the box - and - whisker plot below represents some data set. what percentage of the data values are less…

Explanation:

Step1: Recall box - and - whisker plot properties

In a box - and - whisker plot, the box represents the inter - quartile range (IQR) with the left side of the box being the first quartile ($Q_1$), the line inside the box being the median ($Q_2$), and the right side of the box being the third quartile ($Q_3$). The whiskers extend to the minimum and maximum values.

Step2: Identify the position of 82

82 is between the third quartile ($Q_3$) and the maximum value. The third quartile represents the 75th percentile. Values from the minimum to $Q_3$ make up 75% of the data, and values between $Q_3$ and the maximum make up the remaining 25% of the data. Since 82 is greater than $Q_3$ (which is 81 in this case), we know that 75% of the data values are less than or equal to 81. To find the percentage of data values less than or equal to 82, we note that since the data is uniformly distributed between $Q_3$ and the maximum (a simplifying assumption for a continuous data set), we consider the proportion. The range from 81 to 87 is 6 units, and the value 82 is 1 unit above 81. The proportion of the data in the upper 25% that is less than or equal to 82 is $\frac{82 - 81}{87 - 81}=\frac{1}{6}$. The percentage of data less than or equal to 82 is $75%+\frac{1}{6}\times25%$. $75%+\frac{25}{6}%=\frac{450 + 25}{6}%=\frac{475}{6}%\approx79.17%$. But if we assume the data is discrete and we just consider the quartile - based percentages, since 82 is in the upper half of the upper quartile range, we can say that approximately 87.5% of the data values are less than or equal to 82. This is because we can think of splitting the upper 25% into two equal parts (since we are past $Q_3$). So we have 75% (up to $Q_3$) plus 12.5% (half of the remaining 25%).

Answer:

87.5%