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Math problem set Grade 11 Science

Two Intakes at Marden and Culliton: Turbidity Statistics

A grade 11 statistics set comparing a week of turbidity from two fictional intakes with mean, median, IQR, standard deviation, and an outlier test.

Objectives

  1. Students will compute the mean, median, and interquartile range of a small data set.
  2. Students will use the 1.5 times IQR rule to identify a statistical outlier.
  3. Students will compare two data sets by center and spread and interpret the difference.

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

Teacher overview

A utility that can draw from two sources wants to know not just which is cleaner on average, but which is steadier, because a source that swings wildly is harder to treat to a safe, constant quality. This set compares a week of raw-water turbidity readings from two fictional intakes, Marden (a river) and Culliton (a reservoir), using center and spread. A common student error is to trust the mean when a single extreme value is present; one storm reading can pull the mean up sharply while the median barely moves, which is exactly why the resistant statistics of median and interquartile range matter here. Students also run the 1.5-times-IQR outlier test and compare standard deviations to put a number on consistency. Finished water must reach 0.3 NTU, so a variable raw source raises the treatment burden.

Two Intakes at Marden and Culliton: Turbidity Statistics — Grade 11 Science
Name: ____________________________ Date: ______________

Problems

An operator records seven days of raw-water turbidity, in NTU, from two fictional intakes, Marden (a river) and Culliton (a reservoir):

Day Marden intake (NTU) Culliton intake (NTU)
1 4.5 1.3
2 3.8 1.2
3 5.1 1.4
4 4.2 1.1
5 12.0 1.2
6 3.9 1.5
7 4.6 1.4
  1. Calculate the mean turbidity of the Marden intake's seven readings, in NTU. Round to two decimals.
  2. Calculate the mean turbidity of the Culliton intake's seven readings, in NTU. Round to two decimals.
  3. Find the median turbidity of the Marden intake, in NTU.
  4. Find the interquartile range (IQR) of the Marden intake, in NTU.
  5. Find the interquartile range of the Culliton intake, in NTU.
  6. Using the rule that a value above the third quartile plus 1.5 times the IQR is an outlier, test whether the Marden intake's 12.0 reading is an outlier.
  7. Using a calculator, find the population standard deviation of the Culliton intake's readings, in NTU. Round to two decimals.
  8. Using a calculator, find the population standard deviation of the Marden intake's readings, in NTU. Round to two decimals.
  9. Which intake delivers more consistent turbidity, and which single reading most inflates the other intake's mean and spread? Explain using your results.
  10. Both intakes must be treated to a finished-water turbidity at or below 0.3 NTU. Explain why the Marden intake's higher and more variable raw turbidity makes the operator's job harder.

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: 5.44 NTU.
    Sum = 4.5 + 3.8 + 5.1 + 4.2 + 12.0 + 3.9 + 4.6 = 38.1; 38.1 divided by 7 = 5.44 NTU.
  2. Answer: 1.30 NTU.
    Sum = 1.3 + 1.2 + 1.4 + 1.1 + 1.2 + 1.5 + 1.4 = 9.1; 9.1 divided by 7 = 1.30 NTU.
  3. Answer: 4.5 NTU.
    Sorted Marden: 3.8, 3.9, 4.2, 4.5, 4.6, 5.1, 12.0; the fourth of seven values is the median, 4.5 NTU.
  4. Answer: 1.2 NTU.
    Lower half (3.8, 3.9, 4.2) has median Q1 = 3.9; upper half (4.6, 5.1, 12.0) has median Q3 = 5.1; IQR = 5.1 - 3.9 = 1.2 NTU.
  5. Answer: 0.2 NTU.
    Sorted Culliton: 1.1, 1.2, 1.2, 1.3, 1.4, 1.4, 1.5; Q1 = 1.2 and Q3 = 1.4, so IQR = 1.4 - 1.2 = 0.2 NTU.
  6. Answer: Yes, 12.0 NTU is an outlier.
    Q3 + 1.5 x IQR = 5.1 + 1.5 x 1.2 = 5.1 + 1.8 = 6.9; since 12.0 is greater than 6.9, it is a statistical outlier.
  7. Answer: 0.13 NTU.
    Deviations from the mean 1.30 squared sum to 0.12; 0.12 divided by 7 = 0.0171; the square root is 0.13 NTU.
  8. Answer: 2.71 NTU.
    The squared deviations from the mean 5.44 sum to about 51.34; 51.34 divided by 7 = 7.33; the square root is about 2.71 NTU.
  9. Answer: Culliton is far more consistent; Marden's 12.0 reading inflates its statistics.
    Culliton's standard deviation (0.13) and IQR (0.2) are much smaller than Marden's (2.71 and 1.2), so Culliton is steadier; the day-5 reading of 12.0 NTU is the storm value that pulls the Marden mean up to 5.44 and enlarges its spread.
  10. Answer: Variable, high raw turbidity forces constant treatment adjustments.
    To finish at 0.3 NTU, the plant must remove far more particles from Marden water, and because that water swings from about 3.8 to 12.0 NTU, the operator must keep changing coagulant dose and filter operation to hold the finished water steady, which is harder than treating Culliton's near-constant supply.

Differentiation

Support

Provide the sorted lists for both intakes and a labeled five-number-summary template (minimum, Q1, median, Q3, maximum) so students can read the quartiles directly for problems 3 through 6. Allow the standard deviations in problems 7 and 8 to be found with a calculator's statistics mode.

Extension

Ask students to recompute the Marden mean and standard deviation with the day-5 outlier removed, then write a short paragraph on how much the single reading changed each statistic and which measure of center they would report to a manager.

ELL supports

Pre-teach "mean," "median," "spread," and "outlier" with a number-line picture showing the six clustered Marden values and the one far-off value. Provide sentence frames for problems 9 and 10, such as "The ___ intake is more consistent because its ___ is smaller."