Water for Schools
Menu

Lesson plan Grade 11 Math

Harlow Springs and Odell Flats: Comparing Hardness Spread

A grade 11 statistics lesson comparing the center and spread of two fictional source-water hardness data sets and interpreting how an outlier shifts the mean.

Objectives

  1. Students will compute the mean, median, and range of a data set.
  2. Students will compare the center and spread of two data sets to judge consistency.
  3. Students will interpret how a single outlier affects the mean more than the median.

Vocabulary

Key words: mean, median, range, outlier, spread

Teacher overview

This lesson uses two fictional source wells, Harlow Springs and Odell Flats, that feed one blended water system. Students compare hardness readings from each source to decide which one delivers more consistent water, practicing the choice between mean and median and between range and other spread measures. The lesson is built around a single outlier in the Odell Flats data so students can see, with real numbers, how one unusual reading pulls the mean while leaving the median almost untouched. A common misconception this lesson addresses is assuming the mean always describes the typical value, even when a data set has an extreme outlier.

Procedure

Warm-up (5 min): Project two short lists of numbers with the same mean but very different spreads, and ask students which list is "more consistent." Collect a few answers to surface the idea that center alone does not describe a data set.

Direct instruction (12 min): Define mean, median, range, and interquartile range, and explain the guidance that the median and interquartile range are more resistant to outliers than the mean and range. Explain that comparing two data sets means comparing both center and spread, not just one number. Introduce the fictional blended system fed by Harlow Springs and Odell Flats, and that hardness is measured in milligrams per liter.

Model (10 min): Work the Harlow Springs data together: readings of 120, 124, 118, 122, 126, 121, 119, and 123 milligrams per liter. Show that the mean is 121.6 milligrams per liter, the median is 121.5, and the range is 126 minus 118, or 8. Point out how tight the spread is, and record these three Harlow values on the board for later comparison.

Guided and independent practice (20 min): Distribute the handout with the Odell Flats data. Have students work items 1 through 6 with a partner, then finish items 7 through 9 on their own. Circulate and check that students subtract the smallest reading from the largest for the range, and that they re-sort the data after removing the outlier.

Closure and exit ticket (8 min): Ask for a thumbs up from students who agree that Odell Flats is less consistent than Harlow Springs; if fewer than three-quarters signal agreement, review item 8 as a class. Collect the exit response to item 9 and scan for whether students name the outlier as the reason the mean misleads.

Harlow Springs and Odell Flats: Comparing Hardness Spread — Grade 11 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Harlow Springs and Odell Flats wells supply one blended system, and both are measured for hardness in milligrams per liter. Harlow Springs has a mean of 121.6, a median of 121.5, a range of 8, and an interquartile range of 4 milligrams per liter. The eight Odell Flats readings are 118, 122, 116, 125, 119, 124, 210, and 121 milligrams per liter.

  1. Find the mean of the Odell Flats readings, in milligrams per liter. Round to the nearest tenth.
  2. Find the median of the Odell Flats readings.
  3. Find the range of the Odell Flats readings.
  4. Identify the outlier in the Odell Flats data.
  5. Recompute the mean of Odell Flats with the outlier removed. Round to the nearest tenth.
  6. Recompute the median of Odell Flats with the outlier removed.
  7. Which measure of center changed more when the outlier was removed, the mean or the median?
  8. Find the interquartile range of the Odell Flats readings. Then compare Harlow Springs and Odell Flats using both the range and the interquartile range. Which source delivers more consistent hardness, and which spread statistic best supports your answer?
  9. The plant blends both sources. Explain why reporting only the mean of Odell Flats could mislead an operator about the water's typical hardness.

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: 131.9 mg/L. Mean = (118 + 122 + 116 + 125 + 119 + 124 + 210 + 121) ÷ 8 = 1,055 ÷ 8 = 131.875, about 131.9 milligrams per liter.
  2. Answer: 121.5 mg/L. Sorted, the readings are 116, 118, 119, 121, 122, 124, 125, 210; the median is the average of the middle two, (121 + 122) ÷ 2 = 121.5.
  3. Answer: 94 mg/L. Range = 210 − 116 = 94 milligrams per liter.
  4. Answer: 210 mg/L. The value 210 sits far above the other readings, which cluster near 120, so it is the outlier.
  5. Answer: 120.7 mg/L. Removing 210 leaves seven readings summing to 845; 845 ÷ 7 = 120.714, about 120.7 milligrams per liter.
  6. Answer: 121 mg/L. The seven remaining sorted readings are 116, 118, 119, 121, 122, 124, 125; the middle value is 121.
  7. Answer: the mean. The mean fell from 131.9 to 120.7, a shift of about 11.2, while the median moved only from 121.5 to 121, showing the mean is far more sensitive to the outlier.
  8. Answer: Harlow Springs; the interquartile range best supports it. For Odell Flats, Q1 is the median of 116, 118, 119, 121, which is 118.5, and Q3 is the median of 122, 124, 125, 210, which is 124.5, so the IQR is 124.5 − 118.5 = 6 milligrams per liter. Harlow's range of 8 is far smaller than Odell Flats' range of 94, but that gap comes almost entirely from the 210 outlier. The IQR resists the outlier, and Harlow's IQR of 4 is still smaller than Odell Flats' 6, so Harlow delivers more consistent hardness even though the two medians are nearly equal.
  9. Answer: The single 210 reading pulls the Odell Flats mean up to 131.9, well above the roughly 121 that most readings sit near, so an operator using only the mean would overestimate the water's typical hardness.

Differentiation

Support

Provide the Odell Flats data already sorted from least to greatest so students can find the median and identify the outlier without re-ordering, and give a filled example of a mean calculation with a different data set at the top of the handout.

Extension

Ask students to build a ninth Odell Flats reading that would make the mean and median match, and to explain what that says about the shape of a data set whose mean and median are equal.

ELL supports

Provide a labeled number line for the Odell Flats data so the clustered readings and the distant outlier are visible at a glance. Pair "center" with mean and median and "spread" with range, using a simple picture for each.