Introduction to Statistics

📊 Introduction to Statistics

Part 1 of 5 — Data, Variables, and Good Questions

Topics in This Part

🔑 Key Concept: Statistics is the science of collecting, organizing, describing, and drawing conclusions from data. Before we ever compute an average, we have to know what kind of data we have and what question we are trying to answer.

What Is Statistics?

Every day you are surrounded by data — facts and numbers collected about the world:

• The heights of everyone in your class
• Your favorite ice cream flavors
• The number of pets each student owns
• How many minutes you spent on homework last night

A single fact, like "Maria is 152 cm tall," is one data value. A whole collection of values, like the heights of all 25 students, is a data set.

💡 Statistics turns a pile of numbers into a story. Instead of staring at 25 separate heights, statistics lets us say things like "most students are about 150 cm tall" or "the tallest student is 20 cm taller than the shortest."

We do this in four stages: collect data, organize it, describe it, and draw conclusions from it.

Two Kinds of Data

Every data set is built from a variable — the thing we are measuring or recording. Variables come in two main types.

A quick test: "Can I find the average of it and have it make sense?"

• Average eye color? ❌ — eye color is categorical.
• Average height? ✅ — height is numerical.

⚠️ Watch out: Numbers are not always numerical data. A jersey number or a ZIP code is just a label — averaging jersey numbers is meaningless, so those are categorical.

Categorical or Numerical? 🔽

Sort each variable into the correct type.

Statistical Questions

A statistical question is one you expect to answer with data that varies — the answers are not all the same.

🔑 Key Idea: A statistical question anticipates variability — different responses you will need to summarize. A question with exactly one answer is not statistical.

Concept Check 🎯

The Data Cycle

Statisticians follow a repeating data cycle:

1. Ask a statistical question.
2. Collect data that can answer it.
3. Organize & display the data (tables, plots — Part 4).
4. Analyze the data using numbers like averages (Parts 2–3).
5. Interpret the results and answer the question — which often leads to a new question.

💡 In the rest of this lesson we focus on stages 3 and 4: how to summarize a data set with a single typical value (center), how to describe how spread out it is, and how to display it so the story is easy to see.

📊 Introduction to Statistics

Part 2 of 5 — Measures of Center: Mean, Median, Mode

🔑 The Big Idea: A measure of center is a single number that represents a "typical" value for a whole data set. The three most common are the mean, the median, and the mode.

The Mean (Average)

The mean is what most people call the "average." To find it:

$\text{mean} = \frac{\text{sum of all the values}}{\text{how many values there are}}$

Worked Example

Five students scored: $7, 9, 10, 6, 8$.

$\text{sum} = 7 + 9 + 10 + 6 + 8 = 40$ $\text{mean} = \frac{40}{5} = 8$

The mean score is $8$.

💡 Think of it as fair sharing. If all $40$ points were split equally among the $5$ students, each would get $8$. The mean is the value everyone would have if the total were shared evenly.

Find the Mean 🧮

Add up the values, then divide by how many there are.

1) Data: $4, 8, 6, 2$. Mean $= \,?$ 2) Data: $10, 10, 13, 7$. Mean $= \,?$ 3) Data: $3, 5, 9, 6, 2$. Mean $= \,?$

The Median (Middle Value)

The median is the middle value when the data is put in order from least to greatest.

Odd number of values

Data: $3, 7, 4, 9, 5$. First put it in order: $3, \; 4, \; \boxed{5}, \; 7, \; 9$ There are $5$ values, so the 3rd one is the middle. The median is $5$.

Even number of values

When there is no single middle, the median is the mean of the two middle values.

Data in order: $2, \; \boxed{6, \; 8}, \; 10$. The two middle values are $6$ and $8$: $\text{median} = \frac{6 + 8}{2} = 7$

⚠️ Always sort first! Finding the "middle" of an unsorted list gives the wrong answer.

Concept Check 🎯

The Mode (Most Frequent)

The mode is the value that appears most often.

• Data $2, 4, 4, 4, 7, 9$ → the mode is $4$ (it appears three times).
• A data set can have more than one mode if two values tie for "most frequent."
• A data set can have no mode if every value appears the same number of times.

🔑 Mode is the only measure of center that works for categorical data. You can find the most common eye color even though "average eye color" makes no sense.

Match the Measure 🔽

For the data set $5, 8, 8, 8, 11$, choose each value. (Notice the values are already in order.)

📊 Introduction to Statistics

Part 3 of 5 — Spread, Range, and Outliers

🔑 The Big Idea: A center tells you the "typical" value, but two data sets with the same mean can look completely different. Spread describes how far apart the values are, and outliers are values that sit far from the rest.

The Range

The simplest measure of spread is the range:

$\text{range} = (\text{largest value}) - (\text{smallest value})$

Worked Example

Daily high temperatures (°C): $18, 22, 15, 24, 20$.

