Mean, Median, and Mode

📊 Mean, Median, and Mode

Part 1 of 5 — Measures of Center

Topics in This Part

🔑 Key Concept: A whole list of numbers is hard to picture. A measure of center squeezes that list down to one number that stands in for the "typical" value. The three you'll master are the mean, the median, and the mode.

What "Average" Really Means

When someone says a class has an average score of $85$, they mean: one number that summarizes how the whole class did. That's a measure of center — it points to the middle, or typical, value of a data set.

There are three common ones, and they answer slightly different questions:

💡 The word average usually means the mean, but median and mode are also kinds of averages. This lesson teaches all three — and, just as importantly, when to use each one.

Concept Check 🎯

Reading a Data Set

A data set is just a list of values you collected. For example, the number of pages five students read last night:

$7,\;\; 4,\;\; 9,\;\; 4,\;\; 6$

Two words you'll use constantly:

• Count ($n$): how many values there are. Here $n = 5$.
• Sum: all the values added together. Here $7 + 4 + 9 + 4 + 6 = 30$.

⚠️ Don't confuse count with sum. The count is how many numbers you have ($5$ of them); the sum is what you get when you add them ($30$). The mean needs both.

Count and Sum 🧮

For the data set $8,\; 3,\; 5,\; 3,\; 6,\; 5$:

1) How many values are there? (the count $n$) 2) What is the sum of all the values?

Where We're Headed

You now know the three measures of center and the two ingredients — count and sum — that the mean depends on. Next:

• Part 2: the mean, step by step (including decimals).
• Part 3: the median, and the special rule for even-sized lists.
• Part 4: the mode, plus how to choose the right measure.
• Part 5: mixed practice and an exit quiz.

🔑 Keep this in mind: the mean uses every value, the median only cares about position, and the mode only cares about frequency. That difference is the whole story.

Match the Definition 🔽

Match each measure of center to how you compute it.

📊 Mean, Median, and Mode

Part 2 of 5 — The Mean (the Arithmetic Average)

🔑 The Idea: The mean answers "if everyone shared equally, how much would each get?" You find it by adding all the values and dividing by how many there are.

How to Find the Mean

$\text{Mean} = \frac{\text{sum of all values}}{\text{number of values}} = \frac{\text{sum}}{n}$

Worked Example: $7, 4, 9, 4, 6$

Step 1 — Add the values (the sum): $7 + 4 + 9 + 4 + 6 = 30$

Step 2 — Count the values: there are $n = 5$.

Step 3 — Divide: $\text{Mean} = \frac{30}{5} = 6$

✅ Check: A mean of $6$ should land somewhere inside the data — and it does ($6$ is between the smallest, $4$, and the largest, $9$). The mean can never be larger than the biggest value or smaller than the smallest.

When the Mean Isn't a Whole Number

The sum doesn't always divide evenly — and that's fine. The mean is often a decimal or fraction, even when every value is a whole number.

Worked Example: $10, 12, 15$

$\text{Mean} = \frac{10 + 12 + 15}{3} = \frac{37}{3} = 12.33\overline{3}\ldots$

So the mean is about $12.33$. A mean between two of the data values is completely normal.

💡 The mean does not have to be a value that appears in the list. The mean of $10, 12, 15$ is $\approx 12.33$ — a number that isn't even in the data.

Concept Check 🎯

A Two-Step Habit

For every mean, run the same two steps in order:

1. Sum — add all the values.
2. Divide — by $n$, the count.

💡 If your answer comes out bigger than the largest value or smaller than the smallest, you've made an arithmetic slip — go back and recheck the sum or the division.

Compute the Mean 🧮

Find the mean of each set. Decimals are fine.

1) $4,\; 8,\; 6,\; 2$ 2) $12,\; 15,\; 9$ 3) $5,\; 6,\; 8,\; 6$ (decimal)

Working Backwards from the Mean

Because $\text{mean} = \dfrac{\text{sum}}{n}$, you can rearrange to get $\text{sum} = \text{mean} \times n$. This lets you solve missing-value problems.

Example: Find the missing test score

Maria's four test scores are $80, 90, 70,$ and an unknown $x$. Her mean is $85$. Find $x$.

Step 1 — Total points needed: $\text{sum} = \text{mean} \times n = 85 \times 4 = 340$.

Step 2 — Subtract the known scores: $x = 340 - (80 + 90 + 70) = 340 - 240 = 100$.

✅ Check: $\dfrac{80 + 90 + 70 + 100}{4} = \dfrac{340}{4} = 85$ ✓

Find the Missing Value 🧮

1) Three numbers have a mean of $20$. Two of them are $15$ and $18$. What is the third? 2) A student's five quiz scores average $8$. Four of the scores are $7, 9, 6,$ and $10$. What is the fifth score?

📊 Mean, Median, and Mode

Part 3 of 5 — The Median (the Middle Value)

🔑 The Idea: The median is the value sitting in the middle once the data is lined up in order. Half the values are below it, half above. The single most important rule: sort first.

Finding the Median — Odd Number of Values

Step 1 — Sort the values from least to greatest. Step 2 — Find the middle value. With an odd count, exactly one value sits in the center.

Worked Example: $7, 4, 9, 4, 6$

Sort: $\;4,\; 4,\; 6,\; 7,\; 9$

There are $5$ values, so the middle one is the 3rd:

$4,\; 4,\; \underline{6},\; 7,\; 9 \quad\Rightarrow\quad \text{median} = 6$

⚠️ The #1 median mistake is forgetting to sort. If you read straight off the unsorted list $7,4,9,4,6$, the "middle" looks like $9$ — which is wrong. Always sort first.

