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Math problem set Grade 11 Science

Ellersby's Lead Samples: The 90th Percentile and the Action Level

A grade 11 statistics problem set where students rank a fictional system's lead samples, find the 90th percentile, and compare it to the current and 2027 action levels.

Objectives

  1. Students will compute the median, mean, and interquartile range of a set of lead sampling results.
  2. Students will find the 90th-percentile result and compare it to the current and future lead action levels.

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.

Also: 90th percentile, median, interquartile range, outlier

Teacher overview

The Lead and Copper Rule judges a system by its high end rather than its average, comparing the 90th-percentile result of the samples to an action level. This set has students work a fictional system's sampling round through the median, mean, and spread, then find that 90th percentile and test it against both the current action level of 15 ppb and the 10 ppb level that takes effect in 2027. A common misconception is that a low average proves a system is fine; students should see that a single very high tap can leave the mean comfortable while the 90th percentile fails.

Ellersby's Lead Samples: The 90th Percentile and the Action Level — Grade 11 Science
Name: ____________________________ Date: ______________

Problems

The Ellersby water system is a fictional utility. In one sampling round it collected these 10 lead results, in parts per billion (ppb): 1, 2, 3, 4, 5, 6, 8, 12, 14, 22. The current lead action level is 15 ppb, and under the updated rule it drops to 10 ppb in 2027.

  1. Put the 10 results in order and find the median, in ppb.

  2. Find the mean of the 10 results, in ppb.

  3. Find the first quartile (Q1), the third quartile (Q3), and the interquartile range (IQR), in ppb.

  4. Explain why the mean is higher than the median for this set of results.

  5. The rule ranks the results from low to high and takes the value at position 0.9 x n as the 90th percentile. With n = 10, find that position and the 90th-percentile value, in ppb.

  6. Compare the 90th percentile from item 5 to the current action level of 15 ppb. Does it exceed the level, and by how much is it above or below?

  7. Now compare the same 90th percentile to the 10 ppb action level that takes effect in 2027. By how much is it above or below?

  8. Explain why the Lead and Copper Rule compares the 90th percentile to the action level instead of the mean.

  9. Ellersby tests one more home and adds an 11th result of 30 ppb. Recompute the 90th percentile for the 11 results, rounding the position up when it is not a whole number.

  10. If a system's 90th percentile exceeds the action level, name two actions the rule requires the utility to take.

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: 5.5 ppb. Ordered, the middle two values (5th and 6th) are 5 and 6, so the median is (5 + 6) / 2 = 5.5 ppb.

  2. Answer: 7.7 ppb. The sum is 1 + 2 + 3 + 4 + 5 + 6 + 8 + 12 + 14 + 22 = 77, and 77 / 10 = 7.7 ppb.

  3. Answer: 9 ppb IQR. The lower half is 1, 2, 3, 4, 5, so Q1 = 3; the upper half is 6, 8, 12, 14, 22, so Q3 = 12. IQR = Q3 - Q1 = 12 - 3 = 9 ppb.

  4. Answer: The single 22 ppb result is much higher than the rest, so it pulls the mean upward while barely affecting the median. This right skew from one high value is why the mean (7.7) sits above the median (5.5).

  5. Answer: 14 ppb. The position is 0.9 x 10 = 9, so the 90th percentile is the 9th value in the ordered list, which is 14 ppb.

  6. Answer: 1 ppb below (no exceedance). The 90th percentile is 14 ppb and the action level is 15 ppb, so 15 - 14 = 1 ppb; it sits 1 ppb below the level and does not exceed it.

  7. Answer: 4 ppb above. Against the 2027 level of 10 ppb, the same 90th percentile of 14 ppb is 14 - 10 = 4 ppb above the level, so it would exceed it.

  8. Answer: The action level is about limiting the worst exposures, not the typical one. A system could have a low mean yet still have some homes with dangerous lead; the 90th percentile targets that high end, so a few very high taps cannot be hidden by many low ones.

  9. Answer: 22 ppb. With n = 11 the position is 0.9 x 11 = 9.9, which rounds up to the 10th value. The ordered list is now 1, 2, 3, 4, 5, 6, 8, 12, 14, 22, 30, so the 10th value is 22 ppb, which would exceed the 15 ppb action level.

  10. Answer: Any two required responses, such as optimizing or installing corrosion-control treatment, delivering public education to customers, and accelerating replacement of lead service lines, which the updated rule requires for nearly all lines within ten years.

Differentiation

Support

Provide the 10 results already ordered on a number line, and give the quartile and percentile-position rules on a reference card. Let students find the median and mean first, then tackle the 90th percentile with the position formula written out.

Extension

Have students build a box plot of the original 10 results, mark the 90th percentile and both action levels on it, and write a short paragraph on what the plot shows that the mean alone would hide.

ELL supports

Pre-teach "median," "percentile," "action level," and "exceed" with a simple ranked strip of dots showing where the 90th percentile falls. Allow items 4 and 8 to be answered with a short sentence built from a provided frame, such as "The mean is higher because ___."