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.
Understanding Statistics: Computing Quartiles and the IQR
The main challenge is deciding exactly which values belong in each half of a data set. Start by locating the central value or central pair. When there is an odd number of values, many school courses leave the overall median out of both halves before finding the other quartiles.
For example, with nine ordered values, the fifth value is the median. The first four values form the lower half, while the last four form the upper half. Find the middle of each of those halves.
This gives a consistent result. Some textbooks use a different convention and include the overall median in both halves. Follow the method your teacher or textbook specifies, since different conventions can produce slightly different quartiles.
An even number of values needs careful averaging. Suppose a lower half contains the values 4, 7, 9, and 12. Its middle lies between 7 and 9, so the lower quartile is 8.
The same idea applies when finding the overall median or upper quartile. Averaging is only appropriate when the middle lies between two positions. Do not average values just because they are near each other.
A useful habit is to mark the positions in the ordered list before doing any arithmetic. This prevents a common mistake where a value is placed in the wrong half or counted twice.
The interquartile range describes the width of the central section, where many of the typical values lie. A small interquartile range means these values are packed fairly close together. A large one means they vary more.
Two classes could have the same median test score but very different interquartile ranges. One class may have scores clustered near the median, while the other has a much wider mix of scores. This makes the interquartile range useful for comparing groups.
It gives information that the median alone cannot show. It is especially helpful for data such as house prices, travel times, rainfall totals, and incomes, where a few unusually high or low results can make the full range misleading.
Quartiles are often shown in a box plot. The box runs from the lower quartile to the upper quartile, and a line inside the box marks the median. The length of each part gives a quick visual clue about spread and possible unevenness in the data.
Teachers may ask you to use one point five times the interquartile range to set lower and upper fences. Values beyond these fences are flagged as possible outliers. A flagged value is not automatically an error.
It might be a real result caused by an unusual event. Check the original data first. An outlier could reveal a recording mistake, or it could show something important that deserves explanation.
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.