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Worksheet Grade 12 Math

Ackworth Tap Sampling: The 90th Percentile and the Lead Rule

A grade 12 worksheet using twenty fictional tap results to build a five-number summary, find the 90th percentile, and test it against the lead action level.

Objectives

  1. Students will build a five-number summary and box plot from a set of tap sampling results.
  2. Students will compute a 90th percentile using the ranking method written into the lead rule.
  3. Students will report a mean at a precision that matches the laboratory method.

Vocabulary

Action level
A trigger number in lead and copper testing. If too many homes test above this level, the utility must take steps like improving treatment and telling the public.
First-draw sample
A water sample taken from a faucet after the water has sat still in the pipes for hours, usually first thing in the morning. It shows how much lead can build up in water that sits in a home's pipes.

Also: 90th percentile, five-number summary, interquartile range, outlier

Teacher overview

Lead compliance is not judged on the average of a system's tap samples. Under the Lead and Copper Rule, a system compares the 90th percentile of its first-draw tap results against the action level, which is 15 ppb and drops to 10 ppb in 2027. This worksheet gives students twenty results from the fictional town of Ackworth and walks them from the raw list to the percentile the rule actually uses. The error to watch for at this grade is treating the 90th percentile as a value 90 percent of the way from the smallest reading to the largest, rather than a ranked position in the ordered list.

Ackworth Tap Sampling: The 90th Percentile and the Lead Rule — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

The fictional Ackworth Water Department collected one first-draw tap sample from each of twenty homes with lead service lines. Every result is reported in parts per billion (ppb), rounded by the laboratory to the nearest 0.1 ppb.

Site Result Site Result
S-01 4.8 S-11 1.6
S-02 1.4 S-12 5.5
S-03 11.5 S-13 2.9
S-04 2.5 S-14 14.2
S-05 0.8 S-15 1.1
S-06 6.2 S-16 3.6
S-07 3.3 S-17 9.9
S-08 31.0 S-18 2.2
S-09 2.0 S-19 7.4
S-10 8.6 S-20 4.1

Use this ranking method for the 90th percentile, which is the method written into the rule: order the results from lowest to highest, multiply the number of samples by 0.9, and report the result at that position counting up from the lowest.

  1. Order the twenty results from lowest to highest. Find the range, meaning the largest result minus the smallest.

  2. Find the median of the twenty results.

  3. Find the first quartile, Q1, using the median of the lower ten results.

  4. Find the third quartile, Q3, using the median of the upper ten results.

  5. Find the interquartile range, where IQR = Q3 - Q1.

  6. Find the upper outlier fence, where the fence = Q3 + 1.5 x IQR.

  7. Which single result sits above that fence?

  8. List the five-number summary for this data set in order: minimum, Q1, median, Q3, maximum.

  9. Draw a box plot of the twenty results on a number line running from 0 to 32. Plot the item 7 result as a separate outlier point, and end the upper whisker at the largest result that is not an outlier. Draw a vertical line at 10. Which parts of your box plot lie to the right of that line?

  10. Find the mean of all twenty results.

  11. The laboratory reports each result to the nearest 0.1 ppb. A student writes the mean from item 10 as 6.2300 ppb. Rewrite that mean at a precision the laboratory method can support.

  12. Recompute the mean with the outlier from item 7 removed, using the other nineteen results. Round to the nearest 0.01 ppb.

  13. Apply the ranking method above to find Ackworth's 90th percentile result.

  14. Compare your item 13 answer with the current action level of 15 ppb and with the 10 ppb level that applies from 2027. Which of the two does Ackworth exceed?

  15. Items 10 and 13 give two different summaries of the same twenty homes. Explain in two or three sentences why the rule uses the 90th percentile instead of the mean.

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: 30.2 ppb.
    Ordered: 0.8, 1.1, 1.4, 1.6, 2.0, 2.2, 2.5, 2.9, 3.3, 3.6, 4.1, 4.8, 5.5, 6.2, 7.4, 8.6, 9.9, 11.5, 14.2, 31.0. Range = 31.0 - 0.8 = 30.2 ppb.

