9.5 Comparing Data Sets & Circle Graphs

9.5 Comparing Data Sets & Circle Graphs

Quick Answer: Comparing two data sets requires both a measure of center and a measure of spread - two classes can share a mean of 78 and be nothing alike. Pair them correctly: mean with mean absolute deviation (MAD), median with interquartile range (IQR). A circle graph displays parts of a single whole, so sectors must total 100%, and each central angle equals the sector's percent times 360°. Circle graphs are honest about proportion and silent about sample size, which is exactly how they mislead.

---

Where This Sits in the Blueprint

Grade 7's Data Analysis and Probability category is a full quarter of the test, and comparison items are its centerpiece.

---

Why Center Alone Is Never Enough

Two seventh-grade classes take the same 100-point test.

Identical centers. Completely different classes. Class A is uniformly near the average; Class B contains students who are struggling badly and students who are far ahead. A teacher who saw only "mean 78" for both would plan the wrong lesson for one of them.

Computing the MAD for Class B: deviations from 78 are 18, 8, 0, 8, 18, summing to 52; $52 \div 5 = 10.4$. For Class A the deviations are 2, 1, 0, 1, 2, summing to 6; $6 \div 5 = 1.2$.

The MAD answers a concrete question: on average, how far is a typical value from the mean? Class B's typical student sits about 10 points from the average; Class A's sits about 1 point away.

---

Pairing Center with Spread

[!IMPORTANT] Do not mix the pairs. Reporting a median alongside a MAD, or a mean alongside an IQR, gives a resistant summary of one thing and a non-resistant summary of another. FAST comparison items often supply exactly one matched pair for each data set, and choosing the mismatched combination from the answer choices is a designed trap.

Interquartile range. $\text{IQR} = Q_3 - Q_1$. It measures the width of the middle 50% of the data and ignores the tails entirely, which is precisely why an outlier cannot move it.

---

Comparing with Box Plots

Box plots make two data sets comparable at a glance. Read them in a fixed order: median line, box width, whisker lengths, overlap.

Daily minutes of reading, two classes

Class A   |------[====|====]------|
          20    35   45   55     70          median 45, IQR 20

Class B   |----------[==|==]---|
                     50 55 60  68            median 55, IQR 10
          20                   70

• Center: Class B's median (55) exceeds Class A's (45), so B's typical student reads more.
• Spread: Class A's IQR is 20, twice Class B's 10, so A's students vary far more.
• Overlap: the boxes overlap between 50 and 55, so plenty of Class A students out-read plenty of Class B students. The difference is in the typical value, not in every individual.

That last point is what comparison items reward. A correct answer says the typical value differs; a wrong answer says every student in one class beats every student in the other.

The Language of a Good Comparison

A complete comparison statement has three parts: center, spread, and context.

Class B's median reading time (55 minutes) is 10 minutes greater than Class A's (45 minutes), and Class B is also more consistent, with an IQR of 10 minutes compared to Class A's 20 minutes. So a typical Class B student reads longer, and Class B students are more alike in how long they read.

---

Circle Graphs

A circle graph shows how one whole divides into parts. It is the right display when the question is "what share?" and the wrong display when the question is "how many?" or "how has it changed?"

Building One

Each sector's central angle is its share of the full 360°:

$\text{central angle} = \frac{\text{part}}{\text{whole}} \times 360°$

Example. A survey of 200 students asks their preferred lunch:

Both totals are the check: percents must sum to 100 and angles must sum to 360. If they do not, a value was misread or a category was dropped.

Reading One Backward

Given a sector of 126° out of a 350-student survey, how many students does it represent?

$\frac{126}{360} = 0.35 \quad \Rightarrow \quad 0.35 \times 350 = 122.5 \approx 123 \text{ students}$

Two proportional steps: angle to share, then share to count.

What a Circle Graph Cannot Show

[!WARNING] The classic FAST distractor compares two circle graphs by sector alone: "More students at Lincoln chose soccer than at Jefferson, because Lincoln's sector is larger." That conclusion needs the enrollment numbers, and a circle graph never supplies them. If a comparison item offers no totals, the honest answer is that the comparison cannot be made.

---

Common Exam Traps & Misconceptions

[!WARNING]

Trap 1: Declaring two data sets identical because their means match

Equal means say nothing about spread. Always check MAD, IQR, or range before concluding the groups are alike.

[!WARNING]

Trap 2: Treating a wider box as more data

A box plot's box width shows the spread of the middle 50% of values, not how many values there are. Two box plots from samples of 20 and 2,000 can look the same.

[!WARNING]

Trap 3: Overstating a comparison

"Class B reads more than Class A" is defensible about typical values. "Every Class B student reads more than every Class A student" is refuted the moment the boxes or whiskers overlap.

[!WARNING]

Trap 4: Reading a circle graph sector as a count

A 25% sector is a quarter of that survey's respondents. Without the total, it is not a number of people.

[!WARNING]

Trap 5: Forgetting that angles must total 360

When a circle graph item supplies all sectors but one, the missing sector is $360°$ minus the others - or 100% minus the others. Subtracting from 180 or from 100 degrees is a frequent slip.