Mean, Median, Mode, and Range

📊 Mean, Median, Mode, and Range

Part 1 of 5 — Meet the Four Measures

Topics in This Part

🔑 Key Concept: A list of numbers can be hard to understand all at once. Mean, median, mode, and range are four tools that squeeze a whole list down into a single useful number.

What Is a Data Set?

A data set is just a collection of numbers you've gathered. Each number is a data value.

Imagine you asked five friends how many books they read last month and got:

$7,\quad 2,\quad 9,\quad 4,\quad 3$

That list of five numbers is your data set. By itself it's a little messy. The four measures in this lesson each answer a different question about it:

💡 The mean, median, and mode all describe the center of the data. The range describes the spread.

Match the Word to Its Meaning 🔽

Pick the measure that answers each question.

Center vs. Spread

Think about a basketball team's heights.

• A center number (mean, median, or mode) tells you a typical height — about how tall a usual player is.
• A spread number (range) tells you how different the players are — is everyone about the same height, or are there both very short and very tall players?

You almost always want both: one number for the center, and one for the spread. Together they paint the whole picture.

⚠️ Watch out: "Average" in everyday speech usually means the mean, but median and mode are also "averages" in math class. Always read carefully to see which one is being asked for.

Concept Check 🎯

Which Measure Fits? 🔽

For each real-life question, choose the single best measure to use.

What's Next

You now know the four words and what each one is for:

🔑 Mean = average · Median = middle · Mode = most · Range = spread.

In the next four parts we'll learn exactly how to calculate each one, starting with the mean. Then in Part 5 you'll find all four for the same data set and choose which one best describes it.

📊 Mean, Median, Mode, and Range

Part 2 of 5 — The Mean (Average)

🔑 The Idea: The mean is the "fair share." If everyone put their stuff into one big pile and split it back out equally, each person's equal share is the mean.

How to Find the Mean

There are just two steps:

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

1. Add up every number.
2. Divide by how many numbers there are.

Worked Example

Find the mean of $12,\ 8,\ 10,\ 6,\ 14$.

Step 1 — Add: $12 + 8 + 10 + 6 + 14 = 50$

Step 2 — Divide by how many numbers (there are $5$): $\text{Mean} = \frac{50}{5} = 10$

✅ The mean is $10$. Notice it sits right in the middle of the list — that's typical of the mean.

Why "Fair Share" Works

Picture five jars holding $12, 8, 10, 6,$ and $14$ marbles. That's $50$ marbles in all. If you poured them together and split them evenly among the $5$ jars, each jar would get:

$\frac{50}{5} = 10 \text{ marbles}$

So the mean is the amount each jar would hold if everything were shared equally. That's why the mean can land on a number that isn't even in the original list!

💡 The mean does not have to be one of the original values. The mean of $5$ and $8$ is $\frac{5+8}{2} = 6.5$ — a number that wasn't in the list at all.

Concept Check 🎯

Find the Mean 🧮

Add, then divide by how many numbers there are.

1) Mean of $5,\ 5,\ 8,\ 2$ 2) Mean of $90,\ 85,\ 100,\ 95,\ 80$ 3) Mean of $6,\ 8,\ 10,\ 12,\ 14$

Work Backward 🧮 (challenge)

Sometimes you know the mean and have to find a missing value.

Four numbers have a mean of $10$. Three of them are $8,\ 12,$ and $7$. What is the fourth number?

📊 Mean, Median, Mode, and Range

Part 3 of 5 — The Median (Middle)

🔑 The Idea: The median is the number smack in the middle of the list — after you put the numbers in order. Half the data is below it, half is above it.

Step Zero: Always Put the Numbers in Order

The single most common median mistake is forgetting to sort the numbers first. The middle of an unsorted list means nothing.

Worked Example — Odd Number of Values

Find the median of $6,\ 2,\ 9,\ 4,\ 7$.

Step 1 — Order them (smallest to largest): $2,\quad 4,\quad \boxed{6},\quad 7,\quad 9$

Step 2 — Find the middle. There are $5$ numbers, so the middle one is the $3$rd:

$\text{Median} = 6$

✅ With an odd count, exactly one number sits in the middle. Two numbers are below it ($2, 4$) and two are above it ($7, 9$).

When There's an Even Number of Values

If there are an even number of values, two numbers share the middle. The median is the mean of those two middle numbers (add them and divide by $2$).

