Question

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (22.4,30.1,36.3,42.5,50.7)

Short answer

The distribution of the five number summary (22.4,30.1,36.3,42.5,50.7) is most likely right-skewed.

Step-by-step solution

Identify the five-number summary

For this exercise, the five-number summary provided is (22.4,30.1,36.3,42.5,50.7) respectively representing the minimum, Q1, median, Q3, and maximum of the distribution.

Analyzing skewness based on five-number summary

The skewness of a distribution is determined by comparing the distance of Q1, median, and Q3 from the minimum and maximum respectively. If the median is closer to Q1 than Q3, the distribution is right-skewed. If it's closer to Q3, it is left-skewed. If the distances are approximately equal, the distribution is symmetric.

Compare distances and determine skewness

Here, the distance between the minimum and the median (|36.3 - 22.4| = 13.9) is less than the distance between the median and the maximum (|50.7 - 36.3| = 14.4). Moreover, median (36.3) is slightly closer to the first quartile (30.1) than to the third quartile (42.5). Thus, the distribution is most likely skewed to the right.

Key concepts

Skewed Distribution

In data analysis, understanding the shape of a data distribution is crucial. A skewed distribution differs from a symmetric distribution as most data values are concentrated on one side of the distribution graph. Specifically, a distribution is considered skewed if the tail on one side of the data's peak is longer than the tail on the other side.

There are two types of skewness:

• Right-skewed (positive skew): The tail of the distribution is longer on the right side, and the bulk of values, including the median and quartiles, lie to the left of the graph.
• Left-skewed (negative skew): Conversely, the tail is longer on the left side with most data points to the right.

Examining the five-number summary of a dataset, which includes the minimum, first quartile (Q1), median, third quartile (Q3), and the maximum, helps us determine where the data is likely skewed. If the median is closer to Q1, the data is right-skewed, and if it's closer to Q3, it's left-skewed. If the median is relatively centered between Q1 and Q3, the data is likely symmetrical.

By applying these principles to the five-number summary given in the exercise (22.4,30.1,36.3,42.5,50.7), we see that the median (36.3) is slightly closer to Q1 (30.1) than to Q3 (42.5), suggesting a right-skewed distribution.

Data Analysis

Data analysis is a comprehensive process that consists of inspecting, cleaning, transforming, and modeling data to discover useful information, inform conclusions, and support decision-making. It is a fundamental aspect of various fields such as business, science, and social science to make sense of complex data sets.

To perform effective data analysis, certain steps are crucial:

• Collection of relevant data
• Detailed inspection to ensure accuracy
• Data cleaning to remove errors or inconsistencies
• Data transformation to structure it suitably for analysis
• Statistical analysis to quantify patterns or trends
• Interpretation of results to make informed decisions

A key element in the data analysis toolkit is the use of visual and quantitative methods to understand the distribution of data. The five-number summary is one such tool that provides a quick snapshot of the distribution's shape and spread. In our exercise, this summary has been used as an early analytical step to infer the skewness of the data set without delving into complicated calculations or software tools.

Statistical Concepts

Statistical concepts are foundational to various fields of study, providing methods and principles for collecting, analyzing, interpreting, and presenting data. Statistics enable us to not only describe and summarize data but also to make inferences about a larger population based on a sample. Some fundamental statistical concepts include:

• Descriptive statistics: Summarizing data using tools like mean, median, mode, range, and the five-number summary.
• Inferential statistics: Drawing conclusions about a population based on a statistical sample.
• Probability: Quantifying the likelihood of events.
• Hypothesis testing: Making decisions or predictions based on data analysis.

These concepts are used to analyze data distributions effectively. When we consider the exercise provided, it utilizes the statistical concept of a five-number summary to give an insight into the spread and center of the data. Following that, we then analyze the data's skewness as a preliminary step in understanding more complex statistical concepts like mean deviation and standard deviation.