Measures of Center

📊 Measures of Center

Part 1 of 5 — What "Center" Means

Topics in This Part

🔑 Key Concept: A measure of center is a single number that represents a "typical" value in a data set. The three you must know are the mean, the median, and the mode.

Why Summarize Data?

Imagine a teacher records every quiz score in a class:

$7,\; 8,\; 8,\; 9,\; 10,\; 6,\; 9,\; 7,\; 8,\; 10$

A list of ten numbers is hard to talk about. Instead we ask: "What is a typical score?" That one summary number is a measure of center.

The three standard measures answer slightly different questions:

💡 They often disagree — and which one to trust depends on the data. That's the whole point of this lesson.

The Mean (Average)

The mean is the sum of all the values divided by how many values there are:

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

Worked Example

Find the mean of $\;4,\; 8,\; 15,\; 16,\; 23,\; 42$.

Step 1 — Add them up: $4 + 8 + 15 + 16 + 23 + 42 = 108$

Step 2 — Count the values: there are $6$ numbers.

Step 3 — Divide: $\text{mean} = \frac{108}{6} = 18$

🔑 Key Idea: The mean uses every number, so it is pulled toward unusually large or small values. Hold onto that — it becomes important in Part 3.

Concept Check 🎯

Compute the Mean 🧮

Find the mean of each data set. (Decimals are fine.)

1) $\;6,\; 10,\; 14$ 2) $\;3,\; 5,\; 8,\; 12$ 3) $\;20,\; 20,\; 30,\; 40,\; 40$

Where We're Headed

You can now compute the mean of any list. But the mean is only one of three measures.

In Part 2 we find the median (the middle) and the mode (the most common value) — and you'll see they're often easier to read straight off a sorted list. Before we move on, lock in which name goes with which idea.

💡 Almost every center problem starts the same way: sort the numbers first. Keep that habit.

Match the Measure 🔽

Each measure answers a different question. Match each description to the right measure of center.

📊 Measures of Center

Part 2 of 5 — Median & Mode

🔑 Golden Rule: To find the median or mode, always sort the data from least to greatest first. An unsorted list will fool you every time.

The Median (Middle Value)

The median is the value in the middle of a sorted list.

Odd number of values → there is one exact middle value.

$3,\; 7,\; \boxed{8},\; 12,\; 19 \quad\Rightarrow\quad \text{median} = 8$

Even number of values → there are two middle values; the median is their average.

$2,\; \boxed{5,\; 7},\; 10 \quad\Rightarrow\quad \text{median} = \frac{5+7}{2} = 6$

⚠️ Don't skip the sort! The median of $9,\; 2,\; 5$ is not $2$. Sorted, it's $2,\; 5,\; 9$, so the median is $5$.

Concept Check 🎯

The Mode (Most Common Value)

The mode is the value that appears most often.

$2,\; 3,\; 3,\; 5,\; 7,\; 7,\; 7,\; 9 \quad\Rightarrow\quad \text{mode} = 7 \;\;(\text{appears } 3 \text{ times})$

A data set can have special cases:

💡 The mode is the only measure of center that works for non-numeric data — like the most common eye color or favorite sport.

Concept Check 🎯

Median & Mode Drill 🧮

Sort first, then answer.

1) Median of $\;9,\; 3,\; 6,\; 1,\; 6\;$ (5 values) 2) Median of $\;10,\; 2,\; 6,\; 8\;$ (4 values) 3) Mode of $\;4,\; 7,\; 4,\; 9,\; 4,\; 7$

📊 Measures of Center

Part 3 of 5 — Outliers & Which Measure to Use

🔑 The Big Question: When the mean and median disagree, which one should you report? The answer depends on whether the data has an outlier.

What Is an Outlier?

