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. (15,25,30,35,45)

Short answer

The distribution of the given five number summary is symmetric.

Step-by-step solution

Identify the Five Number Summary

The five number summary given is (15,25,30,35,45), where 15 is the minimum, 25 is the first quartile (Q1), 30 is the median (Q2), 35 is the third quartile (Q3), and 45 is the maximum.

Analyze Position of Q1, Q2, and Q3

The distance from Q1 to Q2 is 5, and from Q2 to Q3 is also 5. This indicates that the median is equally distant from the first quartile as it is from the third quartile.

Final Conclusion

Since the median is at the same distance from both quartiles, the distribution of numbers is most likely symmetric.

Key concepts

Data Distribution

Understanding data distribution is fundamental in statistics as it tells us how values in a dataset are spread out. A five number summary is an excellent tool to quickly gain insight into a dataset's distribution. It includes five critical values: the minimum, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum.

A visual representation of these values can be found in a box plot, where the spread of the data can be examined at a glance. The proximity or distance between these numbers helps us classify the distribution of data. For instance, if Q1 and Q3 are equidistant from Q2, the distribution could be said to be fairly uniform around the median. Moreover, if the minimum and maximum values are equally distant from Q1 and Q3 respectively, it suggests a uniform range of data. These interpretations provide an easy-to-understand overview of a dataset's distribution without overwhelming the student with numerical complexity.

Skewness

Skewness refers to the asymmetry in the distribution of values in a dataset. To identify skewness, one must look at the position of the median between Q1 and Q3. If the median is closer to Q1 than to Q3, the distribution is skewed to the right, indicating a longer tail on the right side of the distribution's 'peak'. Conversely, if the median is closer to Q3, we have a skew to the left, showing that there's a longer tail to the left.

Understanding skewness is crucial because it impacts the mean of the distribution. In a right-skewed distribution, the mean is greater than the median, while in a left-skewed distribution, the mean is less than the median. Recognizing the direction and extent of skewness helps in choosing the right statistical methods for analysis and provides insights into the nature of the data, enabling students to make more informed decisions about the data they are working with.

Symmetric Distribution

A symmetric distribution occurs when the left and right sides of the graph are mirror images of each other along the central vertical axis. In a perfectly symmetrical dataset, the median cuts the data into two equal halves, and the values of Q1 and Q3 are equidistant from the median. This also means the mean and the median of the dataset are equal, or very close to each other.

In relation to a five number summary, symmetry can often be deduced when the two interquartile ranges (the distances from Q1 to Q2 and Q2 to Q3) are the same. Such an equality implies that the data are evenly distributed around the median. For students, recognizing a symmetric distribution is helpful as many statistical tests assume normality, which is a form of symmetric distribution. It simplifies the selection of the statistical method for further analysis and enhances understanding of data behavior.