Quartiles help you describe how a data set is spread out after the values are placed in order. They split the ordered data into four parts, so you can see where the lower, middle, and upper portions of the data fall. The interquartile range, or IQR, measures the spread of the middle 50 percent of the data.
This is useful because it is less affected by extreme values than the full range.
Key Facts
- Order the data from least to greatest before finding quartiles.
- The median, Q2, is the middle value of the ordered data set.
- Q1 is the median of the lower half of the ordered data, and Q3 is the median of the upper half.
- IQR = Q3 - Q1.
- Lower fence = Q1 - 1.5(IQR) and upper fence = Q3 + 1.5(IQR).
- Five-number summary = minimum, Q1, median, Q3, maximum.
Vocabulary
- Quartile
- A quartile is a value that divides ordered data into four groups with about equal numbers of data values.
- Median
- The median is the middle value of an ordered data set, or the average of the two middle values when there is an even number of values.
- Interquartile Range
- The interquartile range is the distance between the first and third quartiles and measures the spread of the middle half of the data.
- Five-Number Summary
- The five-number summary lists the minimum, Q1, median, Q3, and maximum of a data set.
- Outlier Fence
- An outlier fence is a cutoff found using 1.5 times the IQR to identify values that are unusually low or high.
Common Mistakes to Avoid
- Forgetting to sort the data first: quartiles depend on position in the ordered list, so using the original order gives incorrect results.
- Including the median in both halves without checking the method: for many classroom methods, when the data set has an odd number of values, the median is not included when finding Q1 and Q3.
- Confusing range with IQR: the range uses maximum minus minimum, while the IQR uses Q3 minus Q1 and describes only the middle 50 percent.
- Calling every value outside the box an outlier: in a box plot, potential outliers are values below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR), not just values beyond Q1 or Q3.
Practice Questions
- 1 Find the five-number summary and IQR for the data set: 4, 7, 8, 10, 12, 13, 15, 18, 21.
- 2 For the data set 5, 6, 7, 9, 10, 12, 13, 14, 18, 30, find Q1, Q3, IQR, the lower fence, and the upper fence. Identify any outliers.
- 3 A data set has Q1 = 20, median = 28, Q3 = 36, minimum = 4, and maximum = 60. Explain what the IQR says about the spread of the middle half of the data and why the minimum and maximum do not directly determine it.