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Worksheet Grade 11 Health & Consumer Science

Comparing Home Chlorine Readings on Two Brookfield Streets

A grade 11 statistics worksheet on a fictional town's home chlorine readings: compare two streets by IQR and standard deviation, then test an outlier.

Objectives

  1. Students will compute the mean, median, interquartile range, and population standard deviation of two small data sets.
  2. Students will compare the center and spread of two streets' chlorine readings and interpret the difference in context.
  3. Students will identify an outlier with the 1.5 times IQR rule and explain its effect on each statistic.
  4. Students will judge readings against the chlorine MRDL and the typical distribution range, including how compliance with the MRDL is decided.

Vocabulary

Chlorine residual
Chlorine residual is the small amount of chlorine still left in the water after it leaves the treatment plant. It keeps protecting the water from germs all the way to your faucet.

Also: mean, median, interquartile range, standard deviation, outlier

Teacher overview

Home test strips let a resident check the chlorine residual at their own tap, and a set of readings tells more than any single number. In this worksheet students work with readings from the fictional town of Brookfield, where residents on two streets each measured their tap water for five days. Students compare the two streets using the median, mean, interquartile range, and standard deviation, then add a sixth, unusually high Maple Street reading and examine which statistics it distorts and which it barely moves. The last item asks whether that one reading means the town broke the federal limit, which depends on how the maximum residual disinfectant level (MRDL) of 4.0 mg/L is actually applied.

Students may use the statistics functions on a graphing calculator, but they should show the deviation work for items 5, 6, and 12 at least once by hand. Calculators report two standard deviations: σx divides by n and matches the key, while sx divides by n − 1 and gives slightly larger values (listed in the key so you can recognize them).

Comparing Home Chlorine Readings on Two Brookfield Streets — Grade 11 Health & Consumer Science
Name: ____________________________ Date: ______________

Student Worksheet

In the fictional town of Brookfield, residents recorded their tap chlorine residual, in milligrams per liter (mg/L), for five days. The federal MRDL for chlorine is 4.0 mg/L, and systems using free chlorine typically keep the residual between 0.2 and 2.0 mg/L in the distribution system.

Maple Street readings (mg/L): 0.8, 1.0, 1.2, 1.0, 1.5

Oak Street readings (mg/L): 0.4, 2.0, 1.8, 0.6, 1.2

Methods to use on this worksheet:

  • Quartiles: put the readings in order and split them into a lower half and an upper half. When the number of readings is odd, leave the median out of both halves. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
  • Standard deviation: use the population formula σ = √( Σ(x − x̄)² ÷ n ), where x̄ is the mean and n is the number of readings. On a graphing calculator this is the value labeled σx.
  • Outliers: a reading is an outlier if it is greater than Q3 + 1.5 × IQR or less than Q1 − 1.5 × IQR.

Part A: Comparing the two streets

  1. Write the five Maple Street readings in order from least to greatest, and find their mean and median.
  2. Write the five Oak Street readings in order from least to greatest, and find their mean and median.
  3. Find the range of each street's readings.
  4. Find Q1, Q3, and the interquartile range (IQR) for each street.
  5. Find the population standard deviation of the Maple Street readings. Show each deviation from the mean, its square, and the sum of the squares. Round to the nearest hundredth.
  6. Find the population standard deviation of the Oak Street readings, showing the same steps. Round to the nearest hundredth.
  7. Which street's readings are more spread out? Use the range, the IQR, and the standard deviation to support your answer, and explain in words what Oak Street's standard deviation tells a resident about a typical day's reading.
  8. Are all ten readings inside the typical distribution range of 0.2 to 2.0 mg/L? Which street has readings at or near the edges of that range?
  9. A resident says Oak Street's water has "more chlorine on average." Using your mean values, is that claim correct, and does a higher mean also mean the readings are more consistent? Explain.

Part B: A sixth Maple Street reading

On the sixth day, a Maple Street resident reported a reading of 4.4 mg/L. The other five Maple Street readings are unchanged, so the Maple Street data set now has six readings.

