A box plot (also called a box-and-whisker plot) is a graph that displays the distribution of a data set using five values: the minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), and maximum. The box spans from Q1 to Q3, and the line inside the box marks the median. The IQR = Q3 − Q1 measures the spread of the middle 50% of the data. Box plots appear on Florida FSA statistics standards (MAFS.912.S-ID) and the SAT Math “Problem Solving & Data Analysis” section.
A distribution of medians.
Formal definition: A box plot — also called a box-and-whisker plot — is a standardized graphical display of a data set’s distribution based on its five-number summary: minimum, Q1 (first quartile), median, Q3 (third quartile), and maximum. The “box” spans from Q1 to Q3, showing where the middle 50% of data points fall. The “whiskers” extend from the box to the minimum and maximum values (excluding outliers). A vertical line inside the box marks the median. Box plots allow rapid visual comparison of center, spread, and skewness across data sets.
Where you’ll see it: Box plots appear in Florida MAFS.912.S-ID.1 and MAFS.912.S-ID.2 standards, FSA Algebra 1 and Statistics assessments, SAT Math “Problem Solving & Data Analysis” domain (1–2 questions per test), AP Statistics, and ACT Mathematics.
| COMPONENT | WHAT IT IS | IN THE DIAGRAM ABOVE | SAT/FSA USE |
|---|---|---|---|
| Minimum | The smallest value in the data set (excluding outliers) | Left whisker endpoint = 5 | Range calculation: Max – Min |
| Q1 (First Quartile) | The median of the lower half of the data — 25% of data falls below this | Left edge of box = 10 | IQR calculation, outlier detection |
| Median (Q2) | The middle value — 50% of data falls below, 50% above | Line inside box = 18 | Center comparison between box plots |
| Q3 (Third Quartile) | The median of the upper half of the data — 75% of data falls below | Right edge of box = 26 | IQR calculation, outlier detection |
| Maximum | The largest value in the data set (excluding outliers) | Right whisker endpoint = 35 | Range: Max – Min = 35 – 5 = 30 |
| IQR | Q3 – Q1 = the span of the box = the range of the middle 50% | Q3 – Q1 = 26 – 10 = 16 | Spread comparison, outlier detection |
Every box plot problem on the SAT or Florida FSA requires one or more of these three calculations. The five-number summary builds the plot. The IQR measures its spread. The outlier formula determines which data points fall outside the whiskers.
How to find each:
The IQR is the width of the box – it measures the spread of the middle 50% of the data.
A large IQR → data is spread out (high variability).
A small IQR → data is clustered (low variability).
SAT questions frequently ask you to compare IQRs between two box plots – the wider box has the larger IQR.
IQR ≠ range. Range = Max – Min (includes all data).
A data point is an outlier if it falls below the lower fence or above the upper fence:
Outliers are plotted as individual dots beyond the whiskers – the whiskers stop at the last non-outlier value.
SAT Hard questions: "Which value, if added to the data set, would be classified as an outlier?" – calculate both fences first, then test each answer choice.
This rule is NOT given on the SAT reference sheet – it must be memorized.
Follow these steps in order. A common error is skipping Step 1 (ordering the data) — if the data is unordered, every subsequent step produces the wrong quartile values.
| # | STEP | COMMON ERROR |
|---|---|---|
| 1 | Order the data from least to greatest. Every calculation below depends on the data being sorted. | Skipping this step and finding quartiles from unsorted data – the most common construction error. |
| 2 | Find Min and Max. These become the endpoints of the whiskers (unless outliers are present). | Using the first and last value from the original unsorted list instead of the sorted list. |
| 3 | Find the Median (Q2). If odd count: the middle value. If even count: the average of the two middle values. | Confusing median with mean. The median is positional – always count from both ends to find the center. |
| 4 | Find Q1. The median of the data values below (not including) Q2. | Including Q2 in the lower half when the data count is odd – Q2 is excluded from both halves. |
| 5 | Find Q3. The median of the data values above (not including) Q2. | Same error as Step 4 on the upper half. Q2 is never part of the Q1 or Q3 calculation. |
| 6 | Draw the plot. Draw a number line → mark Min, Q1, Median, Q3, Max → draw a box from Q1 to Q3 → draw a vertical line at Median → extend whiskers from Q1 to Min and from Q3 to Max. | Drawing whiskers from the center of the box (the median) instead of from the edges (Q1 and Q3). |
A box plot has the following values: Min = 12, Q1 = 20, Median = 28, Q3 = 35, Max = 50. Find (a) the IQR, (b) the range, and (c) the percentage of data between Q1 and Q3.
Class A has a box plot with Q1 = 60, Median = 72, Q3 = 85. Class B has Q1 = 55, Median = 68, Q3 = 80. Which class has greater variability in scores? Which class performed better overall?
