Quartile Formula - Q1, Q2, Q3 with Examples

Key Points

• The quartile formula locates quartiles by position in the ordered data, with $Q_1$ at the $0.25(N+1)$th value and $Q_3$ at the $0.75(N+1)$th.

• The NIST e-Handbook of Statistical Methods uses these positions and interpolates between neighbouring values when the position is not a whole number.

• $Q_2$ is the median, so the three quartiles split an ordered data set into four groups of equal count.

• The data must be sorted before any quartile position means anything, because quartiles are defined by rank rather than by value.

• More than one quartile convention is in use, so two defensible methods can give different quartiles for the same small data set.

What Is the Quartile Formula?

The quartile formula finds the three values that cut an ordered data set into four equal-sized groups. The NIST/SEMATECH e-Handbook of Statistical Methods places them at these positions:

$$Q_1 \text{ at the } 0.25(N+1)\text{th ordered value}, \qquad Q_3 \text{ at the } 0.75(N+1)\text{th}$$

$Q_2$ sits at the $0.5(N+1)$th position, the median. Where a position falls between two ranks, interpolate.

The Full Definition

A quartile is a cut point, not a count. $Q_1$ has a quarter of the data at or below it, $Q_2$ has half, and $Q_3$ has three quarters. Because the definition is positional, the formula returns a rank first and a value second, which is why sorting is part of the method rather than a preliminary to it.

Variable Key

Examples of the Quartile Formula

Example 1

Find the quartiles of 2, 4, 5, 7, 8, 11, 12, 15, 18, 19, 27.

Here $N = 11$, so $N+1 = 12$ and the positions are $0.25(12) = 3$, $0.5(12) = 6$ and $0.75(12) = 9$:

$$Q_1 = 5, \qquad Q_2 = 11, \qquad Q_3 = 18$$

Final answer: $Q_1 = 5$, $Q_2 = 11$, $Q_3 = 18$

Example 2

Find $Q_1$ of 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 by taking a quarter of the way along the values.

The data runs from 10 to 100, so a quarter of that span looks like the natural cut: $10 + 0.25(90) = 32.5$.

That is wrong, and the reason matters. The figure 32.5 is a quarter of the way through the range of values, but a quartile is a quarter of the way through the count of observations, and the two agree only when the data is evenly spaced.

Using positions instead, $N+1 = 11$, so $Q_1$ sits at $0.25(11) = 2.75$. Interpolate 75 per cent of the way from the 2nd value to the 3rd:

$$Q_1 = 20 + 0.75(30 - 20) = 27.5$$

Final answer: $Q_1 = 27.5$

Example 3

Find $Q_3$ of the same ten values.

The position is $0.75(11) = 8.25$, so interpolate a quarter of the way from the 8th value to the 9th:

$$Q_3 = 80 + 0.25(90 - 80) = 82.5$$

Final answer: $Q_3 = 82.5$

Example 4

Find the quartiles of 3, 9, 4, 15, 7 and then the interquartile range.

Sort first: 3, 4, 7, 9, 15. With $N = 5$, positions are $0.25(6) = 1.5$, $0.5(6) = 3$ and $0.75(6) = 4.5$:

$$Q_1 = \frac{3+4}{2} = 3.5, \qquad Q_2 = 7, \qquad Q_3 = \frac{9+15}{2} = 12$$ $$\text{IQR} = 12 - 3.5 = 8.5$$

Final answer: $Q_1 = 3.5$, $Q_2 = 7$, $Q_3 = 12$, $\text{IQR} = 8.5$

Example 5

A data set has $Q_1 = 20$, $Q_2 = 24$ and $Q_3 = 36$. Find its Galton skewness.

The NIST e-Handbook gives Galton skewness as $\frac{Q_1 + Q_3 - 2Q_2}{Q_3 - Q_1}$:

$$\frac{20 + 36 - 48}{36 - 20} = \frac{8}{16} = 0.5$$

Final answer: $0.5$, a positive value indicating a right-skewed distribution. Quartiles therefore describe shape as well as spread.

Don't Confuse Quartiles With Quarters or Percentiles

Saying a value "is in the first quartile" usually means it falls in the lowest quarter, while $Q_1$ is the boundary value itself. Both usages are common.

The Mistakes Students Make Most Often

Mistake 1: Using N instead of N plus 1 in the position

Where it slips in: The data has $N$ values, so $N$ feels like the right multiplier.

Don't do this: Position of $Q_1 = 0.25N$.

The correct way: Position of $Q_1 = 0.25(N+1)$. The $N+1$ counts the gaps between ranks, which is why it also gives the familiar median position $\frac{N+1}{2}$.

Students who check $Q_2$ against the median they already know catch a wrong multiplier immediately, since $Q_2$ must equal the median by definition.

Mistake 2: Reading quartiles off unsorted data

Where it slips in: The formula names positions, so data in collection order looks usable.

Don't do this: Take the 3rd value as $Q_1$ without sorting.

The correct way: Sort ascending, then count ranks. A $Q_1$ larger than $Q_2$ is the signature of unsorted data.

Mistake 3: Assuming every method gives the same quartiles

Where it slips in: Spreadsheets, calculators and textbooks implement different conventions.

Don't do this: Compare a quartile from one method against one from another.

The correct way: Use one convention throughout and name it. On small samples the disagreement can exceed the quantity being measured.

Conclusion

• The quartile formula gives positions, not values: $Q_k$ sits at $\frac{k}{4}(N+1)$ in the ordered data.

• $Q_2$ is always the median, which makes it a free check on any quartile calculation.

• Sorting is part of the method, because quartiles are defined by rank rather than by value.

• Quartile conventions differ between tools, so any reported quartile should name the method behind it.

• To work through data handling with a teacher, explore a 6th grade math tutor or math classes online.

Five Minutes of Practice

Find all three quartiles of 5, 8, 12, 13, 17, 20, 22, 29. Then confirm your $Q_2$ equals the median, and compute the interquartile range.

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Read More

• Interquartile Range Formula

• Outlier Formula

• Percentile Formula

• Mean Median Mode Formula

• What Is Median