Worksheet Grade 9 Math
Quillfen's Two Wells: Nitrate Box Plots and an Outlier
A grade 9 worksheet comparing eleven months of nitrate readings from two fictional wells using box plots, an outlier fence, and the mean against the median.
Objectives
- Students will compute a five-number summary and an interquartile range for a set of water-quality readings.
- Students will identify an outlier using the upper fence and explain its effect on the mean and the median.
- Students will compare the spread of two data sets displayed as box plots on a common scale.
Vocabulary
- Nitrate
- Nitrate is a form of nitrogen that plants use as food. It gets into water from fertilizer, animal waste, and septic systems, and too much of it is unhealthy for both people and streams.
Also: quartile, interquartile range, outlier, resistant statistic
Teacher overview
Nitrate reaches groundwater from fertilizer, manure, and failing septic systems, so utilities sample their wells on a schedule and watch the pattern over the year. This worksheet gives students eleven readings from each of two fictional wells and asks them to summarize both, find the outlier in one of them, and see how much that single reading moves the mean. A common error here is treating the mean as the automatic summary of any data set; students should finish the worksheet able to say why the median is the safer choice when one reading sits far from the rest.
Student Worksheet
The Quillfen Water District is a fictional utility with two supply wells. Each well was sampled once a month for eleven months. All readings are nitrate concentrations in milligrams per liter.
Quillfen North: 2.3, 1.4, 2.8, 1.9, 2.6, 1.2, 2.1, 3.0, 1.6, 2.4, 1.8
Quillfen South: 2.6, 9.4, 1.9, 3.4, 2.2, 1.5, 3.1, 2.0, 2.4, 2.9, 1.7
Order each list before you begin. With eleven values, the median is the sixth value, the first quartile is the third value, and the third quartile is the ninth value.
- Find the median for Quillfen North.
- Find the first quartile for Quillfen North.
- Find the third quartile for Quillfen North.
- Find the interquartile range for Quillfen North.
- Find the median for Quillfen South.
- Find the interquartile range for Quillfen South.
- Find the upper outlier fence for Quillfen South, using the third quartile plus 1.5 times the interquartile range.
- Name the single Quillfen South reading that lies above the fence you found in item 7.
- Find the mean for Quillfen South using all eleven readings. Round to the nearest hundredth.
- Find the mean for Quillfen South with the outlier removed. Round to the nearest hundredth.
- Find the median for Quillfen South with the outlier removed.
- State how much the mean changed and how much the median changed when the outlier was removed, then say which statistic is more resistant to an outlier.
- Draw box plots for both wells on one number line scaled from 0 to 10 milligrams per liter. Describe one difference in spread that the plots make visible.
- In this fictional scenario the Quillfen board sets an internal alert level of 5.0 milligrams per liter. Using your two box plots, say which well needs more frequent sampling and give the reason.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: 2.1 mg/L. Sorted, Quillfen North is 1.2, 1.4, 1.6, 1.8, 1.9, 2.1, 2.3, 2.4, 2.6, 2.8, 3.0, and the sixth value is 2.1.
- Answer: 1.6 mg/L. The third value in the sorted list is 1.6.
- Answer: 2.6 mg/L. The ninth value in the sorted list is 2.6.
- Answer: 1.0 mg/L. Interquartile range = 2.6 - 1.6 = 1.0.
- Answer: 2.4 mg/L. Sorted, Quillfen South is 1.5, 1.7, 1.9, 2.0, 2.2, 2.4, 2.6, 2.9, 3.1, 3.4, 9.4, and the sixth value is 2.4.
- Answer: 1.2 mg/L. The first quartile is 1.9 and the third quartile is 3.1, so the interquartile range is 3.1 - 1.9 = 1.2.
- Answer: 4.9 mg/L. Fence = 3.1 + 1.5 x 1.2 = 3.1 + 1.8 = 4.9.
- Answer: 9.4 mg/L. It is the only reading above 4.9, and it is far above it.
- Answer: 3.01 mg/L. The eleven readings sum to 33.1, and 33.1 / 11 = 3.0091, which rounds to 3.01.
- Answer: 2.37 mg/L. Removing 9.4 leaves a sum of 23.7 across ten readings, and 23.7 / 10 = 2.37.
- Answer: 2.3 mg/L. With ten values, the median is the average of the fifth and sixth, which is (2.2 + 2.4) / 2 = 2.3.
- Answer: The mean fell 0.64 and the median fell 0.10. The mean dropped from 3.01 to 2.37, a change of 0.64, while the median moved only from 2.4 to 2.3. The median is the more resistant statistic, because it depends on the position of the middle value rather than on the size of every value.
- Answer: Both boxes are narrow and sit near the low end, but the South plot has a long whisker or a marked outlier point reaching 9.4. Quillfen North runs from 1.2 to 3.0 with a box from 1.6 to 2.6. Quillfen South has a box from 1.9 to 3.1 and, aside from the outlier, a maximum of 3.4. The two middle halves overlap heavily, so the visible difference is the extreme high value in the South well rather than a shift in the center.
- Answer: Quillfen South. Its readings sit slightly higher overall, and one sample already passed the district's alert level of 5.0, which suggests an occasional source of nitrate near that well. Quillfen North came no closer than 2 milligrams per liter to the alert level, so its schedule can stay as it is while the South well is watched more closely.
Differentiation
Support
Give students both fictional lists already sorted, and provide a labeled diagram of a box plot with the minimum, first quartile, median, third quartile, and maximum marked, so they can match each computed value to a position on the plot. Work items 1 through 4 together as a class.
Extension
Ask students to suppose the district collects a twelfth South reading and to determine what value that reading would need for the median to rise above the third quartile of the Quillfen South readings. Have them explain why no single added value can do this.
ELL supports
Pre-teach "quartile," "spread," and "outlier" using a number line with sticky notes that students slide into position. Provide a sentence frame for item 12: "When the outlier was removed, the mean changed by ___ and the median changed by ___, so the ___ is more resistant."
Extensions
Have students find the published annual water-quality report for a nearby water system and identify which statistics that report uses to summarize a year of sampling. Ask them to explain, in two or three sentences, whether the report's chosen summary would hide or reveal a single high reading like the one in the Quillfen South data.