Finding Quartiles in a Set of 15 Values
In Range and Why It Is a Limited Measure, you considered how far apart the smallest and largest observations are. The range uses only those endpoints. The interquartile range, or IQR, describes a different part of the spread: the width of the middle half of the ordered data. To find it, we first need the first quartile, \(Q_1\), and the third quartile, \(Q_3\).
As established in Five-Number Summary and Boxplot Construction, for 15 ordered observations the median is in position 8, \(Q_1\) is in position 4, and \(Q_3\) is in position 12. This tutorial focuses on finding those quartiles carefully and using them to calculate the IQR. The positions refer to the observations after they have been sorted from smallest to largest—not to the order in which the data were originally recorded.
A reliable way to locate the quartiles is to order all 15 values, count their positions, and then read the values in positions 4 and 12. The median, in position 8, is not included in either half when using this method. In other words, the lower seven values occupy positions 1 through 7, and their middle value is in position 4. The upper seven values occupy positions 9 through 15, and their middle value is in position 12.
Notice the distinction between a position and a value. Position 4 identifies the fourth observation in the ordered list; \(Q_1\) is the numerical value of that observation. If multiple observations have the same value, count each one separately. Repeated values still occupy separate positions.
A Four-Step Method
Confirm that the data set contains 15 observations, counting repeats as separate observations.
Write the values from smallest to largest. Check that none have been left out or counted twice.
Read the value in position 4 as \(Q_1\), and the value in position 12 as \(Q_3\). Position 8 is the median.
Calculate \(Q_3-Q_1\). Report the IQR in the variable’s units and describe it as the width of the middle half in context.
For 15 values, the position pattern is especially straightforward:
Position 8 is the median. The middle observation among positions 1 through 7 is position 4, and the middle observation among positions 9 through 15 is position 12. Once the data are ordered, these positions give a direct way to locate \(Q_1\) and \(Q_3\), even when some values repeat.
Worked Examples: Calculating \(Q_1\), \(Q_3\), and the IQR
Worked Example: Trail Loop Times
A fictional group records the completion times, in minutes, for 15 attempts at a short trail loop. The times are already ordered: 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, and 25. Find \(Q_1\), \(Q_3\), and the IQR, then interpret the result.
Find the quartiles. There are 15 values. Position 4 contains 10, so \(Q_1=10\) minutes. Position 12 contains 18, so \(Q_3=18\) minutes. Position 8, the median, is 14 minutes; it is not included when locating either quartile.
Calculate the IQR. Subtract the first quartile from the third quartile:
Interpret in context. The width of the middle half of these trail-loop times is 8 minutes, from \(Q_1=10\) minutes to \(Q_3=18\) minutes. The IQR is a width, not a statement that every time differs from every other time by 8 minutes or that a typical time is 8 minutes.
Check the calculation. The subtraction is consistent with the order of the quartiles: \(Q_3\) is the larger value, and \(18-10=8\). The result is positive and has the same units as the times.
Worked Example: App Loading Times in an Unordered List
A fictional student group records the loading times, in seconds, of an app on 15 devices. The times as recorded are 18, 12, 15, 14, 12, 21, 16, 17, 13, 19, 15, 22, 14, 20, and 16. Find and interpret the quartiles and IQR.
Order the observations. Sorting the values, while keeping repeated observations, gives:
Locate positions 4 and 12. Counting from the left, position 4 contains 14 seconds, and position 12 contains 19 seconds. Therefore, \(Q_1=14\) seconds and \(Q_3=19\) seconds. The two values of 12 seconds occupy positions 1 and 2, and the two values of 14 seconds occupy positions 4 and 5. Repeats count as separate observations, so position 4 is still 14.
Calculate the IQR.
Interpret in context. The middle half of the recorded app loading times has a width of 5 seconds, extending from 14 seconds at \(Q_1\) to 19 seconds at \(Q_3\). This summarizes the spread between the quartiles; it does not describe the full span from the shortest to the longest loading time.
A position check helps catch sorting mistakes: the ordered list must contain all 15 observations, including both copies of 12, 14, 15, and 16. If one repeat were accidentally omitted, the positions—and potentially the quartiles—could change.
