12, 17, or even 20. Getting none at all,<\/i> however, is virtually impossible (though, theoretically, the probability isn’t zero).<\/p>\n
Drennan’s work can be thought of as rolling a die, where the desired outcome — a “one” (a deal) — comes up roughly every sixth roll. Roughly,<\/i> because the historically observed conversion rate (excluding Drennan’s results) is 380 deals out of 2,361 calls, or 380\/2,361 = 16.1%. The expected value from his 89 “rolls” (calls) is 89 × 0.161 = 14.3 “ones” (deals), but intuitively, we understand there could be slightly more or slightly fewer.<\/p>\n
If we calculate (we’ll learn how shortly) the exact probabilities of “landing” a specific number of deals in 89 calls and plot them on a graph, the most likely outcome (the “expected value”)<\/i> of 14 deals will be in the center, with the remaining possible outcomes (13 and 15 deals, 12 and 16 deals, etc.) becoming progressively less likely each time, giving the graph its bell shape:<\/p>\n
\n\"\"\n<\/div>\n
We can’t say that 9 deals is entirely<\/i> improbable — the bar is visible, if modest, with a probability of 0.037. That is, it happens in 1 out of 27 cases — which may be unlikely, but not extremely<\/i> unlikely.<\/p>\n
All that’s left is to figure out how we arrived at “1 in 27,” and how this connects to difficult, “staff”, decisions in a sales department.<\/p>\n
Calculating the Binomial Distribution<\/h2>\n
Both Excel and Google Sheets have a built-in binomial distribution function.<\/b> It’s exactly what will give us the answer to whether it’s time to fire the unlucky Drennan.<\/p>\n
In the cell next to his 10.1% conversion rate, we calculate the function:<\/p>\n
=BINOM.DIST(D7,C7,$E$11,1)<\/code><\/pre>
\n\"\"\n<\/div>\n
In this function, we specify, in order: the number of successes (deals), the number of trials (calls), and the probability of success (the 16.1% conversion rate). The last, 4th parameter, we set to “1.”<\/p>\n
\n\"\"\n<\/div>\n
What is this 0.0763 we got? 0.0763 is the probability of getting no more than 9 deals out of 89 calls given a deal probability of 16.1%.<\/b> In other words, it’s the probability of getting anywhere from 0 to 9 deals inclusive<\/i> with these parameters. Conversely, 1 − 0.0763 = 0.9237 is the probability of getting 10 or more<\/i> deals.<\/p>\n
(By the way, if you change the 4th parameter to “0,” you get the probability of getting exactly<\/i> 9 deals.)<\/p>\n
\n\"\"\n<\/div>\n
You could say that, taking the sum of all bars on the graph as 1, the sum of bars “0” through “9” equals 0.0763, or 7.63%. As you can see, it’s far more likely to land in the blue portion of the bell curve than in the red one (92.37% vs. 7.63%).<\/p>\n
Conclusion:<\/b> the probability that Drennan, while actually converting at 16.1%, would randomly (remember the possibility of a “rough patch”?) end up with no more than 9 deals out of 89 calls is 7.63%.<\/p>\n
Conversely, there’s a 92.37% probability that he would have gotten 10 or more deals. Roughly speaking, there’s a 7.63% chance he was simply unlucky, and a 92.37% chance that bad luck alone doesn’t explain it, and he is most likely converting at a rate below 16.1%.<\/p>\n
Therefore, if a 90% confidence level<\/b> is enough for the VP of Sales, then Drennan can be let go at the end of his probation — the manager is, indeed, not meeting his target. If, however, a 95% confidence level<\/b> is required, the data is not yet sufficient, and it would be advisable to observe a bit longer.<\/p>\n
So what confidence level should we choose? There is no right answer here.<\/p>\n
If we set it too low,<\/b> we risk accidentally firing good managers, but we won’t lose deals by continuing to work with bad ones.<\/p>\n
If we set it too high,<\/b> we’ll have to accumulate data for too long before making a mathematically justified decision to fire a bad manager, but the probability of accidentally firing a good one will be lower. In my opinion, a 90% level is optimal for this case. Drennan can be fired.<\/p>\n
Wait, What About Whitfield’s 19.7%?<\/h2>\n
Indeed, if there are “bad” managers for whom there’s a 92.37% probability that their conversion is below the required 16.1%, then logically, there can also be “good” ones.<\/p>\n
\n\"\"\n<\/div>\n
Dan Whitfield is the quiet opposite of Drennan. He’s been with the company for six months, never makes a fuss, never asks for a raise. He just dials, talks, and closes. Over that time he put up 56 deals on 284 calls — a 19.7% conversion rate — compared to the projected 0.161 × 284 = 46 deals. Could he have been lucky this whole time? Could it be that, while actually performing like everyone else at a 16.1% conversion rate, he randomly ended up with more deals than the VP of Sales projected?<\/p>\n
The binomial distribution function returns 0.9563 — meaning that with a 95.63% probability, working like everyone else, he would have gotten no more than 56 deals... but he didn’t get more<\/i> than 56 deals... He got exactly<\/i> 56!<\/p>\n
Let’s simply refine the function by recalculating it for 56 − 1 = 55 deals:<\/p>\n
\n\"\"\n<\/div>\n
For 55 deals, the result is 0.9402. That is, with a 94.02% probability, Whitfield (working at a 16.1% conversion rate) would have gotten no more than<\/i> 55 deals. This means the probability of getting more than<\/i> 55 deals is the remaining 5.98%! It turns out our Whitfield landed in a sliver of his own bell curve — just on the opposite end — and the probability of landing there is only about 6%.<\/p>\n
\n\"\"\n<\/div>\n
The VP of Sales has already decided that before making staffing decisions, he wants to be at least 90% confident in the results. And in Whitfield’s results, he’s 94.02% confident! That leaves only 5.98% for the possibility that Whitfield was simply lucky.<\/p>\n
So either he got lucky, even though he actually performs like everyone else (at a 16.1% conversion rate) and doesn’t deserve a higher salary, or he truly converts at a rate above 16.1% and it would be fair to reward him.<\/p>\n
6% clearly loses to 94%, so Whitfield gets the promotion.<\/p>\n",
            "summary": "Your new sales rep is converting at 10% instead of the expected 16%. Fire him? Not so fast. Using the binomial distribution function built into Excel and Google Sheets, you can calculate the exact probability that poor results are just a cold streak — and make staffing decisions you can actually defend with math.",
            "date_published": "2019-03-29T18:03:00+03:00",
            "date_modified": "2026-04-12T03:09:44+03:00",
            "tags": [
                "binomial",
                "Sales Department",
                "sales funnel"
            ],
            "image": "https:\/\/ivanbaldin.blog\/pictures\/funnel01.png",
            "_date_published_rfc2822": "Fri, 29 Mar 2019 18:03:00 +0300",
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