An outlier is a value that is much larger or much smaller than the rest of the data. Watch what one outlier does:

$\text{Data: } \;3,\; 4,\; 5,\; 6,\; 7$ $\text{mean} = \frac{25}{5} = 5, \qquad \text{median} = 5$

Now add one huge value, $\;100$:

$\text{Data: } \;3,\; 4,\; 5,\; 6,\; 7,\; 100$ $\text{mean} = \frac{125}{6} \approx 20.8, \qquad \text{median} = \frac{5+6}{2} = 5.5$

The mean jumped from $5$ to about $20.8$ — but the median barely moved (from $5$ to $5.5$).

⚠️ Key Insight: The mean is sensitive to outliers; the median is resistant to them. One extreme value can drag the mean far from "typical."

Pick the Right Measure 🔽

For each situation, choose the measure of center that best represents a typical value.

A Quick Rule of Thumb

💡 This is why news reports usually quote the median household income, not the mean — a few billionaires would inflate the mean and make it misleading.

Concept Check 🎯

See the Outlier's Effect 🧮

Data set: $\;2,\; 4,\; 6,\; 8,\; 80$

1) Find the mean. (decimal is fine) 2) Find the median.

📊 Measures of Center

Part 4 of 5 — Working Backward & Weighted Means

🔑 Level Up: Real problems rarely just say "find the mean." They give you the mean and ask for a missing value, or they weight some values more than others. This part handles both.

Finding a Missing Value

If you know the mean, you can find a missing data value. The trick: mean × count = total sum.

Worked Example

Four quiz scores have a mean of $80$. Three of them are $75,\; 82,\; 88$. Find the fourth.

Step 1 — Find the required total using $\text{sum} = \text{mean} \times \text{count}$: $\text{sum} = 80 \times 4 = 320$

Step 2 — Add the known scores: $75 + 82 + 88 = 245$

Step 3 — Subtract to find the missing score: $320 - 245 = 75$

✅ Check: $\dfrac{75 + 82 + 88 + 75}{4} = \dfrac{320}{4} = 80$ ✓

Work Backward 🧮

1) Five numbers have a mean of $12$. What is their total sum? 2) Three test scores have a mean of $90$. Two of them are $85$ and $92$. Find the third score.

The Weighted Mean

Sometimes values count for different amounts. A weighted mean multiplies each value by its weight, then divides by the total weight.

Worked Example: A Course Grade

Your grade is 40% homework and 60% exams. You earned $85$ on homework and $90$ on exams.

$\text{grade} = (0.40)(85) + (0.60)(90)$ $= 34 + 54 = 88$

Because the weights ($0.40 + 0.60 = 1$) already add to $1$, you do not divide again.

💡 A plain mean is just a weighted mean where every value has the same weight. Weights let some values "count more."

Build the Weighted Mean 🔽

A grade is 30% quizzes and 70% final exam. A student scored $80$ on quizzes and $90$ on the final.

Weighted Mean Practice 🧮

A class grade is 20% labs and 80% tests. A student earned $95$ on labs and $85$ on tests.

1) Lab contribution: $(0.20)(95) = \,?$ 2) Test contribution: $(0.80)(85) = \,?$ 3) Final grade (sum of the two contributions): $\,?$

📊 Measures of Center

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) compute the mean, median, and mode, (2) decide which one fits the data, and (3) work backward and use weighted means. Let's put it all together.

Quick Reference

⚠️ Always sort first for median and mode. And remember: $\text{sum} = \text{mean} \times \text{count}$ is the key to every "find the missing value" problem.

Mixed Practice 🧮

Data set: $\;12,\; 15,\; 15,\; 18,\; 20$

1) Mean (decimal is fine) 2) Median 3) Mode

From Numbers to Word Problems

On a test, measures of center usually arrive dressed up as a story: a missing grade, a misleading average, a weighted course score. The math is identical — the skill is translating the words into the right move.

🔑 Read the question and ask: Am I being told the mean (so I can find a total), warned about an outlier (so I should pick the median), or given weights (so I build a weighted mean)? The next two checks practice exactly that.

Apply It 🎯

Exit Quiz ✅

Answer all three to finish the lesson.