Kid-friendly explanation
Imagine you line up test scores from lowest to highest. A quartile is a special score that helps cut this line into four fair groups. Each group has about the same number of scores. Quartiles help show where the lower, middle, and higher scores are in a data set!
Step-by-step worked examples
Example 1 (middle school level): Finding quartiles in a small data set
Suppose these are the number of pages 9 students read in a week:
Data: 4, 7, 9, 10, 12, 13, 15, 18, 20
Step 1: Make sure the data are in order from least to greatest. They already are.
Step 2: Find the minimum and maximum.
- Minimum (smallest value) = 4
- Maximum (largest value) = 20
Step 3: Find the median, which is the middle value.
There are 9 values, so the median is the 5th value: 12.
Step 4: Find the lower quartile (first quartile, often written Q1).
Look at the lower half of the data, below the median: 4, 7, 9, 10.
The median of these 4 numbers is the average of the 2 middle numbers: (7 + 9) / 2 = 16 / 2 = 8.
So the lower quartile Q1 = 8.
Step 5: Find the upper quartile (third quartile, often written Q3).
Look at the upper half of the data, above the median: 13, 15, 18, 20.
The median of these 4 numbers is the average of the 2 middle numbers: (15 + 18) / 2 = 33 / 2 = 16.5.
So the upper quartile Q3 = 16.5.
Summary of the 5-number set:
- Minimum = 4
- Lower quartile Q1 = 8
- Median = 12
- Upper quartile Q3 = 16.5
- Maximum = 20
Example 2 (upper grade level): Using quartiles and interquartile range
A class records the number of minutes spent on homework in one night:
Data: 5, 10, 12, 15, 18, 20, 22, 25, 30, 35
Step 1: Check that the data are ordered. They are already in order from least to greatest.
Step 2: Find the minimum and maximum.
- Minimum = 5
- Maximum = 35
Step 3: Find the median.
There are 10 values, so the median is the average of the 5th and 6th values.
The 5th value is 18 and the 6th value is 20, so median = (18 + 20) / 2 = 38 / 2 = 19.
Step 4: Find the lower quartile Q1.
The lower half of the data is 5, 10, 12, 15, 18.
The median of these 5 numbers is the middle one, which is 12, so Q1 = 12.
Step 5: Find the upper quartile Q3.
The upper half of the data is 20, 22, 25, 30, 35.
The median of these 5 numbers is the middle one, which is 25, so Q3 = 25.
Step 6: Find the interquartile range, which is the spread of the middle half of the data.
Interquartile range = Q3 − Q1 = 25 − 12 = 13.
So the quartiles show that the middle half of homework times are between 12 and 25 minutes, and that middle spread is 13 minutes wide.
Going deeper with quartiles
Quartiles are part of what is called the 5-number summary: minimum, lower quartile, median, upper quartile, and maximum. This summary is often shown in a box plot, where the box stretches from the lower quartile to the upper quartile and the line inside the box marks the median.
In later grades, quartiles help describe how data are spread out, not just where the center is. For example, the interquartile range uses quartiles to measure how spread out the middle 50% of the data is, which is less affected by extreme values than the full range.
Quartiles also connect to other statistics such as mean, variance, and standard deviation. Together, these measures give a fuller picture of a data set: where it is centered, how spread out it is, and whether most values are close together or widely scattered.
Common mistakes with quartiles
- Mistake: Forgetting to put the data in order before finding quartiles. For the unordered list 20, 5, 30, 10, 25, a student might take the 3rd number in the list, 30, as the median, then use the first two numbers to get Q1 = (20 + 5) / 2 = 12.5 and the last two to get Q3 = (10 + 25) / 2 = 17.5, which are all wrong.
Correction: Sort the data first: 5, 10, 20, 25, 30. The median is 20, Q1 = (5 + 10) / 2 = 7.5, and Q3 = (25 + 30) / 2 = 27.5. - Mistake: Including the median in both halves when finding Q1 and Q3 for an odd number of data points, which can give Q1 = 9 and Q3 = 15 for 4, 7, 9, 10, 12, 13, 15, 18, 20.
Correction: When there is a single middle value, use it as the median only, then split the remaining values below and above it to find Q1 and Q3. - Mistake: Mixing up Q1 and Q3 and labeling the lower quartile as the upper quartile on a box plot, which might lead to saying Q1 = 25 and Q3 = 12 in Example 2 above.
Correction: Q1 is always the quartile closer to the minimum, and Q3 is always the quartile closer to the maximum.
How quartiles show up in everyday life
Quartiles appear whenever data are grouped into four parts. A common example is test scores or grades, where schools might talk about students in the "top quartile" or "bottom quartile" of performance. Quartiles can also describe things like income levels, game scores, or running times, helping compare who is in the lower, middle, or higher groups.