Median of an Odd List 🧮

Sort each list in your head, then enter the median.

1) $5,\; 2,\; 8,\; 1,\; 9$ 2) $12,\; 7,\; 7,\; 15,\; 10,\; 7,\; 20$

Finding the Median — Even Number of Values

With an even count there's no single middle value — there are two middle values. The median is the mean of those two (add them, divide by $2$).

Worked Example: $4, 4, 6, 7, 9, 12$

This list is already sorted and has $6$ values. The two middle ones are the 3rd and 4th: $6$ and $7$.

$\text{median} = \frac{6 + 7}{2} = \frac{13}{2} = 6.5$

💡 Just like the mean, the median can be a value not in the list — here $6.5$ isn't one of the data points. That's expected for even-sized data sets.

Concept Check 🎯

Finding the Middle Position Quickly

For a sorted list of $n$ values, the middle position is $\dfrac{n+1}{2}$.

• Odd $n$ (say $n = 7$): $\dfrac{7+1}{2} = 4$, so the median is the 4th value.
• Even $n$ (say $n = 6$): $\dfrac{6+1}{2} = 3.5$ — between the 3rd and 4th values, so average those two.

💡 The half-position is a quick reminder that even-sized lists put the median between two data points.

Median of an Even List 🧮

Sort first, then average the two middle values. Decimals are fine.

1) $3,\; 9,\; 5,\; 1$ 2) $20,\; 14,\; 18,\; 11,\; 16,\; 13$

Why the Median Is "Tougher" Than the Mean

The median only cares about position, not size. So one wildly large or small value — an outlier — barely moves it.

Compare the data set $2, 3, 4, 5, 6$ with the same set but a typo turns the last value into $600$:

The single outlier dragged the mean from $4$ to $122.8$, but the median didn't budge. We say the median is resistant to outliers — a key idea we'll use in Part 4 to choose between them.

📊 Mean, Median, and Mode

Part 4 of 5 — The Mode & Choosing a Measure

🔑 The Idea: The mode is the value that appears most often. Unlike the mean and median, a data set can have no mode, one mode, or several modes — and the mode is the only measure that works on words and categories, not just numbers.

Finding the Mode

The mode is the value that appears the most times. A quick way: sort the list, then look for repeats.

Worked Example: $7, 4, 9, 4, 6$

Sort: $\;4,\; 4,\; 6,\; 7,\; 9$. The value $4$ appears twice; every other value appears once.

$\text{mode} = 4$

Special cases

💡 The mode is the only measure of center you can use for non-numeric data, like favorite colors or shoe sizes. You can't take the "mean color," but you can find the most common one.

Concept Check 🎯

A Reliable Way to Spot the Mode

When a list is long, it's easy to miscount. A safe method:

1. Sort the list (repeats end up side by side).
2. Tally how many times each distinct value appears.
3. The value with the highest tally is the mode.

⚠️ Don't confuse the mode with how many times it appears. For $3, 3, 3, 7$, the mode is $\textbf{3}$ (the value), not $3$ as in "three times" — even though they happen to match here, they mean different things.

Find the Mode 🧮

Enter the mode of each list. (Each of these has exactly one mode.)

1) $8,\; 5,\; 8,\; 2,\; 8,\; 5$ 2) $11,\; 14,\; 11,\; 9,\; 14,\; 11,\; 9$

Choosing the Right Measure

Which measure best describes the "typical" value? It depends on the data.

Classic example — house prices on a street:

$180k, $190k, $200k, $210k, $5,000k

That one mansion is an outlier. The mean (about $1,156k) makes the street look far pricier than it is; the median ($200k) describes a typical home much better.

⚠️ Outliers wreck the mean but not the median. When income, home prices, or any data has extreme values, the median is usually the fairer summary.

Pick the Best Measure 🔽

Choose the most appropriate measure of center for each situation.

Four Properties Worth Memorizing

Before the final practice, lock in how the three measures behave:

🔑 These four rows explain why you'd ever pick one measure over another — exactly the judgment the next check asks for.

True Facts About the Measures 🔽

Complete each statement.

📊 Mean, Median, and Mode

Part 5 of 5 — Mixed Practice & Mastery Check

You can now find all three measures of center, handle even-sized lists, work backwards from a mean, and choose the right measure for a situation. Let's put it together — one data set at a time.

Quick Reference

🔑 One data set, three answers. For $4, 4, 6, 7, 9$: mean $= \frac{30}{5} = 6$, median $= 6$, mode $= 4$. They needn't agree — each describes the "center" in its own way.

⚠️ Top three slip-ups: (1) not sorting before finding the median, (2) forgetting to average the two middles on an even list, and (3) confusing count ($n$) with sum in the mean.

All Three at Once 🧮

For the data set $3,\; 7,\; 3,\; 9,\; 8$, find each measure of center.

1) Mean 2) Median 3) Mode

Before the Final Check

Run through this mental checklist on any "find the center" problem:

• Mean? Sum, then divide by $n$. (Sanity-check: it lands between the min and max.)
• Median? Sort first, then take the middle — average the two middles if $n$ is even.
• Mode? The most frequent value (could be none, one, or several).
• Which to report? Outliers or categories? Lean median or mode. Otherwise, the mean.

💡 The two questions ahead mix a direct computation with a work-backwards problem — the same skills from Parts 2–4, now without scaffolding.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.