Question

Construct a stem and leaf plot for these\(\ 50\) measurements: $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\\2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\\3.8 & 6.2 &2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\\2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\\4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ a. Describe the shape of the data distribution. Do you see any outliers? b. Use the stem and leaf plot to find the smallest observation. c. Find the eighth and ninth largest observations.

Short answer

Based on the step-by-step solution provided: 1. The stem and leaf plot created from the given data set is: 1 | 68 2 | 125578899 3 | 11455667788999 4 | 012233567899 5 | 11667 6 | 12 2. The shape of the data distribution appears to be somewhat symmetric and roughly bell-shaped, which suggests normal distribution. There do not seem to be any outliers. 3. The smallest observation in the data set is 1.6. 4. The eighth and ninth largest observations in the data set are both 5.1.

Step-by-step solution

Create a stem and leaf plot for the given data set

First, we should order the data set in ascending order, as follows: 1.6 1.8 2.1 2.2 2.5 2.5 2.5 2.7 2.8 2.8 2.9 2.9 3.1 3.1 3.4 3.5 3.5 3.6 3.6 3.6 3.7 3.7 3.7 3.7 3.8 3.9 3.9 3.9 4.0 4.0 4.0 4.1 4.2 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.9 4.9 5.1 5.1 5.6 5.6 5.7 6.1 6.2 Next, we can create a stem and leaf plot with a stem unit of 1 and a leaf unit of 0.1. The final plot will look like this: 1 | 68 2 | 125578899 3 | 11455667788999 4 | 012233567899 5 | 11667 6 | 12

Describe the shape of the data distribution and identify any outliers

Looking at the stem and leaf plot, we can see that the data is somewhat symmetric and roughly bell-shaped. This indicates that the data might be normally distributed. There are no obvious gaps or isolated points in the plot, so there do not appear to be any outliers.

Use the stem and leaf plot to find the smallest observation

The smallest observation in the data set can be easily found by looking at the first (top) row of the stem and leaf plot. In this case, the smallest observation is 1.6.

Find the eighth and ninth-largest observations

To find the eighth and ninth-largest observations, we can count the data points from the largest value (bottom right) to the left and up through the plot. The eighth and ninth largest observations are 5.1 and 5.1, respectively.

Key concepts

Data Distribution Shape

Understanding the shape of data distribution is essential in grasping the nature and behavior of a dataset. When we look at a stem and leaf plot, we're able to quickly identify the form of the distribution, which might be symmetric, skewed, bimodal, or irregular. Symmetric distributions, often shaped like a bell, suggest that the data is balanced around the mean. If a stem and leaf plot reveals such a pattern, we might infer a normal distribution, which is a key concept in probability and statistics.

In the given exercise, after constructing the plot, the data appeared to be roughly bell-shaped, indicating symmetry and allowing us to assume normality. This insight into the data's shape is a foundational step in descriptive statistics, as it helps predict probabilities and understand the central tendencies (such as mean, median, and mode) within the dataset.

Statistical Outliers

Statistical outliers are observations that lie an abnormal distance from other values in the data set. Identifying these can help highlight errors in data collection or indicate that an observation is exceptional. Outliers can be spotted in a stem and leaf plot as digits that stand far from the cluster of other figures within the stems.

Outliers impact the mean of a dataset and can skew its distribution, which makes understanding their nature crucial in descriptive statistics. For instance, if the dataset we're examining has an outlier, we may need to consider using the median instead of the mean for a better central tendency measure. However, in our exercise, we analyze the stem and leaf plot and find no obvious outliers, resulting in no significant impact on the distribution's shape.

Descriptive Statistics

Descriptive statistics summarize and describe the features of a dataset through measures like the mean, median, mode, range, variance, and standard deviation. These tools help us transform raw data into information suitable for interpretation. A stem and leaf plot serves as an effective tool for descriptive statistics because it provides a visual distribution of the data while preserving the original data values.

Understanding how to use this plot is key to conveying information about the location, spread, and shape of the data in question. In the context of our exercise, the stem and leaf plot allowed us to see that the data is fairly symmetrical and gave us a visual to estimate the mean and median, which are likely to be close together. It also presented a clear way to identify the range of the dataset, from the smallest to the largest observation.

Probability and Statistics Education

Probability and statistics education is necessary for equipping students with the skills to analyze data and make informed decisions based on that data. Understanding stem and leaf plots is often an introductory step in learning how to visualize statistical information. These plots blend numerical data comprehension with graphical representation skills, thus forming a solid foundation for more complex statistical concepts.

By mastering how to construct and interpret stem and leaf plots, students can better comprehend data distribution shapes, central tendency, and variability. Moreover, this skill is a stepping stone towards understanding other essential aspects in statistics, such as probabilities, hypothesis testing, and confidence intervals. Our exercise showcases how engaging with these concepts through hands-on practice can provide students with meaningful insights into data analysis techniques.