Worked Example — Even Number of Values

Find the median of $3,\ 8,\ 5,\ 2,\ 9,\ 6$.

Step 1 — Order them: $2,\quad 3,\quad \boxed{5,\ 6},\quad 8,\quad 9$

Step 2 — The two middle numbers are $5$ and $6$. Average them:

$\text{Median} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5$

💡 So the median of an even-length list often lands "between" two numbers — like $5.5$ here. That's perfectly normal.

Find the Median Step by Step 🔽

You're finding the median of $10,\ 4,\ 7,\ 1$.

Find the Median 🧮

Order first, then find the middle. (Decimals are fine.)

1) Median of $12,\ 5,\ 8,\ 3,\ 15,\ 9$ 2) Median of $14,\ 3,\ 9,\ 21,\ 7$

Concept Check 🎯

📊 Mean, Median, Mode, and Range

Part 4 of 5 — The Mode and the Range

🔑 Two quick ones: The mode is the value that appears most often. The range is the highest value minus the lowest value. Neither one needs any dividing.

The Mode — Most Often

The mode is the number that shows up the most times. A handy trick: "MOde" and "MOst" both start with MO.

Worked Example

Find the mode of $4,\ 7,\ 4,\ 9,\ 2,\ 4$.

Count how many times each number appears:

The number $4$ appears most (three times), so the mode is $4$.

Special Cases

• No mode: If every number appears the same number of times, there is no mode. Example: $1, 2, 3, 4$ — each appears once.
• More than one mode: If two numbers tie for most, the set has two modes. Example: in $2, 3, 3, 5, 5, 8$, both $3$ and $5$ appear twice — so the modes are $3$ and $5$.

⚠️ The mode is the only one of the four that must be an actual value from the list (you're literally picking a number that's there).

Mode Check 🎯

The Range — How Spread Out

The range measures how far apart the data is. It's the simplest formula in the whole lesson:

$\text{Range} = \text{highest value} - \text{lowest value}$

Worked Example

Find the range of $14,\ 3,\ 9,\ 21,\ 7$.

• Highest value: $21$
• Lowest value: $3$

$\text{Range} = 21 - 3 = 18$

💡 A small range means the numbers are bunched close together. A large range means they're widely spread out. The range is a single number, not a pair — say "the range is $18$," not "from $3$ to $21$."

Find the Mode and Range 🧮

For each set, give the answer asked for.

1) Mode of $5,\ 9,\ 5,\ 12,\ 9,\ 5$ 2) Range of $68,\ 72,\ 65,\ 80,\ 75,\ 70$ 3) Range of $6,\ 7,\ 7,\ 8,\ 8,\ 8,\ 9$

Mode or Range? 🔽

For the data set $2,\ 3,\ 3,\ 5,\ 5,\ 8$, choose the correct value.

📊 Mean, Median, Mode, and Range

Part 5 of 5 — All Four Together & Mastery Check

You can now find each measure on its own. The real skill is finding all four for the same data set and knowing which one best describes it. Let's put it all together.

Quick Reference

One Full Example — Plant Heights (cm)

A class measured five seedlings: $3,\ 5,\ 5,\ 7,\ 10$ (already in order).

$\text{Mean} = \frac{3+5+5+7+10}{5} = \frac{30}{5} = 6$ $\text{Median} = 5 \;(\text{the middle of } 3,5,\boxed{5},7,10)$ $\text{Mode} = 5 \;(\text{appears twice})$ $\text{Range} = 10 - 3 = 7$

⚠️ Mean vs. Median: One very large value (like a single super-tall plant) pulls the mean up but barely moves the median. When data has an unusual extreme, the median is often the more "typical" description.

All Four at Once 🧮

Use the data set $4,\ 8,\ 6,\ 8,\ 4$. Find each measure.

1) Mean 2) Median 3) Range

(For the mode of this set, note there are actually two — we'll handle that in the quiz.)

Read the Question Carefully

In a mixed set of problems, the hardest part is no longer the arithmetic — it's noticing which measure each question is asking for. Slow down and underline the key word:

💡 The same data set can have a mean, median, mode, and range that are all different numbers. Match each answer to the word the question used.

Choose the Right Measure 🎯

Mixed Review 🔽

Different sets, different measures — read each label carefully.

Exit Quiz ✅

Answer all three to finish the lesson.