The median of a set of nine data values - an odd number - is the value that appears in the middle when the values are ranked. For 8 to be the median, 8 must be in the middle - that is, four values must appear before 8, and four values must appear after 8.

In the given data set, four values - 4, 5, 5, 6 - are known to be appear before 8, so  must appear after 8. Since  is an integer from 1 to 10,  can only be 8, 9, or 10. This makes three the correct response.

The median of a set of eleven data values - an odd number - is the value that appears in the middle when the values are ranked. For 5 to be the median, 5 must be in the middle - that is, five values must appear before 5, and five values must appear after 8.

We can answer this question by looking at three cases.

Case 1: 

Without loss of generality, assume ; this reasoning holds for any lesser value of . The data set becomes

,

and the median is 4.

Case 2: 

The data set becomes

The middle value - the median - is 5.

Case 3: 

Without loss of generality, assume ; this reasoning holds for any greater value of . The data set becomes

Again, the median is 5. 

Therefore, we can set  equal to 5, 6, 7, 8, 9, or 10 - any of six different values - and make the median of the set 5.

The median of a set of ten data values - and even number - is the arithmetic mean of the two values that appear in the middle when the values are ranked. For  to be the median, it would have to hold that either the two middle values are both . The data set, in ascending order, would be as follows:

.

 must be between 6 and 8 inclusive. Since it is given that  is an integer, there are three possibilities: 6, 7, and 8. 

Three is the correct response.

The mean of a data set is equal to the sum of the entries divided by the number of entries. There are ten entries in the set, so 

The median of a data set with an even number of values is the arithmetic mean of the values of the two entries in the middle when the values are arranged in ascending order:

This value is .

The midrange of a data set is the arithmetic mean of the least and greatest elements in the set. This element is 

The correct response is that 

.

First, we test to see if she is already assured of a 70 average. The worst-case scenario for Janice is that she will score 0 points on the twelfth test,. If this happens, her grade will be the mean of the ten best tests so far. She will drop the 0 and the fifth score, 54, so her average will be the sum of the other ten tests divided by 10:

Janice is already assured of a mean of at least 70, since any score can only raise her grade. 

First, we test to see if she is already assured of a 80 average. The worst-case scenario for Donna is that she will score 0 points on the twelfth test,. If this happens, her grade will be the mean of the ten best tests so far. She will drop the 0 and the fifth score, 54, so her average will be the sum of the other ten tests divided by 10:

Donna has not yet attained her 80. Now, we test to determine what her twelfth test score must be. If she scores 60 or less she will drop this score as well as the 54, so we will assume that she scores more than 60. Calling this score , the mean of her scores will be the the sum of her best nine scores thus far and this unknown score , divided by 10. The mean should be greater than or equal to 80, so we can set up and solve for  in this inequality:

Note that the third (60) and fifth (54) scores have been omitted. Add the known scores to get

Multiply both sides by 10:

Subtract 656 from both sides:

Donna would have to score 144 or more on the twelfth test - an impossible feat. She cannot attain an average of 80 or greater.

The arithmetic mean of a data set is the sum of the items in the set divided by the number of items. There are ten items, so the mean is 

Simplify the numerator by combining the like terms:

Now, split the fraction, and reduce to lowest terms:

,

the correct response.

The mean of a data set is equal to the sum of the entries divided by the number of entries. There are nine entries in the set, so 

The median of a data set with an odd number of values is the value of the entry in the middle when the values are arranged in ascending order:

This value is .

The midrange of a data set is the arithmetic mean of the least and greatest elements in the set. This element is 

The correct response is that 

.

The mode of a data set is the value that appears in the set the most frequently (or a mode is one of several such values). A set is considered to have no mode if and only if no value is repeated. Of the four data sets given,  is the only one that fits this definition.

The mean is found by adding the values in a set and dividing that number by the total number of values.