{"id":6996,"date":"2023-04-26T00:00:00","date_gmt":"2023-04-26T04:00:00","guid":{"rendered":"https:\/\/www.sisense.com\/how-to-calculate-confidence-intervals-in-sql\/"},"modified":"2024-09-23T15:20:54","modified_gmt":"2024-09-23T19:20:54","slug":"how-to-calculate-confidence-intervals-in-sql","status":"publish","type":"post","link":"https:\/\/www.sisense.com\/blog\/how-to-calculate-confidence-intervals-in-sql\/","title":{"rendered":"How to calculate confidence intervals in SQL"},"content":{"rendered":"

<\/p>\r\n

Statistical overconfidence: Dangerous and easy<\/h2>\r\n

Imagine you have a small online business. This month 200 users signed up on your website, and 10 of them bought your $800 service. Great! You\u2019ve made $8k of income. How much should you expect to make this year?<\/p>\r\n

The straightforward answer is $8k * 12 = $96k. But how confident should you be? Will your conversion rate always be so close to 5%? You could pad the estimate \u00b120% for safety, guessing at $77k to $115k. If $77k would cover all your expenses, should you feel secure?<\/p>\r\n

This is a question of binomial probability<\/a>. Using our favorite binomial confidence interval calculator, the 95% confidence interval for your conversion rate is about 2.5% to 9%.<\/p>\r\n

With a confidence interval that wide, you should expect to make somewhere between $48k and $172k. Yikes! You could end up with half of your simple guess, and that\u2019s if your business doesn\u2019t change.<\/p>\r\n

Automating statistics: Calculating confidence intervals in SQL<\/h2>\r\n

These confidence intervals are very informative, but turning to a calculator for every metric is tedious. If you\u2019ve got hundreds of metrics across dozens of dashboards, it\u2019s downright unsustainable.<\/p>\r\n

Fortunately, the math for calculating confidence interval is simple to implement:<\/p>\r\n

