{ "version": "https:\/\/jsonfeed.org\/version\/1.1", "title": "Ivan Baldin, Data Analyst: posts tagged conversion", "_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\/conversion\/", "feed_url": "https:\/\/ivanbaldin.blog\/tags\/conversion\/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": "1", "url": "https:\/\/ivanbaldin.blog\/all\/conversion-chart-with-confidence-intervals\/", "title": "Conversion Chart with Confidence Intervals", "content_html": "

I’ve been happily using a more informative way to visualize conversion rates for a while now, by adding confidence interval bounds<\/b> to my charts.<\/p>\n

Sales office conversion rates<\/h2>\n

Say we’re evaluating the performance of regional sales offices. By performance<\/i><\/b> we mean the ratio of closed deals to incoming leads (lead-to-sale conversion, or simply “conversion”).<\/i><\/b> For example, if the “Midtown” office closed 19 deals out of 33 leads in a quarter, we consider their performance to be 19\/33 = 57.6%.<\/p>\n

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

Clearly, some offices outperform others: conversion rates range from 57.6% down to 17.6%. It is also apparent that the number of leads varies widely across offices: from 33 leads in “Midtown” to 706 leads in “Astoria.”<\/p>\n

Most people would stop here, but there is a straightforward way to use the concept of a “confidence interval” or “standard deviation (SD)” to reveal something that isn’t immediately obvious.<\/b><\/p>\n

Estimating sample size and SD<\/h2>\n

Because the number of leads differs dramatically across offices (“Midtown” and “Astoria” differ by a factor of nearly 22), our confidence in the calculated conversion rate will not be uniform. For “Astoria,” the 36.1% figure is based on a sample of 706 leads and can be considered quite reliable. For “Midtown,” the 57.6% result comes from a small sample of just 33 leads, so there is no guarantee that, given their own 706 leads over time, they would sustain the same conversion rate.<\/p>\n

Naturally, we need to estimate the confidence interval for each sales office based on the number of leads — that is, the sample size.<\/p>\n

Here is the familiar formula for the standard deviation (SD), or σ<\/i>:<\/p>\n

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

where p<\/i> is the conversion rate and n<\/i> is the number of leads.<\/p>\n

We compute it in column E:<\/p>\n

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

The resulting standard deviation shows the margin of error in the conversion estimate. As expected, it is larger where the sample is smaller. The less data we have, the less reliable the calculated result, and the less confident we can be in our assessment of the corresponding office’s performance.<\/p>\n

Computing the bounds of a 90% confidence interval<\/h2>\n

Let us extend our table with the lower and upper bounds of a 90% confidence interval. In other words, we estimate the range of conversion rates for each office such that there is a 90% probability the true<\/i> conversion rate falls within that range.<\/p>\n

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

Knowing that the bounds of a 90% confidence interval lie at ±1.645 SD, we subtract and add 1.645 SD to obtain the lower and upper bounds, respectively. For “Astoria,” this tells us that the true<\/i> conversion rate lies between 33.1% and 39.1%. (One time out of ten it may still fall outside our interval, but nine times out of ten our estimate holds.)<\/p>\n

Enhancing the chart with “candlesticks”<\/h2>\n

In Excel 2013 we can use a “stock chart,”<\/b> substituting the upper and lower confidence interval bounds for the high<\/i><\/b> and low prices,<\/i><\/b> and the conversion rate calculated earlier for the close price:<\/i><\/b><\/p>\n

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

The enhanced chart does not change the conclusions we drew at the outset. However, it subtly reminds an observant manager that the reported conversion rates are not final<\/b> — and are especially “not final” where the confidence bounds are wider.<\/p>\n

“Midtown” appears to have outperformed “SoHo”, but while “SoHo” strong result is reliable thanks to its narrow interval, the result for “Midtown” is highly approximate. Definitive conclusions can therefore only be drawn for some of the offices; for the rest, more data is needed, and until then their rankings should be considered preliminary — or, as noted above, not final.<\/p>\n", "summary": "Not all conversion rates are created equal. A 57% rate from 33 leads is not the same as 36% from 706. Ivan Baldin reflects on how to add confidence interval bounds to your sales conversion charts using standard deviation and Excel stock chart candlesticks.", "date_published": "2021-05-25T02:13:00+03:00", "date_modified": "2026-04-12T03:24:52+03:00", "tags": [ "analytics", "conversion", "diagrams and charts", "Excel", "sales funnel", "statistics" ], "image": "https:\/\/ivanbaldin.blog\/pictures\/conversion_simple.png", "_date_published_rfc2822": "Tue, 25 May 2021 02:13:00 +0300", "_rss_guid_is_permalink": "false", "_rss_guid": "1", "_e2_data": { "is_favourite": false, "links_required": [], "og_images": [ "https:\/\/ivanbaldin.blog\/pictures\/conversion_simple.png", "https:\/\/ivanbaldin.blog\/pictures\/conversion_simple_chart.png", "https:\/\/ivanbaldin.blog\/pictures\/CodeCogsEqn.png", "https:\/\/ivanbaldin.blog\/pictures\/conversion_SD.png", "https:\/\/ivanbaldin.blog\/pictures\/conversion_SD_max-min.png", "https:\/\/ivanbaldin.blog\/pictures\/conversion_simple_chart_type.png", "https:\/\/ivanbaldin.blog\/pictures\/conversion_simple_chart_modified.png" ] } }, { "id": "5", "url": "https:\/\/ivanbaldin.blog\/all\/kupivip-voice-bot-guessing-the-sample-size\/", "title": "KupiVIP’s Voice Robot: Guessing the Sample Size", "content_html": "

