For the company's roofing shingles data, the median (18) and interquartile range (10) are recommended measures of center and variability, respectively, to accurately depict longevity and assess how their shingles compare to other brands.
To assess the longevity of the company's roofing shingles compared to other brands, we can analyze the given data using measures of center and variability. The provided data set is as follows:
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26
Measures of Center:
The median, which is the middle value in a sorted data set, is a robust measure of center that is less affected by extreme values. In this case, the median would provide a representative estimate of the shingles' longevity.
![\[ \text{Median} = 18 \]](https://img.qammunity.org/2024/formulas/mathematics/college/jb06tyx6zwzszpon9ssi29k6zp8mwd8gqd.png)
Measures of Variability:
To understand the spread of the data, the company should use the interquartile range (IQR). The IQR is the range of the middle 50% of the data, providing insights into the variability without being heavily influenced by outliers.
![\[ \text{IQR} = Q3 - Q1 \]](https://img.qammunity.org/2024/formulas/mathematics/college/rkrujo8gefxkxtoyp0gfzfoul0zcruizpt.png)
![\[ \text{IQR} = 24 - 14 = 10 \]](https://img.qammunity.org/2024/formulas/mathematics/college/lnjaug67vpbul425oynaggafd99i1xhwzb.png)
The IQR gives an indication of the spread of longevity values, focusing on the middle range.
The company should use the median as a measure of center and the interquartile range as a measure of variability. These measures are appropriate for this dataset, allowing the company to showcase the central tendency and spread of their shingles' longevity, providing a more accurate representation compared to using mean and standard deviation, which are sensitive to outliers.