  2. Answer: 3.85 ppb.
    With twenty values the median falls between the 10th and 11th: (3.6 + 4.1) / 2 = 3.85 ppb.

  3. Answer: 2.1 ppb.
    The lower ten values are 0.8 through 3.6. Their median falls between the 5th and 6th: (2.0 + 2.2) / 2 = 2.1 ppb.

  4. Answer: 8.0 ppb.
    The upper ten values are 4.1 through 31.0. Their median falls between the 15th and 16th values overall: (7.4 + 8.6) / 2 = 8.0 ppb.

  5. Answer: 5.9 ppb.
    IQR = 8.0 - 2.1 = 5.9 ppb.

  6. Answer: 16.85 ppb.
    Fence = 8.0 + 1.5 x 5.9 = 8.0 + 8.85 = 16.85 ppb.

  7. Answer: 31.0 ppb.
    Only site S-08 at 31.0 ppb exceeds 16.85 ppb. The next highest reading, 14.2 ppb at S-14, sits below the fence and is not flagged.

  8. Answer: 0.8, 2.1, 3.85, 8.0, 31.0 ppb.
    These are the minimum, Q1, median, Q3, and maximum from items 1 through 4.

  9. Answer: the top of the upper whisker and the outlier point.
    The box runs from 2.1 to 8.0 with the median bar at 3.85, so the whole box sits left of the 10 line. The upper whisker ends at 14.2, the largest result that is not an outlier, so its top section from 10 to 14.2 crosses the line, and the plotted outlier at 31.0 sits far to the right of it.

  10. Answer: 6.23 ppb.
    The twenty results total 124.6 ppb. Mean = 124.6 / 20 = 6.23 ppb.

  11. Answer: 6.2 ppb.
    Each measurement is good to the nearest 0.1 ppb, so a mean carried to four decimal places claims accuracy the method cannot deliver. Reporting 6.2 ppb keeps the answer at the precision of the underlying readings.

  12. Answer: 4.93 ppb.
    Removing 31.0 leaves 124.6 - 31.0 = 93.6 ppb across nineteen homes. Mean = 93.6 / 19 = 4.926, which rounds to 4.93 ppb. One home moved the mean by about 1.3 ppb.

  13. Answer: 11.5 ppb.
    There are 20 samples, and 20 x 0.9 = 18, so the 90th percentile is the 18th value counting up from the lowest. Counting the ordered list, the 18th value is 11.5 ppb.

  14. Answer: the 10 ppb level.
    The 90th percentile of 11.5 ppb sits below the current 15 ppb action level but above the 10 ppb level that applies from 2027. Ackworth passes today and would not pass under the lower level without further corrosion control or service line replacement.

  15. Answer: the mean hides the high-lead homes.
    The mean of 6.23 ppb is pulled down by many low readings and says little about the households actually at risk. The 90th percentile is built to describe the high end of the distribution, so a system cannot pass by having many clean taps offset a few badly contaminated ones. It is also less affected by a single extreme reading than the mean is, since the position in the ranking, not the size of the outlier, sets the value.

Differentiation

Support

Give students the fictional Ackworth results already sorted, printed on a strip of paper with positions 1 through 20 numbered, so the quartile and percentile work does not depend on error-free sorting. Pre-mark positions 5, 6, 10, 11, 15, 16, and 18 so students can see which positions each statistic uses. Provide the fence formula worked through on a different set of five numbers before item 6.

Extension

Have students find how far the fictional S-08 reading would have to fall before the data set has no outlier by the fence rule, then check whether that change moves the 90th percentile at all. Ask them to explain, in writing, why the two statistics respond so differently to the same edit.

ELL supports

Pre-teach "first-draw sample," "percentile," "quartile," and "action level" with a labeled diagram of the sorted Ackworth list showing where each statistic lands. Provide sentence frames for items 14 and 15, such as "The 90th percentile is ___ ppb, which is above the ___ level because ___." Allow students to mark positions on the printed strip instead of writing full sentences for items 1 through 9.

Extensions

Have students plan the follow-up sampling round for the fictional town. If Ackworth adds ten more homes for a total of thirty samples, the ranking method puts the 90th percentile at position 27. Ask students what the ten new results would have to look like for the 90th percentile to fall below 10 ppb, and whether that is a realistic result of corrosion control or only of choosing favorable sampling sites. The discussion connects the arithmetic to the reason sampling site selection is itself regulated.