The mean of a data set is 7.8, the mode is 6.6, and the median is 6.8. What is the least possible number of data values in the set? A. 3 data values B. 4 data values C. 5 data values D. 6 data values Can someone just explain to me how I am supposed to solve this.

To find the least possible number of data values in the set given the mean, mode, and median, we need to consider the definitions of these statistical terms and then construct the smallest data set that fits these conditions.

The mean is the sum of all the data values divided by the number of data values.

The mode is the data value that appears most frequently.

The median is the middle value when the data set is ordered from least to greatest. If the number of data points is even, the median is the average of the two middle numbers.

Knowing that the mode is 6.6, we must have at least two occurrences of 6.6 because for a number to be the mode it has to appear more frequently than any other number. Since the median is 6.8, which is greater than the mode, the median needs at least one value greater than or equal to 6.8 to be in the middle (for an odd number of data points) or two values that average to 6.8 (for an even number of data points).

To create the smallest set, let's try to use the fewest number of data points possible while still achieving the given mean, mode, and median:

1. Since the mode is 6.6, and it must appear more than once, we start with two 6.6s. 2. To have a median of 6.8, 6.8 must be the middle number or the average of the two middle numbers. Since we need to keep the data set as small as possible, let's include one 6.8. 3. At this point, we have three data points: 6.6, 6.6, and 6.8. The average (mean) of only these three numbers would be less than 7.8 because all these values are less than 7.8. Therefore, we must add another number that is higher than 7.8 to bring up the mean. 4. If we add just one number, that number would have to be significantly higher than 7.8 to achieve a mean of 7.8 for all the data points. The presence of such a high number would, however, affect the median, which we know is 6.8. Therefore, we add two numbers that are equal and sufficiently high to get a mean of 7.8, without changing the median position.

At this point, we have the following five data points: 6.6, 6.6, 6.8, X, X. To find the value of X that will give us a mean of 7.8, we use the formula for the mean:

(mean) = (sum of all values) / (number of values)

Based on the known mean (7.8) and the number of values (5), we can calculate the sum of all values:

(sum of all values) = (mean) * (number of values) = (7.8) * (5) = 39

Now we subtract the sum of the known values:

(sum of Xs) = 39 - (6.6 + 6.6 + 6.8) = 39 - 20 = 19

Since there are two X values and their sum is 19:

X = (sum of Xs) / 2 = 19 / 2 = 9.5

So the two missing values are both 9.5, and the smallest data set that satisfies all given conditions is:

6.6, 6.6, 6.8, 9.5, 9.5

Thus, the least possible number of data values in the set is 5, which corresponds to option C.