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 4, then there will be four 4's, and two or fewer of any other value, making 4 the only mode.

If that unknown element is 7, then there will be three 4's, three 7's, and one of each of the other elements, making 4 and 7 the modes.

If the unknown element is any other integer, then there will be three 4's and no more than two of any other element, making 4 the only mode.

The correct choice is therefore any integer other than 7.

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.

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.

(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.

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.