The mode of a set of data is the value which occurs most frequently. The value of appears at least four times in the set. So regardless of the value of , none of the other values can appear more than two times. So is the mode.

The mode of a data set is the element that occurs most frequently.

If , the set becomes .

The mode is 35, which occurs three times, more than any other element.

If , the set becomes .

The mode is 25, which occurs three times, more than any other element.

(A) is greater.

At current, 40 appears four times, more than any other value. For a set to be bimodal, it needs to have two modes; that is, another value would have to appear four times as well. Regardless of whether 30, 50, 60, or any other value replaces the box, this is impossible. The set cannot be made bimodal by adding one element.

The mode of a data set is the element that occurs the most frequently in the set.

In the data set ,  if , it appears four times, more than any other element. If , 1 and 2 each appear three times, making the set bimodal, so 2 is not the only mode, but one of two modes. If  is equal to any other number, 2 appears three times, again, more than any other element. Therefore,  can be any number except 1.

In the data set , each of 1, 2, and 3 appears twice. The only value of  that makes 2 appear the most times, and therefore, the only mode , is .

We therefore know that , but we do not know ; we do not know which one is the greater.

The mode of a set is the element that occurs in the set most frequently. A set without any repeated element is considered to have no mode.

For there to be a mode in the set , it must hold that ; the value of  is the mode of this set. Similarly, ..

If , then, in the set , two of the elements in  appear twice, giving the set two modes. If , then the value they are equal to appears three times as opposed to the other values, which appear one time each. That gives the set exactly one mode. 

The mode of a set is the element that occurs in the set most frequently; a set is bimodal if two elements tie for this.

Consider . If , then 1 and  each appear twice, more than any other element; this makes the set bimodal.  If , the set has one mode, 1, since it appears three times, more than any other element. If  has any other value, then the set has one mode, 1.  Therefore, . For similar reasons, . Therefore, .

The "stem" of this data set represents the tens digits of the data values; the "leaves" represent the units digits. 

The range of a data set is the difference of the high and low values. The highest value represented is 87 (7 is the last "leaf" in the bottom, or, 8, row); the low value is 47 (7 is the first "leaf" in the top, or, 4, row). The difference is , which is the range.

Quantity A: The median (middle number) is , and the mode (most common number) is , so the sum of the two numbers is .

Quantity B: The range is the smallest number subtracted from the largest number, which is .

Quantity A is larger.

The median is the average of the two middle values of a set of data with an even number of values. So we have:

The range is the difference between the lowest and the highest values. So we have:

So the range is greater than the median.