How to find mode - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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

The data set has ten elements - one 2, one 3, three 4's, two 7's, one 9, one 10, and an element of unknown value.

If that unknown element is 7, there are two modes of the set, 4 and 7, which would occur three times each, as opposed to the other elements, which would appear once each.

If that unknown element is any other integer, then the only mode of the data set is 4, which would occur three times, as opposed to the other elements, which would appear once or twice each.

The correct choice is that this impossible.

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

Two values of the data set, 2 and 9, are known to appear twice; the others appear exactly once.

If is 6, 7, or 8, this does not change, and 2 and 9 are both modes.

If is 1, 3, 4, 5, or 10, then three different values appear twice, and 2, 9, and the choice of are the three modes.

If is 2 or 9, then that number appears three times, the other number appears twice, and all others appear once. That makes the choice of the only mode.

This makes the correct choice 2 or 9.

(a) The median of a data set with ten elements is the arithmetic mean of the fifth-highest and sixth-highest elements. These elements are 13 and 15, so the median is

(b) The mode of a data set is the element that occurs most frequently. Since 13 is the only repeated element, it is the mode.

This makes (a) the greater quantity.

5 appears in the data set at least three times. Regardless of the value of , no other element (except 5) can appear more than twice. 5 must be the mode.

If there are an odd number of values in a data set, the median is the middle value. In this problem first we should put the numbers in order:

So the median is .

The mode of a set of data is the value which occurs most frequently, which is in this problem, which has occured three times in the given set of numbers. So the median and the mode are equal.

If a data set has an even number of values, the median is the average of the two middle values. First we should put the numbers in order:

So the median is:

The mode of a set of data is the value which occurs most frequently, which is in this problem. So the mode is greater than the median.

If there are an odd number of values in a data set, the median is the middle value. In this problem first we should put the numbers in order:

So the median is .

The mode of a set of data is the value which occurs most frequently, which is . So the median and the mode are equal.

The mode of a set of data is the value which occurs most frequently, which in this case is .

The mean of a set of data is given by the sum of the data, divided by the total number of values in the set. So we can write:

So the mean of is greater than the mode.

The mode of a set of data is the value which occurs most frequently, which in this case is .

The mean of a set of data is given by the sum of the data, divided by the total number of values in the set. So we can write:

So the mean is greater than the mode.

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 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 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.