Sticky post

A/B testing

The A/B test (also known as a randomised controlled trial, or RCT, in the other sciences) is a powerful tool for product development. some motivations: With the rise of digital marketing led by tools including Google Analytics, Google Adwords, and Facebook Ads, a key competitive advantage for businesses is using A/B testing to determine effects of digital marketing efforts. Why? In short, small changes can have big effects. This is why A/B testing is a huge benefit. A/B Testing enables us to determine whether changes in landing pages, popup forms, article titles, and other digital marketing decisions improve conversion rates … Continue reading A/B testing

Sticky post

Binomial Random Variables: Introduction

Binomial Random Variables So far, in our discussion about discrete random variables, we have been introduced to: The probability distribution, which tells us which values a variable takes, and how often it takes them. The mean of the random variable, which tells us the long-run average value that the random variable takes. The standard deviation of the random variable, which tells us a typical (or long-run average) distance between the mean of the random variable and the values it takes. We will now introduce a special class of discrete random variables that are very common, because as you’ll see, they … Continue reading Binomial Random Variables: Introduction

Introduction to Normal Random Variables: Overview

In the Exploratory Data Analysis sections of this course, we encountered data sets, such as lengths of human pregnancies, whose distributions naturally followed a symmetric unimodal bell shape, bulging in the middle and tapering off at the ends. Many variables, such as pregnancy lengths, shoe sizes, foot lengths, and other human physical characteristics exhibit these properties: symmetry indicates that the variable is just as likely to take a value a certain distance below its mean as it is to take a value that same distance above its mean; the bell-shape indicates that values closer to the mean are more likely, and it … Continue reading Introduction to Normal Random Variables: Overview

The Big Picture: Inference

Recall again the Big Picture, the four-step process that encompasses statistics: data production, exploratory data analysis, probability, and inference. We are about to start the fourth part of the process and the final section of this course, where we draw on principles learned in the other units (exploratory data analysis, producing data, and probability) in order to accomplish what has been our ultimate goal all along: use a sample to infer (or draw conclusions) about the population from which it was drawn. The specific form of inference called for depends on the type of variables involved—either a single categorical or quantitative … Continue reading The Big Picture: Inference

Conditional Probability and Independence Introduction

Introduction In the last section, we established the five basic rules of probability, which include the two restricted versions of the Addition Rule and Multiplication Rule: The Addition Rule for Disjoint Events and the Multiplication Rule for Independent Events. We have also established a General Addition Rule for which the events need not be disjoint. In order to complete our set of rules, we still require a General Multiplication Rule for which the events need not be independent. In order to establish such a rule, however, we first need to understand the important concept of conditional probability. This section will be organized as follows: We’ll first … Continue reading Conditional Probability and Independence Introduction

Probability Rules

Basic Probability Rules In the previous section we considered situations in which all the possible outcomes of a random experiment are equally likely, and learned a simple way to find the probability of any event in this special case. We are now moving on to learn how to find the probability of events in the general case (when the possible outcomes are not necessarily equally likely), using five basic probability rules. Fortunately, these basic rules of probability are very intuitive, and as long as they are applied systematically, they will let us solve more complicated problems; in particular, those problems … Continue reading Probability Rules

Sticky post

How To Distribute Sample

Sampling Distributions Introduction Already on several occasions we have pointed out the important distinction between a population and a sample. In Exploratory Data Analysis, we learned to summarize and display values of a variable for a sample, such as displaying the blood types of 100 randomly chosen U.S. adults using a pie chart, or displaying the heights of 150 males using a histogram and supplementing it with the sample mean (X¯) and sample standard deviation (S). In our study of Probability and Random Variables, we discussed the long-run behavior of a variable, considering the population of all possible values taken by that variable. For example, we … Continue reading How To Distribute Sample

Sticky post

Probability A short story

Sample Spaces As we saw in the previous section, probability questions arise when we are faced with a situation that involves uncertainty. Such a situation is called a random experiment, an experiment that produces an outcome that cannot be predicted in advance (hence the uncertainty). Here are a few examples of random experiments: Toss a coin once and record whether you get heads (H) or tails (T). The possible outcomes that this random experiment can produce are: {H, T}. Toss a coin twice. The possible outcomes that this random experiment can produce are: {HH, HT, TH, TT}. Toss a coin 3 … Continue reading Probability A short story

Random Variables

In the previous sections we’ve learned principles and tools that help us find probabilities of events in general. Now that we’ve become proficient at doing that, we’ll talk about random variables. Just like any other variable, random variables can take on multiple values. What differentiates random variables from other variables is that the values for these variables are determined by a random trial, random sample, or simulation. The probabilities for the values can be determined by theoretical or observational means. Such probabilities play a vital role in the theory behind statistical inference, our ultimate goal in this course. Introduction We first … Continue reading Random Variables

Blind and Double-Blind Experiments

Suppose the experiment about methods for quitting smoking were carried out with randomized assignments of subjects to the four treatments, and researchers determined that the percentage succeeding with the combination drug/therapy method was highest, and the percentage succeeding with no drugs or therapy was lowest. In other words, suppose there is clear evidence of an association between method used and success rate. Could it be concluded that the drug/therapy method causes success more than trying to quit without using drugs or therapy? Perhaps. Although randomized controlled experiments do give us a better chance of pinning down the effects of the … Continue reading Blind and Double-Blind Experiments