  1. Find the mean and median of the six Maple Street readings. By how much did each one change from your answers to item 1?
  2. Find Q1, Q3, and the IQR of the six Maple Street readings. Then use the 1.5 × IQR rule to decide whether 4.4 mg/L is an outlier, showing the fence you compared it with.
  3. Find the population standard deviation of the six Maple Street readings, rounded to the nearest hundredth. Compare how much the standard deviation changed with how much the IQR changed.
  4. A neighborhood newsletter will report one measure of center and one measure of spread for the six Maple Street readings. Which two should it use, and why? Refer to the shape of the data in your answer.
  5. The 4.4 mg/L reading is above the 4.0 mg/L MRDL. Does this reading by itself show that Brookfield's water system violated the MRDL? Explain how the MRDL is judged, and describe a sensible next step for the resident who took the reading.

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: 0.8, 1.0, 1.0, 1.2, 1.5 mg/L; mean 1.1 mg/L; median 1.0 mg/L. The five readings sum to 5.5 mg/L, and 5.5 divided by 5 is 1.1 mg/L. The middle (third) value in the ordered list is 1.0 mg/L.
  2. Answer: 0.4, 0.6, 1.2, 1.8, 2.0 mg/L; mean 1.2 mg/L; median 1.2 mg/L. The readings sum to 6.0 mg/L, and 6.0 divided by 5 is 1.2 mg/L. The middle value of the ordered list is 1.2 mg/L.
  3. Answer: Maple Street 0.7 mg/L; Oak Street 1.6 mg/L. Maple: 1.5 minus 0.8 is 0.7 mg/L. Oak: 2.0 minus 0.4 is 1.6 mg/L.
  4. Answer: Maple Street Q1 = 0.9, Q3 = 1.35, IQR = 0.45 mg/L; Oak Street Q1 = 0.5, Q3 = 1.9, IQR = 1.4 mg/L. With five readings the median is left out of both halves. Maple's lower half is 0.8, 1.0, so Q1 = 0.9; its upper half is 1.2, 1.5, so Q3 = 1.35; IQR = 1.35 − 0.9 = 0.45 mg/L. Oak's lower half is 0.4, 0.6, so Q1 = 0.5; its upper half is 1.8, 2.0, so Q3 = 1.9; IQR = 1.9 − 0.5 = 1.4 mg/L.
  5. Answer: 0.24 mg/L. With mean 1.1, the deviations are −0.3, −0.1, 0.1, −0.1, and 0.4 (for 0.8, 1.0, 1.2, 1.0, 1.5). Their squares are 0.09, 0.01, 0.01, 0.01, and 0.16, which sum to 0.28. Then 0.28 ÷ 5 = 0.056, and √0.056 ≈ 0.2366, which rounds to 0.24 mg/L. A student who used sx (dividing by 4) gets 0.26 mg/L.
  6. Answer: 0.63 mg/L. With mean 1.2, the deviations are −0.8, 0.8, 0.6, −0.6, and 0 (for 0.4, 2.0, 1.8, 0.6, 1.2). Their squares are 0.64, 0.64, 0.36, 0.36, and 0, which sum to 2.00. Then 2.00 ÷ 5 = 0.40, and √0.40 ≈ 0.6325, which rounds to 0.63 mg/L. A student who used sx (dividing by 4) gets 0.71 mg/L.
  7. Answer: Oak Street is more spread out. All three measures agree: its range is 1.6 mg/L against Maple's 0.7 mg/L, its IQR is 1.4 mg/L against 0.45 mg/L, and its standard deviation is 0.63 mg/L against 0.24 mg/L. In context, an Oak Street reading on a typical day lies roughly 0.6 mg/L above or below the street's mean of 1.2 mg/L, so a resident could see anything from well under 1 mg/L to about 2 mg/L. A typical Maple Street reading lies only about 0.2 mg/L from its mean of 1.1 mg/L.
  8. Answer: Yes, all ten readings are inside the range, and Oak Street's readings come closest to its edges. The readings run from 0.4 to 2.0 mg/L, which lies within 0.2 to 2.0 mg/L. Oak Street's 2.0 mg/L reading sits exactly at the upper end, and its 0.4 mg/L reading is only 0.2 mg/L above the lower end. Every Maple Street reading falls between 0.8 and 1.5 mg/L, well inside the range.
  9. Answer: Yes, the claim is correct, but a higher mean does not mean more consistent readings. Oak Street's mean of 1.2 mg/L is slightly higher than Maple Street's 1.1 mg/L. Consistency is a question of spread, not center, and Oak Street's larger range, IQR, and standard deviation show its readings are less consistent.
  10. Answer: Mean 1.65 mg/L, up 0.55 mg/L; median 1.1 mg/L, up 0.1 mg/L. The six readings sum to 5.5 + 4.4 = 9.9 mg/L, and 9.9 ÷ 6 = 1.65 mg/L, so the mean rose by 1.65 − 1.1 = 0.55 mg/L. Ordered, the readings are 0.8, 1.0, 1.0, 1.2, 1.5, 4.4; with an even count the median is the mean of the third and fourth values, (1.0 + 1.2) ÷ 2 = 1.1 mg/L, a rise of 1.1 − 1.0 = 0.1 mg/L.
