Question

Calcium Content The calcium (Ca) content of a powdered mineral substance was analyzed ten times with the following percent compositions recorded: $$\begin{aligned}&\begin{array}{lllll}.0271 & .0282 & .0279 & .0281 & .0268\end{array}\\\&\begin{array}{lllll}.0271 & .0282 & .0279 & .0281 & .0268 \\\\.0271 & .0281 & .0269 & .0275 & .0276\end{array}\end{aligned}$$ a. Draw a dotplot to describe the data. (HINT: The scale of the horizontal axis should range from .0260 to \(.0290 .)\) b. Draw a stem and leaf plot for the data. Use the numbers in the hundredths and thousandths places as the stem. c. Are any of the measurements inconsistent with the other measurements, indicating that the technician may have made an error in the analysis?

Short answer

a) Draw a dotplot representing the calcium content values. Horizontal Axis Scale: | .0260 | .0270 | .0280 | .0290 Dots: | 2 | 5 | 6 | 3 | b) Create a stem and leaf plot for the calcium content values. Stem | Leaf -------------- 0.26 | 8 9 0.27 | 1 1 1 5 6 9 9 0.28 | 1 1 1 2 2 c) Identify any inconsistent measurements that might indicate an error in the analysis. In this case, all the values seem to be fairly close to each other, and there doesn't seem to be any outlier or inconsistent value. Hence, no measurement appears to indicate an error in the analysis.

Step-by-step solution

Preparing the data for dotplot and stem and leaf plot

Organize the given calcium content values in ascending order for ease of plotting. The sorted values are: $$ \begin{aligned} &.0268, .0269, .0271, .0271, .0271,\\ &.0275, .0276, .0279, .0279, .0281,\\ &.0281, .0281, .0282, .0282. \end{aligned} $$

Drawing the dotplot

For each calcium content value, put a dot above the value on the horizontal axis that ranges from \(.0260\) to \(.0290\). The dotplot will look like the following: Horizontal Axis Scale: | .0260 | .0270 | .0280 | .0290 Dots: | 2 | 5 | 6 | 3 |

Creating the stem and leaf plot

In this case, the stem consists of the hundredths place numbers, and the leaf is the thousandths place number. The stem and leaf plot will look like the following: Stem | Leaf -------------- 0.26 | 8 9 0.27 | 1 1 1 5 6 9 9 0.28 | 1 1 1 2 2

Check for inconsistent measurements

Look at both the dotplot and stem and leaf plot to see if there are any unusual values that stand out as potentially erroneous. In this case, all the values seem to be fairly close to each other, and there doesn't seem to be any outlier or inconsistent value. Hence, no measurement appears to indicate an error in the analysis.

Key concepts

Dotplot

A dotplot is a simple yet effective graphical tool used to visualize a set of data points. It displays individual data points as dots along a single axis, making it easier to spot patterns, clusters, and possible outliers. To create a dotplot, follow these steps:

• Identify the range of data. For example, in the calcium content data we are working with, the range is from 0.0260 to 0.0290.
• Draw a horizontal line representing the number line for this range.
• Mark evenly spaced increments along this horizontal line to represent potential data values.
• For each data point, place a dot above its corresponding position on the line. Multiple identical data points will stack dots vertically.

This cumulative display highlights the distribution of the data. It's useful for visualizing small datasets and understanding their distribution at a glance.

Stem and Leaf Plot

The stem and leaf plot is another graphical method used to summarize data. It retains the original data values, offering a more detailed view compared to other summary statistics like mean or median. Here's how you can create a stem and leaf plot:

• Identify the place values in your data: the stem is the higher place value (in this case, hundredths), and the leaf is the lower place value (thousandths).
• List the stems (unique numbers from the stem) vertically in a column.
• For each data point, write its leaf next to the corresponding stem.

For example, in the calcium content dataset, stems like 0.26, 0.27, and 0.28 are paired with leaves like 8, 9, 1, etc. This format quickly shows the frequency of the data points within certain intervals.

Statistical Consistency

Statistical consistency refers to how similar the data points are to each other within a dataset. Consistent data lacks outliers or irregular values that deviate significantly from the rest. To assess statistical consistency:

• Visualize the data using tools like dotplots and stem and leaf plots.
• Look for values that are very different from others, also known as outliers, which might be due to errors in data collection or recording.
• Examine the data's spread, concentrating on how tightly data points cluster around a central value such as the mean or median.

In our exercise, all calcium content measurements are close to each other, indicating a consistent data set without errors. Consistency ensures reliability in results and conclusions drawn from data analyses.