Mean, Median, and Range - ISEE Upper Level: Quantitative Reasoning

$0$. Sum $-3+1+2=0$ and divide by $3$ to find the mean, which handles negative values.

$16$. Subtract the minimum $3$ from the maximum $19$ to find the range.

$9$. The sorted list is $2,2,9,9,9$; the middle value in this odd set is the median.

$\text{mean}=\frac{x_1+x_2+\cdots+x_n}{n}$. The mean is the arithmetic average, obtained by dividing the sum of all values by the total number of values.

The median is the middle value after sorting. For an odd number of sorted data points, the median is the central value that divides the set into two equal parts.

Median is the mean of the two middle sorted values. For an even number of sorted data points, the median is the average of the two central values to represent the middle.

$20$. Total sum is $12 imes 5 = 60$; subtract $40$ from $60$ to find the fifth number.

$48$. Multiply the mean $8$ by the number of values $6$ to find the total sum.

$-1$. Sort as $-7,-4,-1,0,3$; the middle value in the odd set is the median.

$5$. Sort as $1,2,5,8,9$; the middle value in this odd set is the median.

$6$. Sort as $3,4,8,10$; average the two middle values $(4+8)/2$ for the even set.

$\frac{42}{5}$. Divide the total sum $42$ by the number of values $5$ to compute the mean.

$\frac{2}{3}$. Subtract minimum $rac{1}{6}$ from maximum $rac{5}{6}$ to get $rac{4}{6}=rac{2}{3}$.

$\frac{1}{2}$. Sort as $rac{1}{4},rac{1}{2},rac{3}{4}$; the middle value is the median for the odd set.

$12$. Desired sum for mean $8$ with $4$ numbers is $32$; subtract current sum $20$ from $32$.

$12$. Subtract the minimum $-7$ from the maximum $5$ to calculate the range.

$2$. Sorted as $1,2,2,2,100$; the middle value in the odd set is the median, unaffected by the outlier.

Correct median: $\frac{7+4}{2}=\frac{11}{2}$. Sorted list $1,4,7,9$ is even; median is average of middle two, not the higher one.

The range is multiplied by $2$. Multiplying by a constant scales the differences between values by the absolute value of that constant, so range scales by $2$.

The mean increases by $5$. Adding a constant to each value increases the sum by that constant times the number of values, thus raising the mean by the constant.

$\text{range}=\text{max}-\text{min}$. The range measures the spread of data by subtracting the smallest value from the largest value.

Sort the data from least to greatest. Sorting arranges the data in order, enabling identification of the middle value for median and the extremes for range.

$\frac{20}{3}$. Sum the values $4+7+9=20$ and divide by $3$ to find the arithmetic mean.

$7$. Sum the values $6+6+6+10=28$ and divide by $4$ to compute the mean.