statistics definition

Statistics is the science of learning from data. It provides the tools to decide whether what we observe is real or accidental, and whether a difference is large enough to matter.

When a scientist runs an experiment, or when a pollster surveys a group of voters, the results always vary. Statistics gives us a way to interpret that variation and to draw conclusions.

The Two Branches of Statistics

• Descriptive Statistics describe and summarize what we see.
  Example: “The average score on the math test was 78.”
• Inferential Statistics use a sample to make conclusions about a larger group.
  Example: “Based on this sample, we estimate the average score for all students in the district.”

Definition:

• Descriptive statistics = picture of the data.
• Inferential statistics = prediction about the population.

Parametric vs. Non-parametric Statistics

There are two main families of tests:

• Parametric tests (such as the t-test or ANOVA) assume certain conditions in the data, like normal distribution and interval/ratio measurement.
• Non-parametric tests (such as Chi-square or Mann–Whitney) require fewer assumptions and are used when data are ranks (ordinal) or categories (nominal).

Simple rule of thumb:

• If data are interval or ratio (e.g., test scores, heights), use parametric tests.
• If data are ordinal or nominal (e.g., ranks, categories), use non-parametric tests.

First Formula in Statistics: The Mean

The mean is our first step toward summarizing data.

Symbolic formula:
$$\bar{X} = \frac{\sum X}{n}$$

Formula in words:
$$\text{Mean} = \frac{\text{sum of scores}}{\text{number of scores}}$$

Where:

• $$\bar{X}$$ = mean (X bar)
• $$\sum X$$ = sum of all scores
• $$n$$ = number of scores

Example: Data: 6, 8, 10

$$\bar{X} = \frac{6 + 8 + 10}{3} = \frac{24}{3} = 8$$

So the mean is 8.

Visual

Figure 1.1 — The First Decision in Statistics. A flowchart: Descriptive vs. Inferential → Parametric vs. Non-parametric, with examples inside each box.

Why This Matters

Before you can choose the right statistical test, you must know:

1. What kind of data you have (descriptive vs. inferential).
2. How those data are measured (nominal, ordinal, interval, ratio).
3. Which family of tests applies (parametric vs. non-parametric).

This chapter sets the stage. The rest of the book builds from here, using only a small set of simple formulas to unlock the logic of statistics.

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