91Ó°ÊÓ

The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the \(\mathrm{pH}\) in samples of bottled water and tap water. $$ \begin{array}{lllllll} \hline \text { Tap } & 7.64 & 7.45 & 7.47 & 7.50 & 7.68 & 7.69 \\ & 7.45 & 7.10 & 7.56 & 7.47 & 7.52 & 7.47 \\ \hline \text { Bottled } & 5.15 & 5.09 & 5.26 & 5.20 & 5.02 & 5.23 \\ & 5.28 & 5.26 & 5.13 & 5.26 & 5.21 & 5.24 \\ \hline \end{array} $$ (a) Determine the mean, median, and mode \(\mathrm{pH}\) for each type of water. Comment on the differences between the two water types. (b) Suppose the \(\mathrm{pH}\) of 7.10 in tap water was incorrectly recorded as \(1.70 .\) How does this affect the mean? the median? What property of the median does this illustrate?

Step by step solution

01

List the pH values for Tap Water

The pH values for Tap Water are: 7.64, 7.45, 7.47, 7.50, 7.68, 7.69, 7.45, 7.10, 7.56, 7.47, 7.52, 7.47.

02

List the pH values for Bottled Water

The pH values for Bottled Water are: 5.15, 5.09, 5.26, 5.20, 5.02, 5.23, 5.28, 5.26, 5.13, 5.26, 5.21, 5.24.

03

Calculate the Mean pH for Tap Water

Add up all the pH values for Tap Water and divide by the number of values: (7.64 + 7.45 + 7.47 + 7.50 + 7.68 + 7.69 + 7.45 + 7.10 + 7.56 + 7.47 + 7.52 + 7.47) / 12 ≈ 7.50.

04

Calculate the Mean pH for Bottled Water

Add up all the pH values for Bottled Water and divide by the number of values: (5.15 + 5.09 + 5.26 + 5.20 + 5.02 + 5.23 + 5.28 + 5.26 + 5.13 + 5.26 + 5.21 + 5.24) / 12 ≈ 5.19.

05

Find the Median pH for Tap Water

Arrange the pH values of tap water in ascending order and find the middle value(s): 7.10, 7.45, 7.45, 7.47, 7.47, 7.47, 7.50, 7.52, 7.56, 7.64, 7.68, 7.69. The median is the average of the 6th and 7th values: (7.47 + 7.50) / 2 = 7.485.

06

Find the Median pH for Bottled Water

Arrange the pH values of bottled water in ascending order and find the middle value(s): 5.02, 5.09, 5.13, 5.15, 5.20, 5.21, 5.23, 5.24, 5.26, 5.26, 5.26, 5.28. The median is the average of the 6th and 7th values: (5.21 + 5.23) / 2 = 5.22.

07

Identify the Mode pH for Tap Water

Find the most frequently occurring pH value in Tap Water: The pH value 7.47 appears 4 times, so the mode is 7.47.

08

Identify the Mode pH for Bottled Water

Find the most frequently occurring pH value in Bottled Water: The pH value 5.26 appears 3 times, so the mode is 5.26.

09

Analyze differences between the two types of water

Tap water is generally more alkaline with mean and median values around 7.5, while bottled water is more acidic with mean and median values around 5.2. The difference illustrates the expected variations in pH between tap and bottled water.

10

Recalculate the Mean pH with Incorrect Tap Water Value

If the pH 7.10 is incorrectly recorded as 1.70, then the new mean is calculated: (7.64 + 7.45 + 7.47 + 7.50 + 7.68 + 7.69 + 7.45 + 1.70 + 7.56 + 7.47 + 7.52 + 7.47) / 12 ≈ 6.88.

11

Recalculate the Median pH with Incorrect Tap Water Value

So the new list is: 1.70, 7.45, 7.45, 7.47, 7.47, 7.47, 7.50, 7.52, 7.56, 7.64, 7.68, 7.69. The median remains the average of the 6th and 7th values: (7.47 + 7.50) / 2 = 7.485.

12

Discuss the Property of the Median

The correct median remains unchanged, which demonstrates its resistance to extreme values (robustness) unlike the mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mean calculation

The mean, often called the average, is a measure of central tendency. It tells you about the central value of a data set.

To calculate the mean, follow these steps:

• Add up all the numbers in the data set.
• Divide the sum by the total number of values.

For example, the mean pH for tap water can be calculated as:
\[ \frac{7.64 + 7.45 + 7.47 + 7.50 + 7.68 + 7.69 + 7.45 + 7.10 + 7.56 + 7.47 + 7.52 + 7.47}{12} \approx 7.50 \] This means the average pH for tap water is about 7.50, slightly alkaline.
The bottled water mean calculation is similar: \[ \frac{5.15 + 5.09 + 5.26 + 5.20 + 5.02 + 5.23 + 5.28 + 5.26 + 5.13 + 5.26 + 5.21 + 5.24}{12} \approx 5.19 \] The average pH for bottled water is about 5.19, which is more acidic.

median calculation

The median is the middle value in a data set arranged in ascending or descending order. It separates the higher half from the lower half of the data.

To find the median, you need to:

• Arrange the data in order.
• Identify the middle value (or average the two middle values if the data set has an even number of values).

For tap water pH values:
1. Arrange values: 7.10, 7.45, 7.45, 7.47, 7.47, 7.47, 7.50, 7.52, 7.56, 7.64, 7.68, 7.69
2. The 6th and 7th values are 7.47 and 7.50.
3. Median: \( \frac{7.47 + 7.50}{2} = 7.485 \) Similarly, for bottled water:
1. Arrange values: 5.02, 5.09, 5.13, 5.15, 5.20, 5.21, 5.23, 5.24, 5.26, 5.26, 5.26, 5.28
2. The 6th and 7th values are 5.21 and 5.23.
3. Median: \( \frac{5.21 + 5.23}{2} = 5.22 \)

mode calculation

The mode is the value that appears most frequently in a data set. It's another measure of central tendency, which gives insight into the frequency of values.

To find the mode, simply:

• Count how often each value occurs.
• Identify the value(s) with the highest frequency.

In the case of tap water pH values:

• The pH value 7.47 appears 4 times.
• The mode is 7.47.

For bottled water:

• The pH value 5.26 appears 3 times.
• The mode is 5.26.

The mode provides information about the most common levels of acidity or alkalinity in the samples.

data set analysis

Analyzing a data set involves evaluating measures of central tendency (like mean, median, and mode) and understanding the spread and distribution of the values.

In this exercise, we look at pH values for tap and bottled water. Tap water has a mean of 7.50, median of 7.485, and mode of 7.47, indicating it is more alkaline. Bottled water has a mean of 5.19, median of 5.22, and mode of 5.26, showing it is more acidic.

Compare these findings to understand the data better:

• Tap water is overall more alkaline with tighter clustering around 7.47-7.50.
• Bottled water is more consistently acidic around 5.20-5.26.

Such analysis helps in understanding water quality variations between sources.

robustness of median

The median's robustness refers to its stability against extreme values or outliers. Unlike the mean, which can be heavily influenced by outliers, the median often remains unaffected.

Consider an error in recording the tap water pH:

• Original mean: 7.50
• Error mean (recorded 7.10 as 1.70): 6.88

However, the median remains: \[ \frac{7.47 + 7.50}{2} = 7.485 \] This shows that despite the erroneous outlier, the median stayed the same, demonstrating its robustness. This makes the median a reliable measure of central tendency, especially when dealing with potential outliers in a data set.