First Quartile

First Quartile
Low Quartile
Lower Quartile
Q1

For a set of data, a number for which 25% of the data is less than that number. The first quartile is the same as the median of the data which are less than the overall median. Same as the 25th percentile.

Example finding Q1 for data set {2,5,6,9,12}: median=6, first quartile=3.5 (median of lower values 2 and 5).

See also

Third quartile, five-number summary

Key Formula

Q_1 = \text{median of the lower half of the data}

Where:

$Q_1$ = The first quartile, also called the lower quartile or 25th percentile
$\text{lower half}$ = All data values below the overall median (the median itself is excluded if the data set has an odd number of values)

Worked Example

Problem: Find the first quartile of this data set: 3, 7, 8, 12, 15, 18, 21, 24, 30.

Step 1: Order the data from least to greatest. The data is already sorted.

3,\; 7,\; 8,\; 12,\; 15,\; 18,\; 21,\; 24,\; 30

Step 2: Find the overall median. There are 9 values, so the median is the 5th value.

\text{Median} = 15

Step 3: Identify the lower half — all values below the median. Since the data set has an odd count, exclude the median itself.

\text{Lower half: } 3,\; 7,\; 8,\; 12

Step 4: Find the median of the lower half. There are 4 values, so average the 2nd and 3rd values.

Q_1 = \frac{7 + 8}{2} = 7.5

Answer: The first quartile is Q1 = 7.5. This means 25% of the data values are less than 7.5.

Another Example

This example uses an even-sized data set, so the lower half is simply the first n/2 values with no need to exclude a middle value. This is the most common source of confusion for students.

Problem: Find Q1 for this data set with an even number of values: 4, 10, 14, 20, 26, 30.

Step 1: The data is already in order from least to greatest.

4,\; 10,\; 14,\; 20,\; 26,\; 30

Step 2: Find the overall median. There are 6 values, so the median is the average of the 3rd and 4th values.

\text{Median} = \frac{14 + 20}{2} = 17

Step 3: Identify the lower half — the first 3 values (everything below the median position).

\text{Lower half: } 4,\; 10,\; 14

Step 4: Find the median of those 3 values. With an odd count, the middle value is the median.

Q_1 = 10

Answer: The first quartile is Q1 = 10.

Frequently Asked Questions

What is the difference between Q1 and Q3?

Q1 (first quartile) is the median of the lower half of a data set, marking the 25th percentile. Q3 (third quartile) is the median of the upper half, marking the 75th percentile. Together, Q1 and Q3 define the interquartile range (IQR = Q3 − Q1), which measures the spread of the middle 50% of data.

How do you find Q1 when the data set has an odd number of values?

When the data set has an odd number of values, find the overall median first. Then exclude the median and take the lower half — all values below it. Q1 is the median of that lower half. For example, in a set of 11 values, the 6th value is the overall median, and Q1 is the median of values 1 through 5.

Is Q1 always a number in the data set?

Not necessarily. When the lower half has an even number of values, Q1 is the average of the two middle values, which may not be an actual data point. For instance, if the lower half is {3, 7, 8, 12}, then Q1 = (7 + 8)/2 = 7.5, which does not appear in the original data.

First Quartile (Q1) vs. Third Quartile (Q3)

	First Quartile (Q1)	Third Quartile (Q3)
Definition	Median of the lower half of the data	Median of the upper half of the data
Percentile	25th percentile	75th percentile
Position	Below the overall median	Above the overall median
Use in IQR	Lower bound of the IQR	Upper bound of the IQR
Box plot location	Left (or bottom) edge of the box	Right (or top) edge of the box

Why It Matters

You will encounter the first quartile in box-and-whisker plots, which are a standard topic in middle and high school statistics. Q1 is one of the five values in a five-number summary (minimum, Q1, median, Q3, maximum) and is essential for calculating the interquartile range, which is widely used to detect outliers. Standardized tests like the SAT and ACT frequently ask students to read or compute quartiles from data sets.

Common Mistakes

Mistake: Forgetting to sort the data before finding Q1.

Correction: Quartiles depend on the position of values in order. Always arrange the data from least to greatest before identifying the lower half.

Mistake: Including the overall median in the lower half when the data set has an odd number of values.

Correction: When the data set has an odd count, the overall median is a single middle value. Exclude it from both halves before computing Q1 and Q3. Including it shifts Q1 higher than it should be.

Related Terms

Third Quartile — Upper quartile (Q3), the 75th percentile counterpart
Median of a Set of Numbers — The middle value; Q1 is the median of the lower half
Percentile — Q1 equals the 25th percentile
Five Number Summary — Q1 is one of the five summary values
Set — A collection of data values used to compute Q1
Interquartile Range — IQR = Q3 − Q1, measures middle 50% spread
Box and Whisker Plot — Visual display where Q1 marks the left edge of the box