Certainly!
Let's start by finding the arithmetic mean (also known as the average) for the given data using both the direct method and the shortcut method.
Direct Method:
Arithmetic Mean = (Sum of all values) / (Number of values)
Sum of values = 23.4 + 15.6 + 22.1 + 20.0 + 26.7 + 31.4 + 18.9 + 22.5 = 180.6
Number of values = 8
Arithmetic Mean (Direct Method) = Sum of values / Number of values = 180.6 / 8 = 22.575
### Shortcut Method:
There's a shortcut formula to find the mean without having to manually add all the values:
Shortcut Mean = (Sum of first and last term) / 2
Shortcut Mean = (15.6 + 31.4) / 2 = 47 / 2 = 23.5
So, the arithmetic mean by the direct method is 22.575 (rounded to three decimal places), and by the shortcut method, it's 23.5.