The normal approximation interval formula for binomial confidence intervals<\/h2>\r\n
n = number of users\r\nx = number of conversions\r\np = probability of conversion = (x \/ n)\r\nse = standard error of p = sqrt((p * (1 - p)) \/ n)\r\nconfidence interval = p \u00b1 (1.96 * se)<\/code><\/pre>\r\n
See\u00a0Normal approximation interval on wikipedia<\/a>. Note the 1.96 constant specifies a 95% interval on a\u00a0two-tailed normal distribution<\/a>.<\/em><\/p>\r\n
Implementing the formula in SQL<\/strong><\/p>\r\n
Let\u2019s start with a table of the total number of users, and how many converted. Any data that represents a rate \u2014 conversions per user, server errors per request, etc. \u2014 will also work.<\/p>\r\n
select \r\n  count(1) as n, \r\n  sum(case when converted then 1 else 0 end) as x\r\nfrom users\r\ngroup by date_trunc('month', created_at);<\/code><\/pre>\r\n
\"Users<\/figure>\r\n
With our basic data in hand, we want to implement the above formula in SQL. To keep things clear, we wrap each step of the calculation separately:<\/p>\r\n
    \r\n
  1. Calculate the conversation rate, p.<\/li>\r\n
  2. Using p, calculate the standard error, se.<\/li>\r\n
  3. Compute the low and high confidence intervals.<\/li>\r\n
  4. Include the original p conversion rate as our mid estimate.<\/li>\r\n<\/ol>\r\n
    select \r\n  rates.n as users, \r\n  rates.x as conversions, \r\n  p - se * 1.96 as low, \r\n  intervals.p as mid, \r\n  p + se * 1.96 as high \r\nfrom (\r\n  select \r\n    rates.*, \r\n    sqrt(p * (1 - p) \/ n) as se -- calculate se\r\n  from (\r\n      select conversions.*, \r\n      x \/ n::float as p -- calculate p\r\n    from ( \r\n      -- Our conversion rate table from above\r\n      select \r\n        count(1) as n, \r\n        sum(case when converted then 1 else 0 end) as x\r\n      from users\r\n      group by date_trunc('month', created_at);\r\n    ) conversions\r\n  ) rates\r\n) intervals<\/code><\/pre>\r\n
    \"adjusted<\/figure>\r\n
    You might be wondering why we\u2019re seeing 8% on the high end, rather than the 9% mentioned in the introduction. We used the Adjusted Wald method in the introduction, which produces more accurate estimates for small amounts of data.<\/p>\r\n
    A refinement for little data: The Adjusted Wald method<\/h2>\r\n
    The math explained above, though quite accurate with hundreds of users and a healthy conversion rate, becomes increasingly biased with less data or extremely high or low rates. A rule of thumb is to avoid using it with fewer than 5 conversions or 100 users.<\/p>\r\n
    One way to adjust for these shortcomings is to use a more robust binomial proportion confidence<\/a> interval technique like the Adjusted Wald method. In short, it adds a bit of fuzziness to the estimated probability to smooth out the extremely high or low rates which are more common with few data points.<\/p>\r\n
    Given the z-score needed to reach a certain confidence level (1.96 for a 95% confidence), add 0.5 * z^2 to the number of conversions, and z^2 to the number of users. This is roughly +2 and +4 for the 1.96 z-score for 95%.<\/p>\r\n
    select \r\n  rates.n as users, \r\n  rates.x as conversions, \r\n  p - se * 1.96 as low, \r\n  intervals.p as mid, \r\n  p + se * 1.96 as high \r\nfrom (\r\n  select \r\n    rates.*, \r\n    sqrt(p * (1 - p) \/ n) as se -- calculate se\r\n  from (\r\n    select \r\n      conversions.*, \r\n      (x + 1.92) \/ (n + 3.84)::float as p -- calculate p\r\n    from ( \r\n      -- Our conversion rate table from above\r\n      select \r\n        count(1) as n, \r\n        sum(case when converted then 1 else 0 end) as x\r\n      from users\r\n      group by date_trunc('month', created_at);\r\n    ) conversions\r\n  ) rates\r\n) intervals<\/code><\/pre>\r\n
    The important adjustment is here, where we add the constants to the numerator and denominator when calculating\u00a0p:<\/p>\r\n
    (x + 1.92) \/ (n + 3.84)::float as p -- calculate p<\/code><\/pre>\r\n
    \"Conversions<\/figure>\r\n
    This isn\u2019t a magical solution to not enough data: If you have an expected 1% conversion rate and only 100 users, this adjustment will triple the estimated conversion rate, giving you a confidence interval of 0-6%. More data is the answer. At 10 conversions and 1,000 users, the interval shrinks to 0.5% to 1.9%.<\/p>\r\n
    In general, the more data you have, the more statistical approaches like these will be helpful to you.<\/p>\r\n
    Who are we?<\/h2>\r\n
    We make a tool that makes data analysis on large SQL databases fast and easy. You could use our Snippets feature to implement this logic once, and apply it to any dataset.<\/p>\r\n
    If you have a database with many millions or billions of rows, and running hundreds of analyses is getting slow and cumbersome, we think you\u2019ll really love it. Sign up for a free demo. We can also set you up with a free trial on the same day!<\/p>\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n<\/p>","protected":false},"excerpt":{"rendered":"
    Statistical overconfidence: Dangerous and easy Imagine you have a small online business. This month 200 users signed up on your website, and 10 of them bought your $800 service. Great! You\u2019ve made $8k of income. How much should you expect…<\/p>\n","protected":false},"author":4,"featured_media":7198,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_searchwp_excluded":"","footnotes":"","_links_to":"","_links_to_target":""},"categories":[44],"tags":[472],"application":[10],"buyer-role":[],"buyer-stage":[],"department":[6],"industry":[],"topic":[],"class_list":["post-6996","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tech-talk","tag-data-team","application-cloud-data-teams","department-it"],"acf":[],"yoast_head":"\nHow to Calculate Confidence Intervals in SQL | Sisense<\/title>\n<meta name=\"description\" content=\"Confidence intervals are informative, but turning to a calculator for every metric is tedious. 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