pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. b. Which type of water has more dispersion in pH using the standard deviation as the measure of dispersion?

Verified step by step guidance

1. Step 1: Organize the data for each type of water separately. For Tap water, list all the pH 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. For Bottled water, list all the pH 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.
2. Step 2: Calculate the mean (average) pH for each type of water. Use the formula for the mean: \(\text{mean} = \frac{\sum x_i}{n}\), where \(x_i\) are the pH values and \(n\) is the number of values.
3. Step 3: Calculate the variance for each type of water. The variance formula is \(s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2\), where \(\bar{x}\) is the mean calculated in Step 2, and \(n\) is the number of observations.
4. Step 4: Calculate the standard deviation for each type of water by taking the square root of the variance: \(s = \sqrt{s^2}\). This gives a measure of dispersion of the pH values around the mean.
5. Step 5: Compare the standard deviations of the Tap water and Bottled water samples. The water type with the larger standard deviation has more dispersion in pH values.

Key Concepts

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points are close to the mean, while a high standard deviation shows data are spread out over a wider range. It is useful for comparing variability between different datasets.

pH Scale and Its Interpretation

The pH scale ranges from 0 to 14 and measures the acidity or alkalinity of a solution. Values below 7 indicate acidity, values above 7 indicate alkalinity, and 7 is neutral. Understanding pH helps contextualize the data and the significance of differences in water samples.

Comparing Dispersion Between Groups

Comparing dispersion involves analyzing measures like standard deviation to determine which group has more variability. In this case, comparing the standard deviations of pH values for tap and bottled water reveals which type has more inconsistent pH levels.