Worked Example: Daily Steps Recorded in Thousands
A fictional student tracks daily step totals, recorded in thousands of steps, for 15 days. The ordered values are 4.2, 4.5, 4.8, 5.0, 5.1, 5.4, 5.6, 6.0, 6.2, 6.4, 6.7, 7.0, 7.3, 7.6, and 8.1. Find \(Q_1\), \(Q_3\), and the IQR.
Identify the relevant positions. For 15 ordered observations, use position 4 for \(Q_1\), position 8 for the median, and position 12 for \(Q_3\). Reading the ordered list, position 4 is 5.0 thousand steps, position 8 is 6.0 thousand steps, and position 12 is 7.0 thousand steps. Thus, \(Q_1=5.0\) thousand steps and \(Q_3=7.0\) thousand steps.
Find the difference.
Interpret in context. The width of the middle half of the recorded daily step totals is 2.0 thousand steps, between 5.0 thousand and 7.0 thousand steps. Because the values are recorded in thousands, the IQR can also be expressed as 2,000 steps.
The calculation uses the quartile values, not the median. The median helps identify the center of the ordered data, while the IQR describes the distance from \(Q_1\) to \(Q_3\). These are different summaries and answer different questions.
Reading the IQR Carefully
The IQR is found from two locations in the ordered data, but its numerical value is a subtraction. For example, if \(Q_1=14\) seconds and \(Q_3=19\) seconds, the IQR is \(19-14=5\) seconds. The quartiles tell us the lower and upper values that mark the middle half; the IQR tells us the width between those two values.
As discussed in Describing Spread in Context, interpret the IQR as the distance from \(Q_1\) to \(Q_3\), which is the width of the middle half of the ordered observations. Name the variable and group, and use the variable’s units. Avoid saying that the IQR is the full spread of all observations: the minimum and maximum may lie outside the interval from \(Q_1\) to \(Q_3\).
The quartiles are based on positions, so the order of the original list does not matter once the observations have been sorted correctly. The IQR calculation itself does not require you to find the minimum, maximum, or median. However, keeping track of the median at position 8 is a useful check that the list has been counted consistently.
The IQR is also used in the 1.5 IQR rule for flagging possible outliers, covered in Applying the 1.5 IQR Rule for Outliers. Here, the goal is simply to find \(Q_1\), \(Q_3\), and their difference. A value flagged by a rule is not automatically an error, and the IQR alone does not establish why observations vary.
Common Mistakes and AP Exam Tips
- Using the original order. Position 4 means the fourth value after sorting, not the fourth value as it appeared in the question. Show or carefully check the ordered list before selecting quartiles.
- Forgetting repeated values. Each observation gets its own position. If a value occurs twice, write it twice in the ordered list and count both copies.
- Including the median in both halves. For 15 observations, the median is position 8. Find \(Q_1\) from positions 1 through 7 and \(Q_3\) from positions 9 through 15; do not count position 8 as part of either group when locating the quartiles.
- Confusing a position with a value. Position 4 is a rank in the ordered list; \(Q_1\) is the data value in that rank. State the value, not just the position.
- Subtracting in the wrong direction. The IQR is \(Q_3-Q_1\), not \(Q_1-Q_3\). Since \(Q_3\) is at a higher position, it should be at least as large as \(Q_1\).
- Interpreting the IQR as a typical pairwise difference. The IQR is the width from the first to the third quartile. It does not say that every pair of observations differs by the IQR.
- Leaving off context or units. A full-credit interpretation identifies the data being described and gives the units. “The IQR is 5” is less informative than “The middle half of the app loading times spans 5 seconds, from 14 to 19 seconds.”
For a clear written solution, show the ordered values or enough position information to verify your choices, identify \(Q_1\) and \(Q_3\), write the subtraction, and interpret the result in context. Keep the quartiles and IQR in the same units as the original observations.
Check Your Understanding
For each question, count positions in the ordered data and include units when interpreting the IQR.
- For the 15 ordered values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16, identify \(Q_1\), \(Q_3\), and the IQR.
- For 15 observations, which positions contain \(Q_1\), the median, and \(Q_3\)?
- Why must repeated values be counted separately when locating quartiles?
- A group’s \(Q_1\) is 18 minutes and its \(Q_3\) is 27 minutes. Find the IQR and write an interpretation that includes context and units.
- Explain why the IQR should not be described as the difference between the minimum and maximum.