Lindsey found the median and mean of the list of numbers below.
4, 6, 8, 8, 15
Which statement is true if Lindsey includes the number 23 in the list?
A.
The median would decrease.
B.
The median would increase.
C.
The mean would increase.
D.
The mean would decrease.
B. The median would increase.
To determine the effect of including the number 23 in the list on the median and mean, let's calculate them first without including the number 23, and then include it.
Without including 23, the list is: 4, 6, 8, 8, 15.
The median is the middle value when the numbers are in order.
Here they are in order: 4, 6, 8, 8, 15.
The median is the middle value, which is 8.
The mean is the average of all the numbers.
Add up all the numbers: 4 + 6 + 8 + 8 + 15 = 41.
Divide the sum by the total number of values, which is 5.
The mean is 41/5 = 8.2.
Now, let's include the number 23 in the list: 4, 6, 8, 8, 15, 23.
The median is the middle value when the numbers are in order.
Here they are in order: 4, 6, 8, 8, 15, 23.
The median is the average of the two middle numbers, which are 8 and 15.
The median is (8 + 15)/2 = 23/2 = 11.5.
The mean is the average of all the numbers.
Add up all the numbers: 4 + 6 + 8 + 8 + 15 + 23 = 64.
Divide the sum by the total number of values, which is now 6.
The mean is 64/6 = 10.67.
Now let's compare the results with and without including the number 23:
A.
The median would decrease. (This is incorrect, because the median would actually increase from 8 to 11.5)
B.
The median would increase. (This is correct, because the median would increase from 8 to 11.5)
C.
The mean would increase. (This is correct, because the mean would increase from 8.2 to 10.67)
D.
The mean would decrease. (This is incorrect, because the mean would actually increase from 8.2 to 10.67)
Therefore, the correct answer is B. The median would increase.
To find the median of a list of numbers, you need to arrange them in order from least to greatest and then find the middle value. In this case, the list is already sorted: 4, 6, 8, 8, 15. The middle value is 8, so the median is 8.
To find the mean, you need to add up all the numbers in the list and divide the sum by the total number of values. In this case, the sum of the numbers is 4 + 6 + 8 + 8 + 15 = 41, and there are 5 numbers in the list. So the mean is 41/5 = 8.2.
Now, let's consider what happens if Lindsey includes the number 23 in the list. The new list would be: 4, 6, 8, 8, 15, 23.
To find the new median, we need to arrange the numbers in order from least to greatest: 4, 6, 8, 8, 15, 23. The middle value in this case is still 8 because it is the third value. Therefore, the median remains the same.
To find the new mean, we need to add up all the numbers in the new list: 4 + 6 + 8 + 8 + 15 + 23 = 64. Since there are now 6 numbers in the list, the new mean is 64/6 = 10.67.
Comparing the new median and mean to the original ones, we can conclude the following:
A. The median would not decrease (it would stay the same).
B. The median would not increase (it would stay the same).
C. The mean would increase.
D. The mean would not decrease (it would increase).
Therefore, the correct statement is:
C. The mean would increase.