1 Role of Statistics and Key Concepts
1.1 Overview
- A decision before definitions
- Role of statistics in business
- Data types and measurement scales
- Populations vs. Samples
- Parameters vs. Statistics
- Study Designs: Observational vs. Experimental
Statistics: Business Question \(\rightarrow\) Data \(\rightarrow\) Analysis \(\rightarrow\) Decision
The goal is to make that pipeline rigorous, not a simple “gut call.” There should be some method behind the process.
- Precisely describe what the data shows. What was the average order value?
- Estimate what’s outside the data with honest uncertainty. What does our sample tell us about customers in general?
- Compare groups within data and decide if differences are real or noise. Do customers exposed to different policies actualy behave differently?
- Model relationships and determine if they’re causal. What variables are related to spending and do they actually cause some sort of change?
1.2 Types of Data
Categorical: qualitative values relating to characteristics.
- Nomimal: categories with no meaniningful order. There’s no mathematical sense in which is greater or less than another. e.g. red, blue, green
- Ordinal: categories with an order to them. Certain categories correspond to higher or lower rankings, places, etc. e.g. one star, two star, three star
Numerical
- Interval: numerical values with meaningful differences but no true zero. A value of zero doesn’t neccessarily mean “none”, rather it’s just a ‘label’ similar to any other number. e.g. 0°F, 53°F
- Ratio: numerical values with a true equal spacing and a true zero. In this case, a value of zero literally means “none” or “nothing there”. e.g. $0, $5 Because zero has an actual meaning, comparisons (or ratios) like “twice as much” are relevant.
1.3 Population vs. Sample
- Population: entire group of interest. This is the true group in question.
- Sample: the limited subset being observed.
- Parameter: a description of the population. Because it involves the entire group of interest, it’s unknown (it’s impossible to actually include everyone).
- Statistic: a computed description of the sample. Because the sample is only a part of the group, meaning it’s an inference that it represents the true group, it’s a predicted value.
| Population | Sample | |
|---|---|---|
| Described by | Parameter | Statistic |
| Center | \(\mu\) | \(\bar{x}\) |
| Proportion | \(p\) | \(\hat{p}\) |
| Spread | \(\sigma\) | \(s\) |
Say in reality there’s 50,000 customers and you survey 40. The average satisfaction score of the survey is 4.2. What you really want to know is \(\mu = ?\) – the average satisfaction of all 50,000 customers.
Because only 40 were surveyed, you’re using \(\text{statistic} \rightarrow \text{estimate parameter}\) to estimate \(\mu\). That’s the foundation of statistical inference.
Taste a spoon of soup (sample) to judge the pot (population), but only if you stirred it first (representative sample).
1.4 Two Broad Branches of Statistics
Descriptive vs. Inferenential – that distinction becomes important.
Descriptive: describe the data actually observed. Things like mean, median, standard deviation, histogram, boxplot.
*The customers in our sample spent an average of $75.*That’s descriptive
Inferential: use your sample to say something about a larger population. Things liek confidence intervals, hypothesis tests, regression, experiments.
*Based on the sample, we estimate RetailCo customers spend...*Assuming reality: inferential.
| Goal | Tools | Output Examples |
|---|---|---|
| Describe | Tables, histograms, boxplots | Center, spread, shape |
| Infer / Predict | Confidence intervals, hypothesis tests, regression, machine learning | Effects, forecasts |
1.5 Statistical Tools
| Question | Data | Tool |
|---|---|---|
| What’s typical/variable? | Quantitative | Mean, SD, IQR, histogram, boxplot |
| How many are in each group? | Categorical | Frequency table, bar chart |
| Do groups differ? | Categorical + quantitative | Side-by-side boxplots, summaries |
| Are two quantities associated? | Two quantitative variables | Scatterplot, correlation |
So you don’t just randomly choose the statistical technique, the business question and variable types determine the appropriate tool.
1.6 Study Designs: Observational vs. Experimental
Suppose a company claims customers who use our app spend 40% more. Does that mean our app causes customers to spend more? Not neccessarily. Many people who download the app could already be the company’s most loyal customers. This is the problem of confounding.
Observational: observe what people already chose/did. Here, you can establish an association, but causality isn’t established merely by observing a difference.
Experiment (A/B): random assignment of subjects. This randomization helps balance other characteristics (lurking variables) between groups. Therefore, differences in outcomes can support a causal interpretation, assuming everything else in the experiement is valid.
Randomized experiments can support cause-and-effect inference, whereas observational studies do not by themselves; random sampling is separately what supports inference to a population.
Random Sampling and random assignment are not the same thing.
- Random sampling answers “Can I generalize this result to the population.”.
- Random Assignment answers “Can I make a causal claim?”
| Concept | What Is Randomized? | When? | Purpose |
|---|---|---|---|
| Random Sampling | Which people from the population are selected for the sample | When selecting a sample | Makes the sample more representative of the population and supports generalizing to the population |
| Random Assignment | Which treatment/group participants receive | When conducting an experiment | Makes treatment groups more comparable and supports stronger causal conclusions |