  11. Answer: Q1 = 1.0, Q3 = 1.5, IQR = 0.5 mg/L; 4.4 mg/L is an outlier because it is above the upper fence of 2.25 mg/L. With six readings the halves are 0.8, 1.0, 1.0 and 1.2, 1.5, 4.4, so Q1 = 1.0 and Q3 = 1.5, giving IQR = 1.5 − 1.0 = 0.5 mg/L. The upper fence is 1.5 + 1.5 × 0.5 = 1.5 + 0.75 = 2.25 mg/L, and 4.4 is greater than 2.25. (The lower fence, 1.0 − 0.75 = 0.25 mg/L, has no readings below it.)
  12. Answer: 1.25 mg/L; the standard deviation rose from 0.24 to 1.25 mg/L, more than five times larger, while the IQR rose only from 0.45 to 0.5 mg/L. With mean 1.65, the deviations are −0.85, −0.65, −0.65, −0.45, −0.15, and 2.75. Their squares are 0.7225, 0.4225, 0.4225, 0.2025, 0.0225, and 7.5625, which sum to 9.355. Then 9.355 ÷ 6 ≈ 1.5592, and √1.5592 ≈ 1.2487, which rounds to 1.25 mg/L. The single squared deviation for 4.4 (7.5625) is about 81 percent of the total, which is why the standard deviation is so sensitive to it. A student who used sx (dividing by 5) gets 1.37 mg/L.
  13. Answer: The median (1.1 mg/L) and the IQR (0.5 mg/L). One high value makes the six readings strongly skewed to the right. The mean of 1.65 mg/L is higher than five of the six readings, so it does not describe a typical day, and the standard deviation of 1.25 mg/L is driven almost entirely by the one outlier. The median and IQR depend only on the middle of the ordered data, so they still describe the typical Maple Street reading. A good newsletter would also mention the 4.4 mg/L reading separately rather than hide it.
  14. Answer: No, a single reading does not by itself show a violation. The MRDL of 4.0 mg/L is judged as a running annual average, computed every quarter, of monthly averages of the utility's own distribution-system samples, so one day's value, and in particular one home test strip result, does not decide compliance. The reading is still worth acting on because it is far above the other five Maple readings and far outside the typical 0.2 to 2.0 mg/L range. A sensible next step is to retest with a fresh strip, following the package's timing directions exactly, and, if the result is still high, to report it to Brookfield's water utility so it can take its own sample at that tap.

Differentiation

Support

Give students a blank deviation table for items 5, 6, and 12 with columns for the reading, the deviation from the mean, and the squared deviation, plus a final row for the sum. Provide the ordered Oak Street list already written out, and work one quartile example with a practice street before students find their own. Allow calculators throughout and have students check each hand-computed standard deviation against the σx value on a graphing calculator.

Extension

Ask students to recompute items 5, 6, and 12 with the sample standard deviation (dividing by n − 1) and explain why statisticians use n − 1 when five or six readings stand for a street's water over a whole month. Then ask them to design a third street whose five readings have the same mean and median as Maple Street's five-day readings but a standard deviation at least twice as large, and to show the calculations that prove all three conditions are met.

ELL supports

Pre-teach "mean," "median," "quartile," "interquartile range," "standard deviation," "outlier," and "residual" with a number line that shows each street's readings as dots, the typical band between 0.2 and 2.0 shaded, and a box plot drawn above each street's dots. Provide sentence frames for items 7, 13, and 14, such as "The more spread-out street is ___ because its IQR is ___ and its standard deviation is ___," "The newsletter should use the ___ because the data are ___," and "One reading is not a violation because the MRDL is judged by ___."