A list of numbers has an average of 48 and an SD of 12. The list is transformed by subtracting 3 from each entry and then dividing the result by 5.
(1) Find the average of the transformed list.
(2) Find the SD of the transformed list.
1)
2) 2.4
1). 9
To find the average of the transformed list, we can follow a few steps:
Step 1: Subtract 3 from each entry in the original list.
Step 2: Divide each result by 5.
Step 3: Find the average of the transformed list.
Now let's solve it step by step:
(1) Finding the average of the transformed list:
Step 1: Subtract 3 from each entry in the original list.
The original list has an average of 48. When we subtract 3 from each entry, the average will decrease by 3 as well. So the new average after subtracting 3 from each entry is 48 - 3 = 45.
Step 2: Divide each result by 5.
Now, divide each entry by 5. Since dividing each entry by the same number will also divide the average by the same number, the new average after dividing each entry by 5 is 45 / 5 = 9.
Therefore, the average of the transformed list is 9.
(2) Finding the standard deviation (SD) of the transformed list:
To calculate the SD of the transformed list, we need to follow a few steps:
Step 1: Subtract the new average (9) from each entry in the transformed list.
Step 2: Square each result obtained in Step 1.
Step 3: Find the average of the squared values obtained in Step 2.
Step 4: Take the square root of the result obtained in Step 3.
Now let's solve it step by step:
Step 1: Subtract the new average (9) from each entry in the transformed list.
Since we already transformed the list by subtracting 3 and dividing by 5, we can directly calculate the SD without going through the original data set.
Step 2: Square each result obtained in Step 1.
Step 3: Find the average of the squared values obtained in Step 2.
In this case, since all the entries in the transformed list are the same (9), squaring each value will yield the same value. Thus, the average of the squared values will also be 9.
Step 4: Take the square root of the result obtained in Step 3.
The square root of 9 is 3.
Therefore, the SD of the transformed list is 3.
To summarize:
(1) The average of the transformed list is 9.
(2) The SD of the transformed list is 3.