A five-number summary describes a data set using the minimum, first quartile, median, third quartile, and maximum. This cheat sheet helps students organize data, find quartiles, and build box plots accurately. These skills are useful for comparing groups, spotting spread, and identifying unusual values in real-world data.
The most important ideas are position, center, spread, and outliers. The median divides ordered data into two halves, while quartiles divide data into four parts. The interquartile range measures the spread of the middle half of the data using .
A box plot shows these values visually using a box, a median line, whiskers, and sometimes separate outlier points.
Key Facts
- A five-number summary is written as , where is the median.
- The range of a data set is .
- The interquartile range is , which measures the spread of the middle of the data.
- The lower outlier fence is , and values below it are possible outliers.
- The upper outlier fence is , and values above it are possible outliers.
- In a box plot, the box runs from to , and the line inside the box marks the median .
- For a modified box plot, whiskers extend to the smallest and largest non-outlier values, not necessarily to the minimum and maximum.
- A longer box or whisker means the data are more spread out over that part of the distribution.
Vocabulary
- Five-number summary
- A summary of a data set using the minimum, first quartile, median, third quartile, and maximum.
- Median
- The middle value of an ordered data set, also called .
- Quartile
- A value that divides ordered data into four parts with about of the data in each part.
- Interquartile range
- The spread of the middle half of the data, found by .
- Box plot
- A graph that displays the five-number summary with a box from to , a median line, and whiskers.
- Outlier
- A data value that is unusually far from the rest of the data, often checked using the rule.
Common Mistakes to Avoid
- Not ordering the data first, which is wrong because the median and quartiles must be found from values arranged from least to greatest.
- Including the median in both halves when the method says to exclude it, which can change and for data sets with an odd number of values.
- Using when asked for , which is wrong because measures only the middle of the data.
- Drawing whiskers to outliers on a modified box plot, which is wrong because outliers should be plotted as separate points.
- Thinking a box plot shows every data value, which is wrong because it summarizes position and spread rather than listing each value.
Practice Questions
- 1 Find the five-number summary for the data set .
- 2 For a data set with and , find , the lower fence, and the upper fence.
- 3 A modified box plot has , median , , minimum non-outlier , and maximum non-outlier . Describe the box, median line, and whiskers.
- 4 Two box plots have the same median, but one has a much larger . Explain what this means about the two data sets.
Understanding Five-Number Summary & Box Plot Reference
Start by putting every value in order from least to greatest. This step is not optional. Quartiles describe locations in the ordered list, so one misplaced value can change the result.
Find the overall median first. If there is an odd number of values, the median is the single middle value. If there is an even number, it is the average of the two middle values.
Then split the list into a lower half and an upper half. The median of the lower half gives the first quartile. The median of the upper half gives the third quartile.
A detail can cause disagreement between correct-looking answers. Some courses include the overall median in both halves when the data set has an odd number of values. Other courses leave it out of both halves.
These methods can produce different quartiles for small data sets. Follow the rule taught by your teacher, textbook, or calculator, then use that same rule throughout a problem.
When comparing work, check the quartile method before deciding that someone made an error. The ordered data list is worth showing because it makes each choice easy to check.
Outlier fences give a consistent way to flag values that sit far from the central group. First find the interquartile range by subtracting the first quartile from the third quartile. Multiply that result by one point five.
Subtract this amount from the first quartile for the lower fence. Add it to the third quartile for the upper fence. A value beyond a fence is a possible outlier, not automatic proof of a mistake.
It could be a recording error, such as an extra zero in a measurement. It could be a real unusual result, such as one runner with a much slower race time after an injury. Context matters before removing any data.
Box plots are especially useful when comparing several groups on the same scale. A class might compare daily screen time for two grades, test scores from different teaching methods, or rainfall across months. Look first at where the median lines fall.
A higher median suggests a generally higher typical value. Next compare box widths and whisker lengths. A group can have a higher median yet much less consistent results if its plot is wide.
Notice whether the median is near one side of the box. This can suggest that values are packed more closely on one side and stretched farther on the other.
A box plot does not show every individual value, so it cannot reveal all patterns. Use it with the original data when exact details matter.