In the article “Case Study: On Black Friday, KupiVIP Replaced Their Call Center with a Robot, and Only 5 Out of 5,000 People Realized They Were Talking to a Neural Network<\/a>” on vc.ru, one phrase caught my attention above all else — the part where the author mentions a conversion rate increase from 6% to 8%:<\/p>\n

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

Naturally, I started wondering: on what call volume was this conversion rate increase observed, and was that volume sufficient to make a reliable claim that the voice robot outperforms live call center agents?<\/p>\n

The question can be formulated precisely as follows: what is the minimum number of calls required to claim, with a confidence level of, say, 95%, that the conversion rate “increased from 6% to 8%”?<\/p>\n

Building the Experiment in Google Spreadsheets<\/h2>\n

Let’s lay out the available data in Google Spreadsheets. For the calculations that follow, we’ll need a “number of calls” parameter — let’s assume for now that both the robot and live agents each made 1,000 calls before the 6% and 8% conversion rates were obtained:<\/p>\n

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

Essentially, this is a standard A\/B split test, and we’ll need to walk through its algorithm to obtain the Z-score<\/i> and calculate the p-value<\/i>.<\/p>\n

Let’s calculate the standard errors (SD, or σ) for both conversion rates, along with the standard error of the difference between them. The formula for the standard error of a conversion rate:<\/p>\n

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

where p<\/i> is the conversion rate (6%, for example) and n<\/i> is the sample size (1,000 calls). Here’s the Google Spreadsheets calculation:<\/p>\n

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

The standard error of the difference between conversion rates is calculated using the formula:<\/p>\n

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

where σ<\/i> is the standard error of each conversion rate, A and B (live agent and robot). We’ll compute it just below in Google Spreadsheets:<\/p>\n

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

How much larger is the difference between conversions A and B compared to the standard error of that difference? This ratio is called the Z-score.<\/i><\/b> In Google Spreadsheets, it’s straightforward:<\/p>\n

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

So the Z-score = 1.7541. On the normal distribution curve, this corresponds to the 96th percentile — meaning the probability of the Z-score randomly exceeding 1.7541 is about 4% (in other words, 96% of the area under the bell curve falls within +1.7541 standard deviations):<\/p>\n

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

Where did we get 96%? The exact probability, the p-value<\/i>, is calculated using the formula:<\/p>\n

=NORM.DIST(1.7541,0,1,TRUE)<\/code><\/pre>
\n\"\"\n<\/div>\n
P-value = 96.03%.<\/p>\n
So here’s our interim conclusion: if, on a sample of 1,000 calls in each of the two groups, we observed conversion rates (promo code activations) of 6% and 8%, then we are 96% confident that this difference is not due to chance.<\/b> (There remains a 4% probability that the observed difference is random. In that case, perhaps the conversion rate is actually the same — say, 7% — and had we made more calls, the difference would have eventually disappeared.)<\/p>\n
Minimum Experiment Size<\/h2>\n
Now, let’s get back to the original question.<\/p>\n
We weren’t trying to prove that 8% is greater than 6%, and the figure of 1,000 calls for the robot and agents was chosen arbitrarily. What we wanted to calculate was the minimum number of calls needed to establish the statistical significance of the difference between 8% and 6% at a 95% confidence level.<\/b><\/p>\n
1,000 calls turned out to be more than enough. Now we just need to decrease this number until the p-value crosses the 95% threshold. (By the way, according to the normal distribution formula, this corresponds to a Z-score of 1.6449 — try verifying this yourself.)<\/p>\n
In theory, it would probably be possible to derive a large formula for the n<\/i> at which the p-value equals 0.95. In practice, it’s faster to manually find the minimum n through trial and error. Or, better yet, install a Goal Seek<\/b> add-on for Google Spreadsheets and use it to find the answer automatically:<\/p>\n
\n\"\"\n
(Just make sure the number of robot calls is exactly the same as the number of live agent calls, i.e., you’ve entered =C6 in cell C7.)<\/div>\n<\/div>\n
\n\"\"\n<\/div>\n
Conclusions<\/h2>\n
So, we’ve calculated the minimum experiment conditions for evaluating the effectiveness of the KupiVIP voice robot.<\/p>\n
You need at least 879 calls<\/b> in each of the two groups to confirm, with a 95% confidence level, that there is a difference between the 6% promo code activation rate in the control group (real live agents) and the 8% rate in the test group (voice robot).<\/p>\n
(The only caveat is that neither 6% nor 8% yields a whole number of activations on a sample of 879 calls, and in reality, of course, the numbers would be different — and the number of calls in the two groups could vary as well. But this doesn’t really matter much, since the conversion rates cited in the article were most likely rounded.)<\/p>\n
See also:<\/h2>\n
https:\/\/abtestguide.com\/calc\/?ua=1000&ub=1000&ca=60&cb=80<\/a><\/p>\n",
            "summary": "A voice robot beats call center with 8% vs. 6% promo code activations. But how many calls do you actually need to prove the difference is real? We walk through the A\/B test math and find the minimum sample size for 95% confidence using Goal Seek in Google Spreadsheets, Z-score and SD.",
            "date_published": "2019-07-20T13:18:00+03:00",
            "date_modified": "2026-04-12T03:31:13+03:00",
            "tags": [
                "A\/B test",
                "call center",
                "conversion",
                "Google Sheets",
                "statistics"
            ],
            "image": "https:\/\/ivanbaldin.blog\/pictures\/vc_quote.png",
            "_date_published_rfc2822": "Sat, 20 Jul 2019 13:18:00 +0300",
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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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