Measures of Center and Variability

📊 Measures of Center and Variability

Part 1 of 5 — The Mean (Average)

Topics in This Part

🔑 Key Concept: A whole list of numbers can be summarized by a single typical value called a measure of center. The most famous one is the mean — what you probably call the "average."

What Is a Measure of Center?

A data set is just a list of numbers — test scores, heights, the number of pets each kid owns.

A measure of center is a single number that describes the "middle" or "typical" value of the whole list. It answers the question:

"If I had to pick one number to stand for this whole group, what would it be?"

In this lesson you'll meet three measures of center — the mean, the median, and the mode — plus measures of variability (how spread out the data is). We start with the mean.

The Mean = the "Fair Share"

To find the mean, add up all the values, then divide by how many values there are:

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

Think of it as a fair share: if everyone pooled their amounts together and split the total evenly, the mean is what each one would get.

Worked Example

Four friends have $7, 9, 11,$ and $13$ stickers.

$\text{mean} = \frac{7 + 9 + 11 + 13}{4} = \frac{40}{4} = 10$

If they pooled all $40$ stickers and split them evenly, each would get $10$. ✓

💡 The mean does not have to be a number that actually appears in the list. No friend has exactly $10$ stickers, yet $10$ is still the mean.

Concept Check 🎯

Find the Mean 🧮

Add the values and divide by how many there are.

1) Mean of $6, 7, 8, 9, 10 = \,?$ 2) Mean of $3, 3, 4, 10 = \,?$ 3) Mean of $12, 15, 9, 8, 6 = \,?$

Working Backward: A Missing Value

If you know the mean, you also know the total — just multiply:

$\text{sum} = \text{mean} \times \text{number of values}$

Worked Example

Five quiz scores have a mean of $20$. Four of the scores are $18, 22, 15,$ and $19$. What is the fifth?

$\text{total} = 20 \times 5 = 100$ $\text{four known scores} = 18 + 22 + 15 + 19 = 74$ $\text{fifth score} = 100 - 74 = 26$

✅ Check: $\dfrac{18+22+15+19+26}{5} = \dfrac{100}{5} = 20$ ✓

Work Backward 🧮

1) Six numbers have a mean of $10$. What is their total? 2) A student's three test scores are $80, 90,$ and $?$. The mean is $85$. What is the missing score?

📊 Measures of Center and Variability

Part 2 of 5 — Median & Mode

🔑 The Idea: The mean isn't the only way to find a "center." The median is the literal middle of a sorted list, and the mode is the value that shows up most often.

The Median = the Middle

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

Step 1 — Always sort first. This is the step students forget most.

Odd number of values: the median is the single value in the middle.

$3,\; 5,\; \boxed{7},\; 9,\; 11 \quad\Rightarrow\quad \text{median} = 7$

Even number of values: there are two middle values — the median is the mean of those two.

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

⚠️ Sort first, every time. The median of $5, 2, 9, 2, 7$ is not $9$. Sorted it is $2, 2, \boxed{5}, 7, 9$, so the median is $5$.

Concept Check 🎯

The Mode = the Most Common

The mode is the value that appears most often.

💡 A data set can have one mode, more than one mode, or no mode at all. That makes the mode different from the mean and median, which a list always has exactly one of.

The mode is the only measure of center that also works for things you can't average — like the most common color of car in a parking lot.

Match Each Data Set to Its Center 🔽

Use the data set $90, 85, 100, 75, 95$ (five test scores).

Median & Mode Drill 🧮

1) Median of $4, 8, 15, 16, 23, 42 = \,?$ (even count — average the middle two) 2) Mode of $2, 7, 7, 7, 9, 12 = \,?$ 3) Median of $5, 2, 9, 2, 7 = \,?$ (sort first!)

📊 Measures of Center and Variability

Part 3 of 5 — Range, Outliers & Choosing the Right Center

🔑 The Idea: Two data sets can have the same center but feel completely different because one is more spread out. The simplest measure of spread is the range, and a single extreme value — an outlier — can change which center to trust.

The Range = Spread from Low to High

The range measures variability — how spread out the data is. It is the largest value minus the smallest value:

$\text{range} = \text{maximum} - \text{minimum}$

Examples

💡 A range of $0$ means every value is the same — there is no spread at all. The larger the range, the more spread out the data.

