Question

$$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\\2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\\3.8 & 6.2 &2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\\2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\\4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 &3.9\end{array}$$ Construct a stem and leaf plot for these 50 measurements: 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

Answer: The shape of the data distribution is approximately symmetric with a slight skew to the right. The smallest observation is 1.6, while both the 8th and 9th largest observations are 4.9.

Step-by-step solution

Creating the Stem and Leaf Plot

To create the stem and leaf plot, we first organize the data into ascending numerical order: \(1.6,1.8,2.1,2.2,2.5,2.5,2.7,2.8,2.8,2.9,2.9,3.1,3.1,3.4,3.5,3.5,3.6,3.6,3.6,3.7,3.7,3.7,3.8,3.9,3.9,3.9,3.9,4.0,4.0,4.1,4.2,4.2,4.3,4.4,4.5,4.6,4.7,4.8,4.9,4.9,4.9,5.1,5.1,5.6,5.6,5.7,6.1,6.2\), Now, create a stem and leaf plot by separating data into 'stem' and 'leaf.' The 'stem' is the whole number part (left-most digits), and the 'leaf' is the decimal part (right-most digit). 1 | 68 2 | 1255899 3 | 11455667788999 4 | 00123356788999 5 | 116667 6 | 12

Describing the Shape of the Data Distribution and Identifying Outliers

Now, we can analyze the shape of the distribution. The stem and leaf plot appears approximately symmetric, with a slight skew to the right. There are no obvious gaps in the data. No significant outliers are present.

Finding the Smallest Observation, 8th, and 9th Largest Observations

a. The smallest observation can be found in the left-most part of the stem and leaf plot: 1.6. b. To find the 8th and 9th largest observations, we count from the tail (largest value) of the sorted data: 1st largest: 6.2 2nd largest: 6.1 3rd largest: 5.7 4th largest: 5.6 5th largest: 5.6 6th largest: 5.1 7th largest: 5.1 8th largest: 4.9 9th largest: 4.9 The 8th largest observation is 4.9, and the 9th largest observation is also 4.9.

Key concepts

Data Distribution

Understanding data distribution is essential when interpreting any dataset. The shape and spread of data give insights into trends and deviations from expected values. In a stem and leaf plot, such as the one described in the exercise, the distribution can be visualized easily. The plot illustrates the data's frequency distribution, showcasing how data points are clustered around the mean or median.
This particular dataset, when plotted, shows that the majority of data points are clustered around the center, with fewer at the extremes. This indicates a symmetric distribution with a slight right skew, meaning there are slightly more data points on the lower end.
A symmetric distribution tells us that the data is balanced on both sides of the mean, often leading to a normal distribution curve. A right skew typically hints at the presence of some higher values extending the tail compared to the lower end. Being aware of these shapes helps in predicting future data behavior and finding systematic patterns.

Outliers

Outliers are data points significantly different from others in a dataset. They can indicate variability, errors, or unusual conditions needing attention. In statistical analysis, identifying outliers is crucial, as they can skew the results and affect interpretations.
In the stem and leaf plot provided, the data is consistently spread with no gaps indicating sudden deviations. This makes spotting outliers rather straightforward, as an outlying value would be isolated from the rest of the cluster. No significant outliers were noted in this exercise, meaning all values fall within an expected range.
Recognizing the absence or presence of outliers helps in understanding data reliability, highlighting any anomalies possibly requiring further investigation. Particularly in large datasets, outliers should be carefully examined to ensure they aren't results of errors or misrecorded data.

Ordered Data

Ordering data is a fundamental step in many statistical processes. Arranging data from smallest to largest, like the 50 measurements in the exercise, provides clarity and aids in further analysis.
By sorting data, we make it easier to perform tasks such as identifying the smallest and largest values or calculating medians and percentiles. This sequential organization is closely related to generating a stem and leaf plot, as it provides the base data structure for creating the plot.
In the given exercise, after ordering the data, finding specific data points like the smallest observation (1.6) and the 8th and 9th largest observations (both 4.9) becomes straightforward. Ordered data thus enhances efficiency and accuracy in statistical operations.

Statistical Analysis

Statistical analysis involves examining data to derive useful insights and make informed decisions. Through tools like stem and leaf plots, data is simplified into a visual form, allowing for easier interpretation.
In the context of the exercise, statistical analysis helps in summarizing the dataset by highlighting the distribution, identifying specific values, and spotting potential outliers. It provides a framework for understanding how data behaves and relates to real-world phenomena. - By analyzing the plot's shape, we can draw conclusions about the dataset's overall characteristics. - Recognizing patterns such as symmetry or skewness offers clues about underlying data trends.
This kind of analysis is the foundation for further statistical methods such as hypothesis testing, regression analysis, and predictive modeling, enabling deeper insights and evidence-based conclusions.