# Website Conversion: how to calculate confidence interval for tomorrow's conversion rate

Let's say I have a website with certain number of visits per day and conversion events:

Date    Visits   Conversion_Events  Conversion_%
12/1/14  3,369   179                 5.3%
12/2/14  3,297   199                 6.0%
12/3/14  3,355   405                 12.1%
12/4/14  2,035   220                 10.8%
12/5/14  4,834   207                 4.3%

Total    16,890  1,210               7.2%


These numbers fluctuate for multitude of reasons that are unknown to us (we can't control for geo location, demographics, etc., and assuming day-of-week, day-of-month, etc. don't make sense to control for). But we do know day-by-day data fluctuates around some mean. I'm interested in coming up with a plausible 95% confidence interval for tomorrow's (or any other day's) conversion based on historical data.

I can think of a few ways to achieve this:

(1) Look at totals only and use "standard error of proportion" formula: Sp = sqrt(p*(1-p)/n) = sqrt(7.2%*(1-7.2%)/16890). This means in this example 6.8%-7.6%. Intuitively, this seems to me to hide the fact that the day-by-day conversions fluctuate, so I'm concerned this loses some valuable information.

(2) Look at every date as a sample of the population, and calculate mean and stdev across samples. I.e. average() and stdev() on the array [5.3%,6.0%,12.1%,10.8%,4.3%]. Now, I get it that mathematically averaging proportions with varying sample sizes is not great, but intuitively assuming that sample sizes are very large it seems to me that this effect should be mitigated by the fact we are looking at the proportion summary statistic. Note that I gave an example of 5 days, but in reality, I can get a high number of "samples", so I can have a pretty good understanding of the distribution function of day-by-day proportions. This means the 95% confidence interval in this example is 0.8%-14.6%, which is quite big (but probably due to the fact that the numbers in this example are just made up).

(3) Calculate 95% confidence interval of Visits and of Conversion_Events, then take the low/high and high/low. I.e. Visits 95% confidence is 1435-5674, Conversion_Events is 61-361, so 61/5674=1.1% and 361/1435=25.2%. This makes some intuitive sense to me, though I suspect that simply dividing in such manner no longer means the outcome confidence is at 95%.

So, my questions:

a. Is my concern that method (1) is "hiding" some information justified?

b. Does method (2), which intuitively makes the most sense to me, make statistical sense?

c. What's the preferred approach (which I suspect is neither of the 3)?

• What is @Conversion"? – kjetil b halvorsen Jan 24 '15 at 19:48
• Conversion is frequency of occurrence of some desired event. For example, if 100 visitors visited the website, and 5 of them clicked on the "Buy" button, the conversion rate for this action is 5%. – AmitA Jan 28 '15 at 5:48
• In your search for solutions it will help to know that limits for tomorrow's conversion rate are prediction limits, not confidence limits. This means your efforts to use confidence limit formulas are doomed. – whuber Feb 11 '15 at 19:08

After deciding on a model for the daily visit counts, then the conversions. Focus on the probability $p$ that the visit result in conversion. If the visits process is Poisson with expected number of daily visits $\lambda$, then the conversion process is poisson with expectation (per day) $p\lambda$. But, this assumes $p$ is a constant and does not vary, which must be investigated. So make a plot of conversion percentage versus daily visits. Is there some relationship?