How to find Q1 and Q3 by hand

The rule used here: median-excluded

$Q_1$, the first quartile, is the median of the lower half of the sorted data, and $Q_3$, the third quartile, is the median of the upper half. About 25% of the values sit at or below $Q_1$ and about 75% sit at or below $Q_3$, which makes them the 25th and 75th percentiles. Together they bound the middle 50% of the data.

The one judgment call is what to do with the median itself when the number of values $n$ is odd. This site always uses the median-excluded convention: the median value is left out of both halves. That is the rule TI-83 and TI-84 calculators use, so their 1-Var Stats output will agree with your hand arithmetic.

The AP course describes $Q_1$ as the median of the lower half of the ordered data, running from the minimum to the position of the median, and $Q_3$ as the median of the upper half. Stating your convention costs one clause and removes any ambiguity from your work.

The four steps

1. Sort the values from smallest to largest. Nothing below works on unsorted data.
2. Find the median. With $n$ odd it is the single middle value. With $n$ even it is the average of the two middle values.
3. Split into halves. With $n$ odd, the lower half is every value before the median and the upper half is every value after it, with the median itself in neither. With $n$ even, the lower half is the first $n/2$ values and the upper half is the last $n/2$ values.
4. Take the median of each half. The median of the lower half is $Q_1$; the median of the upper half is $Q_3$.

Each half contains the same number of values, either $(n-1)/2$ when $n$ is odd or $n/2$ when $n$ is even. If your two halves have different sizes, you split in the wrong place. A quartile can land between two data values, so $Q_1 = 7.5$ is a perfectly good answer even when every value is a whole number.

Why odd and even n feel different

With an even $n$ the median falls in the gap between two data values, so the split point is already between observations and nothing is left over. Every value goes into one half or the other.

With an odd $n$ the median is an actual data value sitting at the split point, and it has to go somewhere. Median-excluded drops it from both halves. Median-included, the other common rule, keeps it in both.

That single choice is the whole difference between the two conventions. For even $n$ the two rules always agree, so the question only arises when $n$ is odd.

From quartiles to the IQR and the 1.5 IQR fences

Once you have the quartiles, two more results follow immediately. The interquartile range is

and it measures the spread of the middle 50% of the data. Because it ignores the smallest and largest quarter of the values, the IQR is resistant: a single extreme value barely moves it, which is why the AP course pairs the IQR with the median for skewed data.

The fences for the 1.5 IQR outlier rule come next:

Any value below the lower fence or above the upper fence is an outlier. Keep full precision through this calculation; rounding $Q_1$ or the IQR early can flip a borderline value from inside the fence to outside it. The 1.5 IQR rule covers the fences in more depth, and how to make a boxplot shows where they end up on a graph.

Other conventions give slightly different quartiles

There is no single universal definition of a quartile, and honest work says which rule it used. Take the 9 sorted values 3, 7, 8, 12, 14, 19, 21, 26, 60 from the first worked example below.

All three are defensible; they answer slightly different questions about where the 25th percentile sits in a finite data set. Software usually interpolates between the two data values nearest the quartile position, which is why a spreadsheet can hand you a quartile that is not in your data at all. On the 10 protein values in the second worked example, the median-excluded rule gives $Q_1 = 7$ and $Q_3 = 14$, while Excel's QUARTILE.INC gives $Q_1 = 7.25$ and $Q_3 = 13.5$.

That gap is not cosmetic. With the median-excluded quartiles the IQR is 7 and the upper fence is $14 + 1.5 \times 7 = 24.5$, so the value 24 is not an outlier. With the interpolated quartiles the IQR is $13.5 - 7.25 = 6.25$ and the upper fence is $13.5 + 1.5 \times 6.25 = 22.875$, so the same value 24 gets flagged. When you report an outlier, say which quartile rule produced it.

Checking your work on a calculator

A graphing calculator with statistical capability is expected on the AP exam, and its one-variable statistics output lists the five-number summary directly: minX, $Q_1$, Med, $Q_3$, and maxX. Because TI calculators use the median-excluded rule, their $Q_1$ and $Q_3$ should match your hand arithmetic exactly.

If they disagree by a small amount, check three things before assuming the calculator is right. Did you sort the data. Did you leave the median out of both halves for odd $n$. Did you enter every value, since one missing entry changes $n$ and can move both quartiles.

You can also paste a data set into the five-number summary calculator, which uses the same median-excluded convention, or drag values around in the descriptive statistics sandbox to watch $Q_1$ and $Q_3$ move.

Common mistakes

• Not sorting first. Quartiles are positional, so unsorted data produces meaningless answers.
• **Keeping the median in both halves when $n$ is odd.** That is a different convention, and it will disagree with a TI calculator.
• **Splitting at the wrong place when $n$ is even.** There is no middle value to remove; the halves are simply the first $n/2$ and the last $n/2$ values.
• **Using position 25% of the way through as $Q_1$.** For $n = 10$, the 2.5th value is not the median-excluded $Q_1$.
• **Averaging $Q_1$ and $Q_3$ to get the median.** That gives the midhinge, which usually is not the median.
• Rounding too early. Carry the exact quartiles into the IQR and the fences.
• Reporting a quartile with no units. $Q_1 = 7.5$ means nothing without "paint cans" or "minutes" after it.

Where this sits in the AP course

Quartiles are topic 1.7 in Unit 1 of the Fall 2026 AP Statistics course, which covers measures of center and position, and the course defines the IQR as $Q_3 - Q_1$ and lists the 1.5 IQR outlier rule in the same topic. They reappear in topic 1.8 as part of the five-number summary and the boxplot, and in topic 1.9 when you compare two distributions by IQR.

Unit 1 carries 20 to 30% of the multiple-choice section, and quartile arithmetic is the kind of small calculation that appears inside larger free-response parts. Once you have the quartiles, use them to describe the distribution in context rather than leaving them as bare numbers. The official course description is at AP Central.