• Largest value: $24$
• Smallest value: $15$ $\text{range} = 24 - 15 = 9$

The temperatures spanned $9$ degrees.

💡 A small range means the values are close together (consistent); a large range means they are spread out. Range uses only the two extremes, so one unusual value can stretch it a lot.

Find the Range 🧮

Range = largest − smallest.

1) Data: $12, 5, 9, 20, 8$. Range $= \,?$ 2) Data: $44, 50, 47, 41$. Range $= \,?$ 3) Test scores: $70, 95, 88, 70, 100$. Range $= \,?$

Outliers

An outlier is a value that is much larger or much smaller than the rest of the data.

Consider weekly allowances ($): $10, 12, 11, 13, 60$.

Four students get around $10–13, but one gets $60. That $60 is an outlier — it stands far apart from the rest.

How an outlier affects the mean

$\text{mean} = \frac{10 + 12 + 11 + 13 + 60}{5} = \frac{106}{5} = 21.2$

The mean is $21.2 — but no one actually gets close to that! The single large outlier dragged the mean way up.

⚠️ The mean is sensitive to outliers. One extreme value can pull it far from where most of the data sits.

Median vs. Mean with Outliers

Compare what each measure does to the allowance data $10, 11, 12, 13, 60$ (now in order):

The median ($12$) describes a typical student far better than the mean ($21.2$) does here.

🔑 Key Idea: The median is resistant — outliers hardly move it. So when a data set has outliers (like incomes or house prices), the median is usually the more honest measure of center.

Concept Check 🎯

Mean, Median, and an Outlier 🧮

Data set (in order): $4, 5, 6, 7, 8, 30$.

1) What is the median? (even count → average the two middle values) 2) What is the mean? (sum ÷ count; decimal is fine)

📊 Introduction to Statistics

Part 4 of 5 — Displaying Data

🔑 The Big Idea: A good display lets you see a data set's story at a glance — what's typical, what's rare, and how spread out things are. We'll read and build three common displays: frequency tables, dot plots, and bar graphs.

Frequency Tables

A frequency table records how many times each value occurs. The frequency is the count.

Suppose 12 students reported how many siblings they have: $1, 0, 2, 1, 3, 1, 0, 2, 1, 2, 0, 1$

Tally them up:

💡 Check your work: the frequencies must add up to the total number of data values. Here $3 + 5 + 3 + 1 = 12$ ✓. The mode is easy to spot — it's the value with the highest frequency, $1$ (five students).

Read the Frequency Table 🧮

Use the siblings table above:

1) How many students have exactly $2$ siblings? 2) How many students have fewer than $2$ siblings? (that's $0$ siblings plus $1$ sibling) 3) How many students are in the whole data set? (add all the frequencies)

Dot Plots

A dot plot stacks one dot above a number line for each data value. Same siblings data:

        ●
        ●
●       ●       ●
●       ●       ●
●       ●       ●       ●
─────────────────────────
0       1       2       3

• The number of dots above each value equals its frequency ($3, 5, 3, 1$).
• The tallest stack marks the mode ($1$).
• Gaps and lone dots far from the pack reveal outliers.

💡 A dot plot shows the shape of the data — where it clumps and where it thins out — much faster than a list of numbers.

Read the Dot Plot 🎯

A dot plot shows the number of books students read last month:

●
●           ●
●   ●       ●           ●
──────────────────────────
0   1   2   3   4   5   6

(Stacks: 0 → 3 dots, 1 → 1 dot, 3 → 2 dots, 6 → 1 dot.)

Bar Graphs (for Categorical Data)

When data is categorical, we use a bar graph. Each bar's height shows the frequency of that category, and bars have gaps between them.

Favorite fruit of 20 students:

The tallest bar (Apple, $8$) is the most popular category — for categorical data the most frequent category is the mode.

⚠️ Bar graph vs. dot plot. Use a bar graph for categories (fruit, color, sport). Use a dot plot for numerical values you can put on a number line (counts, scores). Don't average categories — "mean fruit" is meaningless.

Choose the Right Display 🔽

Pick the better display for each data set.

📊 Introduction to Statistics

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell categorical from numerical data, (2) find the mean, median, and mode, (3) measure spread with the range and judge outliers, and (4) read and build data displays. Let's put it all together.

Quick Reference

🔑 Choosing a center: if the data has a big outlier, the median usually describes "typical" better than the mean.

Mixed Practice 🎯

Put It Together 🧮

A small data set: $3, 7, 4, 7, 9$.

1) Mean $= \,?$ (sum ÷ count; decimal is fine) 2) Median $= \,?$ (sort first!) 3) Mode $= \,?$ 4) Range $= \,?$

Reasoning Check 🔽

A neighborhood's home prices have a few mansions worth far more than the rest.

Exit Quiz ✅

Answer all three to finish the lesson.