Concept Check 🎯

Outliers Pull the Mean

An outlier is a value that is much larger or much smaller than the rest. Outliers affect the mean a lot, but the median barely at all.

Worked Example

A small online store sold these numbers of items on five days: $2, 3, 4, 5, 86$.

Mean: $\dfrac{2+3+4+5+86}{5} = \dfrac{100}{5} = 20$

Median: sorted is $2, 3, \boxed{4}, 5, 86$, so the median $= 4$.

The mean ($20$) is bigger than every value except the $86$. That one huge day pulled the mean way up, so it doesn't describe a typical day. The median ($4$) does.

🔑 Rule of thumb: When a data set has an outlier, the median is usually the better measure of center, because it isn't dragged around by extreme values.

Pick the Better Center 🔽

A class's homework times (in minutes) are $10, 12, 11, 13, 90$. The $90$ is an outlier (someone forgot for weeks!).

Range Drill 🧮

Find the range $= \text{max} - \text{min}$ for each set.

1) $4, 8, 15, 16, 23, 42 \Rightarrow \text{range} = \,?$ 2) $12, 15, 9, 8, 6 \Rightarrow \text{range} = \,?$ 3) $7, 7, 7, 7 \Rightarrow \text{range} = \,?$

📊 Measures of Center and Variability

Part 4 of 5 — Mean Absolute Deviation (MAD)

🔑 Big Idea: The range only looks at the two most extreme values. The mean absolute deviation (MAD) uses every value to measure spread: it's the average distance of the data from the mean.

What Is the MAD?

The mean absolute deviation answers: on average, how far is each value from the mean?

Steps to find the MAD:

1. Find the mean of the data.
2. Find each value's distance from the mean (always positive — that's the "absolute" part).
3. Find the mean of those distances.

$\text{MAD} = \frac{\text{sum of distances from the mean}}{\text{number of values}}$

💡 "Absolute" just means we drop the minus sign — a distance is never negative. A value $4$ below the mean and a value $4$ above the mean are both a distance of $4$ away.

Worked Example: $2, 4, 6, 8, 10$

Step 1 — Mean: $\dfrac{2+4+6+8+10}{5} = \dfrac{30}{5} = 6$.

Step 2 — Distance of each value from $6$:

Step 3 — Mean of the distances:

$\text{MAD} = \frac{4 + 2 + 0 + 2 + 4}{5} = \frac{12}{5} = 2.4$

So on average, each value sits $2.4$ units away from the mean of $6$.

⚠️ Use the distance (always positive), not the signed difference. If you forget the absolute value, the positives and negatives cancel and you'll always get $0$.

Concept Check 🎯

Build the MAD Step by Step 🧮

Data set: $5, 8, 8, 11$.

1) Mean $= \,?$ 2) Sum of the distances from the mean $= \,?$ 3) MAD $= \,?$

Comparing Spread with the MAD

Two groups of five students each have a mean score of $50$, but very different spreads:

Both groups center at $50$, but Group B is much more spread out. The MAD makes that precise: B's values sit on average $6$ points from the mean, while A's sit only $1.2$ points away.

🔑 A larger MAD means more variability (more spread). A smaller MAD means the data clusters tightly around the mean.

MAD Practice 🧮

1) Find the MAD of $3, 5, 7, 9$. (mean is $6$) 2) Find the MAD of $10, 12, 14, 16, 18$. (mean is $14$)

📊 Measures of Center and Variability

Part 5 of 5 — Mixed Practice & Mastery Check

You can now find the mean, median, and mode (center), the range and MAD (variability), and you know when an outlier makes the median the smarter choice. Let's put it all together.

Quick Reference

⚠️ Two habits that prevent most mistakes: (1) always sort before finding a median, and (2) always use positive distances when finding a MAD.

💡 Center vs. variability: mean / median / mode describe the typical value; range and MAD describe how spread out the data is. A full summary of data usually reports one of each.

Mixed Practice 🎯

Final Computation Drill 🧮

Use the data set $1, 3, 5, 7$ for #1–#2, then answer #3.

1) Mean of $1, 3, 5, 7 = \,?$ 2) MAD of $1, 3, 5, 7 = \,?$ 3) Range of $14, 8, 21, 5, 17 = \,?$

Choose the Right Tool 🔽

Match each goal to the measure you would use.

Exit Quiz ✅

Answer all three to finish the lesson.