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Worksheet Grade 11 Careers & CTE

Penncroft Filter Turbidity: A Compliance Report Worksheet

A grade 11 careers worksheet in which a compliance coordinator compares two filters' turbidity readings by median, IQR, and outlier test before filing a monthly report.

Objectives

  1. Students will compute the median, quartiles, interquartile range, and mean for two sets of turbidity readings.
  2. Students will apply the 1.5 times IQR rule to test whether a high reading is an outlier.
  3. Students will compare a week of readings against a stated compliance requirement and identify which readings fail it.

Vocabulary

Turbidity
Turbidity is a measure of how cloudy or murky water looks because of tiny particles floating in it. Clear water has low turbidity, and muddy water has high turbidity.

Also: interquartile range, outlier, compliance, nephelometric turbidity unit

Teacher overview

Every month a water treatment plant files a report to the state showing whether its filters held turbidity where the rule requires. The compliance coordinator who assembles that report works in descriptive statistics: summarizing each filter, deciding whether a single high reading is an isolated event or a sign of a developing problem, and stating plainly whether the plant met its requirement. This worksheet uses one week of readings from two filters at the fictional Penncroft Water Treatment Plant. The misconception this worksheet corrects is the habit of judging a filter by its average; a filter can post a low mean and still breach the never-to-exceed value on a single reading, which is exactly what the rule is written to catch.

Penncroft Filter Turbidity: A Compliance Report Worksheet — Grade 11 Careers & CTE
Name: ____________________________ Date: ______________

Student Worksheet

The fictional Penncroft Water Treatment Plant recorded 12 turbidity readings from each of two filters over one week, in nephelometric turbidity units. The readings are already sorted from lowest to highest.

Filter Readings (NTU)
Filter 3 0.04, 0.05, 0.05, 0.06, 0.07, 0.07, 0.08, 0.09, 0.11, 0.13, 0.18, 0.31
Filter 4 0.06, 0.08, 0.09, 0.12, 0.14, 0.15, 0.17, 0.21, 0.26, 0.34, 0.42, 1.10

Find quartiles by splitting each ordered list of 12 into a lower half of six values and an upper half of six values, then taking the median of each half. The rule requires finished water at or below 0.3 NTU in 95 percent of monthly samples, and it may never exceed 1 NTU. In a real plant these two limits apply to the combined effluent of all the filters, and individual filters are monitored separately with their own follow-up triggers. For this exercise, apply the rule to the pooled readings of both filters.

  1. Find the median turbidity for Filter 3.
  2. Find the first quartile for Filter 3.
  3. Find the third quartile for Filter 3.
  4. Find the interquartile range for Filter 3.
  5. Find the median turbidity for Filter 4.
  6. Find the interquartile range for Filter 4.
  7. Find the mean turbidity for Filter 4. Round to the nearest thousandth.
  8. Find the upper outlier boundary for Filter 4 using the third quartile plus 1.5 times the interquartile range.
  9. Counting both filters together, how many of the 24 readings are above 0.30 NTU?
  10. What percent of the 24 readings are at or below 0.30 NTU? Round to the nearest tenth of a percent.
  11. Does this week's set of 24 readings meet the requirement that at least 95 percent of samples sit at or below 0.3 NTU? Answer yes or no and give the percent that supports your answer.
  12. Which single reading exceeds the never-to-exceed value of 1 NTU, and which filter produced it?
  13. Filter 4's mean is well above its median. Explain in three or four sentences what causes that gap and which of the two numbers you would put in the report as Filter 4's typical performance.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 0.075 NTU.
    With 12 values the median is the average of the 6th and 7th: 0.07 and 0.08 average to 0.075 NTU.
  2. Answer: 0.055 NTU.
    The lower half is 0.04, 0.05, 0.05, 0.06, 0.07, 0.07. Its median is the average of 0.05 and 0.06, which is 0.055 NTU.
  3. Answer: 0.12 NTU.
    The upper half is 0.08, 0.09, 0.11, 0.13, 0.18, 0.31. Its median is the average of 0.11 and 0.13, which is 0.12 NTU.
  4. Answer: 0.065 NTU.
    The interquartile range is 0.12 minus 0.055, which equals 0.065 NTU.
  5. Answer: 0.16 NTU.
    The 6th and 7th values are 0.15 and 0.17, which average to 0.16 NTU.
  6. Answer: 0.195 NTU.
    Filter 4's lower half is 0.06, 0.08, 0.09, 0.12, 0.14, 0.15, giving a first quartile of the average of 0.09 and 0.12, or 0.105. Its upper half is 0.17, 0.21, 0.26, 0.34, 0.42, 1.10, giving a third quartile of the average of 0.26 and 0.34, or 0.30. The interquartile range is 0.30 minus 0.105, which equals 0.195 NTU.
  7. Answer: 0.262 NTU.
    The 12 readings sum to 3.14. Dividing by 12 gives 0.26167, which rounds to 0.262 NTU.
  8. Answer: 0.5925 NTU.
    The boundary is the third quartile plus 1.5 times the interquartile range: 0.30 plus 1.5 times 0.195 equals 0.30 plus 0.2925, or 0.5925 NTU. Filter 4's reading of 1.10 sits well above that boundary, so it qualifies as an outlier by this test.
  9. Answer: 4 readings.
    Filter 3 has one reading above 0.30, which is 0.31. Filter 4 has three, which are 0.34, 0.42, and 1.10. That is 4 readings in all.
  10. Answer: 83.3 percent.
    Twenty-four readings minus the 4 above 0.30 leaves 20 at or below it. Then 20 divided by 24 equals 0.83333, which is 83.3 percent.
  11. Answer: No, because only 83.3 percent of the readings sit at or below 0.3 NTU.
    The requirement is at least 95 percent, and this week falls short by more than 11 percentage points. A full month of data would be needed for an actual compliance determination, but a week running at 83.3 percent tells the coordinator that the plant is not on track.
  12. Answer: 1.10 NTU, produced by Filter 4.
    No other reading in either filter reaches 1 NTU. The next highest is Filter 4's 0.42 NTU. Under the combined-effluent rule, a single reading above 1 NTU is reportable on its own, separate from the 95 percent calculation. For an individual filter, the follow-up report is triggered by readings above 1 NTU on two consecutive measurements taken 15 minutes apart.
  13. Answer: The 1.10 NTU reading pulls the mean up, so the median of 0.16 is the better description of Filter 4's typical performance.
    The mean of 0.262 sits above nine of the twelve readings, which means it is not describing a typical hour on that filter. Dropping the single 1.10 reading and averaging the remaining eleven gives 2.04 divided by 11, or 0.185, so one point is moving the mean by more than 0.07 on its own, while the median simply steps past it. The coordinator should report the median as the filter's typical value and report the 1.10 reading separately as an event with its own explanation, such as a filter coming back online after backwash without enough filter-to-waste time. Reporting only the mean would blur those two very different pieces of information into one number that describes neither.

Differentiation

Support

Give students the two lower and upper halves for the fictional plant's Filter 4 already split out so the quartile work in item 6 does not depend on counting correctly, and provide a worked example of the 1.5 times IQR boundary on a third short data set before item 8.

Extension

Ask students to draw box plots for both filters at the fictional plant on one shared NTU scale, mark the outlier as a separate point rather than extending the whisker to it, and write two sentences on what the picture shows that the summary numbers alone did not.

ELL supports

Pre-teach "turbidity," "quartile," "interquartile range," and "outlier" with a number line showing Filter 4's twelve readings as dots, so the crowded low group and the single far point are visible before any calculation. Provide the sentence frame "The mean is higher than the median because ___" for item 13.