Quartiles, percentiles, and the interquartile range help students describe where data values fall within a distribution. This cheat sheet gives quick rules for ordering data, finding position values, and summarizing spread. Students need these tools to compare data sets, read box plots, and identify unusual values.
The focus is on clear formulas and consistent steps that work for both small lists and larger data sets.
The most important ideas are the median, quartiles, percentiles, and IQR. Quartiles split ordered data into four parts, while percentiles describe the percent of data at or below a value. The interquartile range is , which measures the spread of the middle of the data.
Outlier fences use and to flag values that are unusually low or high.
Key Facts
- Always order the data from least to greatest before finding quartiles, percentiles, or the interquartile range.
- The median, or , is the middle value of an ordered data set, and for an even number of values it is the average of the two middle values.
- The first quartile is the median of the lower half of the data, and the third quartile is the median of the upper half of the data.
- The interquartile range is , and it measures the spread of the middle of the data.
- A percentile tells the percent of data values that are less than or equal to a given value.
- One common percentile position formula is , where is the percentile and is the number of data values.
- The lower outlier fence is , and the upper outlier fence is .
- The five-number summary is the minimum, , median, , and maximum.
Vocabulary
- Quartile
- A quartile is a value that divides an ordered data set into four parts with about of the data in each part.
- Percentile
- A percentile is a location measure that tells what percent of data values are less than or equal to a given value.
- Interquartile Range
- The interquartile range is the difference between the third quartile and first quartile, given by .
- Median
- The median is the middle value of an ordered data set, also called .
- Five-Number Summary
- A five-number summary lists the minimum, , median, , and maximum of a data set.
- Outlier Fence
- An outlier fence is a boundary found with or to help identify unusual values.
Common Mistakes to Avoid
- Forgetting to order the data first is wrong because quartiles and percentiles depend on position in a sorted list.
- Including the median in both halves when the method says to exclude it can change and , so use the method your class or calculator requires consistently.
- Finding the range instead of the interquartile range is wrong because the range is , while the IQR is .
- Treating the th percentile as of the maximum value is wrong because percentiles describe position in the ordered data, not a percent of the largest number.
- Calling every value outside and an outlier is wrong because outliers are usually checked against the fences and .
Practice Questions
- 1 Find , the median, , and for the data set .
- 2 For the data set , find the five-number summary.
- 3 A data set has and . Find the , the lower outlier fence, and the upper outlier fence.
- 4 Two classes have the same median test score, but Class A has a larger than Class B. Explain what this means about the spread of the middle of scores.
Understanding Quartiles, Percentiles & IQR
A useful feature of quartiles is that they resist the pull of extreme values. Imagine test scores where most students score between 65 and 85, but one score is 10. The mean can shift noticeably because it uses every value in its calculation.
The median and the middle half of the data usually change much less. This makes the five number summary especially helpful for data such as incomes, house prices, waiting times, and online follower counts. These data often include a few very large or very small values that do not represent a typical experience.
Different textbooks and calculators can produce slightly different quartiles from the same small data set. The difference comes from a choice about the overall median when there is an odd number of values. Some methods leave that value out before splitting the remaining data into lower and upper halves.
Other methods include it in both halves. Neither choice is automatically wrong if the method is stated and used consistently. In classwork, follow the convention your teacher or textbook gives.
On a test, show the ordered list and the halves you used. That work makes your reasoning easy to check.
A box plot turns the five number summary into a picture. The box runs from the first quartile to the third quartile, so its length shows the spread of the central group. A line inside the box marks the median.
The sections outside the box show how far the data extend beyond that central group. When comparing two box plots, first compare the medians for typical values. Then compare box lengths for consistency.
A shorter box means the middle values are more tightly clustered. Finally, notice uneven whiskers or separate points. These can suggest a skewed distribution or unusual observations, though the graph alone does not explain their cause.
Outlier fences are warning lines, not proof that a value is an error. A very high rainfall total may be real after a storm. A very low race time may belong to an elite runner.
First check whether the value was recorded correctly. Then use context before deciding what it means. Removing a real outlier just because it looks inconvenient can hide an important pattern.
Percentiles need similar care. Being at the 90th percentile means a value is at least as large as about 90 percent of the group.
It does not mean the score is 90 percent correct. Always identify the reference group, the time period, and the method used to calculate positions before drawing conclusions.