Category: statistics

A random variable (also called a stochastic variable) is a variable that can take on different values depending on some random phenomenon. In many statistics textbooks it is simply abbreviated as r.v. It is a numerical value.

When probability values are assigned to all the possible numerical values of a random variable x, the result is a probability distribution.

In even simpler terms: a random variable is a variable whose values are each associated with a probability of being observed. The set of all possible values of a random variable and their associated probabilities is called a probability distribution. The sum of all probabilities is 1.

Continue reading “Probability Distributions: Discrete Distributions and the Binomial”

We are running an A/B test on two title tags for a product page. Variant A gets 127 clicks out of 4,000 impressions (CTR 3.17%), variant B gets 152 clicks out of 4,000 impressions (CTR 3.80%). A difference of 25 clicks. Is this a real lift, or just random noise?

Questions like this — is this difference due to chance? — arise constantly in any data-driven activity. Hypothesis testing is the statistical procedure that allows us to answer with a methodical approach, distinguishing results that reflect a real phenomenon from those that are merely sampling noise.

In the article on hypothesis testing we saw how to check whether a sample mean is compatible with a hypothesis about the population. In the examples we proposed, however, we knew the population standard deviation, sigma — which allowed us to use the normal distribution and the Z-score.

In practice this is a rather rare case. If we do not know the population sigma, or if we are working with small samples, we need a different distribution: the t distribution, or Student’s distribution.

In the article on hypothesis testing we saw how to check whether a sample mean is compatible with a hypothesis about the population. But in practice the most common case is comparing two groups.

Does a new drug lower blood pressure more than a placebo? Does a training course improve test scores? Does a new landing page produce a higher CTR? In each case we have two groups (treatment vs control, before vs after, variant A vs B) and we want to know whether their means differ. The right tool is the two-sample t-test.

In previous articles, we examined how hypothesis testing works and how the t-distribution allows us to work even when we don’t know the population standard deviation. In both cases, we focused on a specific question: “can I reject the null hypothesis, yes or no?”

But there’s another question, equally important, that we ask ourselves constantly in daily practice: what is the approximate value of the parameter I’m estimating? It’s not enough to know whether the mean differs from a certain value; we want to know where it lies, and with what margin of uncertainty.

This is where confidence intervals (often abbreviated as CI) come into play—one of the most useful and, at the same time, most misunderstood tools in all of inferential statistics.

Continue reading “Confidence Intervals: the 95% doesn’t mean what you think (here’s what it really says)”