Answer
See below.
Work Step by Step
The mean of $n$ numbers is the sum of the numbers divided by $n$. The median of $n$ is the middle number of the numbers when they are in order (and the mean of the middle $2$ numbers if $n$ is even). The mode of $n$ numbers is the number or numbers that appear(s) most frequently. Hence here the mean: $\frac{108+ 92+ 102+ 99+ 116+ 92}{6}=101.5$, the median is the mean of the middle items in the sequence $92, 92, 99, 102,108,116$, which is: $(99+102)/2=100.5$, the mode is $92$. The range is the difference between the largest and the smallest data value. The standard deviation of $x_1,x_2,...,x_n$ is (where $\overline{x}$ is the mean of the data values): $\sqrt{\frac{(x_1-\overline{x})^2+(x_2-\overline{x})^2+...+(x_n-\overline{x})^2}{n}}$. Hence here the range is: $116-92=24$ and the standard deviation is: $\sqrt{\frac{(108-101.5)^2+(92-101.5)^2+...+(116-101.5)^2}{6}}\approx8.5586$ When every value of a data set is multiplied by a constant, the new mean, median, mode, range, and standard deviation can be obtained by multiplying each original value by the constant. Here the constant is $4.5$, hence the mean: $101.5\cdot 4.5=456.75$, the median: $100.5\cdot 4.5=452.25$, the mode:$92\cdot 4.5=414$, the range:$24\cdot 4.5=108$, and the standard deviation: $8.5586\cdot 4.5=38.5137$.