Box plots are tested in the SAT Math section under the “Problem Solving & Data Analysis” domain — the domain that accounts for approximately 17 of the 58 questions on the full SAT Math test. This domain is the most under-prepared section for most students because it requires data interpretation skills, not just formula recall. Box plot questions (1–2 per test) consistently appear at Medium–Hard difficulty because students who can calculate IQR still cannot correctly compare two box plots or identify outliers under time pressure. Florida student-athletes targeting NCAA eligibility or the Bright Futures Scholarship Academic Scholars threshold (1290+ SAT) often leave 3–5 entire data analysis questions blank — an avoidable loss of 30–50 SAT Math points.
| BOX PLOT QUESTION TYPE | SAT FREQUENCY | DIFFICULTY |
|---|---|---|
| Read five-number summary from a box plot | 1 per test | Easy–Medium |
| Calculate IQR from a given box plot | 1 per test | Medium |
| Compare median or IQR of two box plots | 1 per test | Medium–Hard |
| Determine if a value is an outlier (1.5×IQR rule) | 1 per 2 tests | Hard |
| Interpret box plot skewness (left vs. right skew) | 1 per 2 tests | Hard |
(a) IQR = Q3 − Q1 = 30 − 15 = 15. (b) Range = Max − Min = 45 − 8 = 37. (c) The box (Q1 to Q3) always contains 50% of the data — this is a definition. Answer: IQR = 15 · Range = 37 · 50% between Q1 and Q3.
IQR = 44 − 20 = 24. Lower fence = 20 − 1.5(24) = 20 − 36 = −16. Upper fence = 44 + 1.5(24) = 44 + 36 = 80. Test 70: 70 < 80 → NOT an outlier. Test 5: 5 > −16 → NOT an outlier. Answer: Neither 70 nor 5 is an outlier (both within fences of −16 and 80).
Step 1: Already sorted. n=9 (odd). Step 2: Min=3, Max=30. Step 3: Median = 5th value = 15. Step 4: Lower half (below 15): {3,7,9,12} → Q1 = (7+9)/2 = 8. Step 5: Upper half (above 15): {18,21,25,30} → Q3 = (21+25)/2 = 23. Answer: Min=3, Q1=8, Median=15, Q3=23, Max=30. IQR = 23−8 = 15.
IQR Team A = 80−65 = 15. IQR Team B = 88−60 = 28. Team B has greater variability (larger IQR = wider box = more spread in middle 50%). Medians: Team A = 72, Team B = 75. Team B has a higher typical score. Answer: Team B has greater variability (IQR 28 vs 15) AND a higher typical score (median 75 vs 72). SAT note: a wider box = more variable, regardless of which team has higher max.
A box plot — also called a box-and-whisker plot — is a graphical display that summarizes a data set using five key values: the minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), and maximum. The rectangular “box” spans from Q1 to Q3, with a vertical line marking the median. “Whiskers” extend from the box to the minimum and maximum values (excluding outliers). Box plots allow quick visual comparison of center (median), spread (IQR), and distribution shape across data sets. They appear in Florida MAFS.912.S-ID standards and on the SAT Math “Problem Solving & Data Analysis” section.
The five-number summary consists of: (1) Minimum — the smallest value; (2) Q1 — the median of the lower half of the data; (3) Median (Q2) — the middle value of the full data set; (4) Q3 — the median of the upper half; (5) Maximum — the largest value. To find them: first sort the data from least to greatest. Then find the median, splitting the data into two halves (excluding the median itself). Find the median of each half to get Q1 and Q3. The IQR equals Q3 − Q1.
The interquartile range (IQR) = Q3 − Q1. It measures the spread of the middle 50% of the data — the width of the box in the plot. To identify outliers, apply the 1.5×IQR rule: calculate the lower fence (Q1 − 1.5×IQR) and the upper fence (Q3 + 1.5×IQR). Any data point below the lower fence or above the upper fence is an outlier, plotted as an individual dot beyond the whisker. This rule is not given on the SAT reference sheet and must be memorized for Hard-tier data analysis questions.
Box plot questions appear 1–2 times per SAT Math test, under the “Problem Solving & Data Analysis” domain — the domain that accounts for approximately 17 of 58 total SAT Math questions. Box plot questions are typically Medium–Hard difficulty because they require data interpretation, not just formula recall. Common question types include: reading the five-number summary from a given plot, calculating IQR, comparing two box plots by median or IQR, and (less frequently) applying the 1.5×IQR outlier rule. Florida students targeting Bright Futures Scholarship Academic Scholars requirements (1290+ SAT) should treat the full data analysis domain — including box plots — as a high-priority preparation area.
Yes. InLighten’s certified math tutors in Orlando specialize in SAT data analysis including box plots, five-number summaries, IQR, outlier detection, and box plot comparison — all at the level of the Florida FSA Statistics assessment and the SAT Math “Problem Solving & Data Analysis” section. We diagnose exactly which data analysis skills your student is missing before building a targeted session plan. Student-athletes working toward NCAA eligibility requirements or Florida Bright Futures Scholarship thresholds receive specialized SAT Math preparation where data analysis is a core module. Book a free math assessment to start.