Question

Consider this set of data: $$\begin{array}{llllll}4.5 & 3.2 & 3.5 & 3.9 & 3.5 & 3.9 \\\4.3 & 4.8 & 3.6 & 3.3 & 4.3 & 4.2 \\\3.9 & 3.7 & 4.3 & 4.4 & 3.4 & 4.2 \\\4.4 & 4.0 & 3.6 & 3.5 & 3.9 & 4.0\end{array}$$ a. Construct a stem and leaf plot by using the leading digit as the stem. b. Construct a stem and leaf plot by using each leading digit twice. Does this technique improve the presentation of the data? Explain.

Short answer

Short Answer: The stem and leaf plot with each leading digit used twice offers a more detailed representation of the data as it spreads the data across more rows, making it easier to identify patterns and trends within the dataset. This improvement in presentation can help in better understanding the distribution of the data.

Step-by-step solution

Organize the data

First, list the data in ascending order. This will make it easier to construct the stem and leaf plots. $$ 3.2, 3.3, 3.4, 3.5, 3.5, 3.6, 3.6, 3.7, 3.9, 3.9, 3.9, 3.9, \\ 4.0, 4.0, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.5, 4.8 $$

Construct the stem and leaf plot with leading digit as the stem

For this plot, we use the leading digit of each data point as the stem and the remaining digits as leaves. So, for the 3s stem, we have 2, 3, 4, 5, 5, 6, 6, 7, 9, 9, 9, 9, and for the 4s stem, we have 0, 0, 2, 2, 3, 3, 3, 4, 4, 5, 8. The stem and leaf plot looks like this: ``` 3 | 2 3 4 5 5 6 6 7 9 9 9 9 4 | 0 0 2 2 3 3 3 4 4 5 8 ```

Construct the stem and leaf plot with each leading digit used twice

For this plot, we use the leading digit of each data point as the stem twice, in separate rows. The first row will be for the digits 0, 1, 2, 3 and the second row will be for the digits 4, 5, 6, 7, 8, 9. So, for the 3s stem, we have 2, 3, 4 and 5, 5, 6, 6, 7 in the first two rows, and 9, 9, 9, 9 in the second two rows. For the 4s stem, we have 0, 0 and 2, 2, 3, 3, 3 in the first two rows, and 4, 4, 5, 8 in the second two rows. The stem and leaf plot looks like this: ``` 3 | 2 3 4 3 | 5 5 6 6 7 3 | 9 9 9 9 4 | 0 0 2 2 3 3 3 4 | 4 4 5 8 ```

Compare the two stem and leaf plots

Comparing the two plots, we can see that the one with each leading digit used twice (part b) is a little more spread out, allowing us to see how the data is distributed slightly more clearly. In part a, the plots are compressed into fewer rows, making it a bit harder to identify any specific patterns in the data. Using each leading digit twice does improve the presentation as it offers a finer level of detail in the distribution of the data and makes it easier to spot patterns and trends within the data.

Key concepts

Data Organization

Data organization is a cornerstone of effective statistical analysis. It's the process of sorting and structuring data in a way that makes it easier to understand and interpret. When you receive a raw set of numbers, like the one in our exercise, it isn't immediately clear what story these numbers tell. By arranging the data in ascending order, as done in the first step of our solution, we set the foundation for further analysis.

Good data organization aids in identifying patterns, anomalies, or trends. It's also a precursor to creating various types of statistical plots. In the context of our exercise, the progression from a list of numbers to a sorted array allows us to construct a stem and leaf plot which visually communicates the data's distribution.

Statistical Plots

Statistical plots are visual representations of data which make complex information more accessible. The stem and leaf plot, as mentioned in our exercise, is a prime example. It's a method that retains the original data values while also showing the shape of the data distribution. Think of it like a histogram that has not forgotten its roots.

Creating a stem and leaf plot involves choosing a 'stem', typically the leading digits of the numbers, and then listing the 'leaves', which are the trailing digits, next to their respective stems. This way, one can quickly see how many numbers fall into each range. It's a technique often used when you want to display every data point and understand the frequency of each interval.

Data Distribution

Understanding how data is distributed is essential for interpreting results and making predictions. Data distribution refers to how data points are spread out or clustered across a value range. With our stem and leaf plot, we can easily observe the concentration of values around the center and the spread or range of the data.

Moreover, by presenting the data twice using the leading digit, we can discern finer details in the distribution. For instance, having separate stems for lower and upper halves of the same leading digit gives us a clearer picture of where the bulk of data points lie, making patterns like clustering or gaps immediately evident. This augmented level of detail is particularly useful when dealing with probabilities and understanding the likelihood of an event based on the frequency of certain data ranges.

Probability and Statistics Education

Educating students in probability and statistics empowers them with the tools to analyze real-world phenomena. Proper instruction includes not just formula and theory, but also the ability to organize, graph, and interpret data. Visualization tools such as stem and leaf plots are instrumental in this process.

A well-constructed stem and leaf plot, like the ones in our exercise, provides a hands-on approach to data analysis. Students learn by doing; they sort, split, and literally plot the figures to reveal the underlying distribution. This direct interaction with numbers fortifies their understanding of the material and builds a solid foundation for more complex statistical concepts. Incorporating exercises that feature clear organization, visualization, and interpretation of data are essential to creating a robust curriculum in probability and statistics education.