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

1. State the formula for the range of a data set in terms of its maximum and minimum values.

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

2. Find and correct the error: A student says median of $1, 4, 7, 9$ is $7$.

   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.

3. Find the mean of $5$ numbers if their sum is $42$.

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

4. What is the median of the numbers $8, 2, 5, 9, 1$?

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

5. State the formula for the mean (arithmetic average) of $n$ numbers $x_1, x_2, \dots, x_n$.

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

6. If each value in a data set is multiplied by $-2$, how does the range change?

   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 $2$.

7. State the rule for finding the median when there are an odd number of data 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.

8. What is the mean of the numbers $6, 6, 6, 10$?

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

9. What is the median of the numbers $2, 2, 9, 9, 9$?

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

10. Identify the first step you should do before finding a median or a range from a list of numbers.

    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.

11. What is the range of the numbers $12, 3, 19, 8$?

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

12. A data set has mean $8$ with $6$ values. What is the sum of the $6$ values?

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

13. What is the mean of the numbers $4, 7, 9$?

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

14. What number must be added to $3, 7, 10$ to make the mean equal to $8$?

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

15. What is the mean of the numbers $-3, 1, 2$?

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

16. Five numbers have mean $12$. If four numbers sum to $40$, what is the fifth number?

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

17. State the rule for finding the median when there are an even number of data values.

    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.

18. What is the median of the numbers $3, 10, 4, 8$?

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

19. What is the mean of the numbers $\frac{1}{2}, \frac{1}{2}, \frac{3}{2}$?

    $\frac{5}{6}$. Sum $\frac{1}{2}+\frac{1}{2}+\frac{3}{2}=\frac{5}{2}$ and divide by $3$ for the mean.

20. What is the median of the numbers $\frac{1}{4}, \frac{3}{4}, \frac{1}{2}$?

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

21. If each value in a data set increases by $5$, how does the mean change?

    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.

22. What is the range of the numbers $\frac{2}{3}, \frac{1}{6}, \frac{5}{6}$?

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

23. What is the range of the numbers $-2, 5, 0, -7$?

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

24. What is the median of the numbers $-4, -1, -7, 0, 3$?

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

25. What is the median of the numbers $1, 2, 2, 2, 100$?

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

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