If 𝑥 and 𝑦 are two variables such that the summation of 𝑥 equals 15, the summation of 𝑦 equals 25, the summation of 𝑥 times 𝑦 equals 75, and 𝑛 equals five, find the linear correlation coefficient, 𝑟, between 𝑥 and 𝑦.
What do we know about the linear correlation coefficient? And this coefficient, 𝑟, can be found by multiplying 𝑛 times the summation of 𝑥 times 𝑦 minus the summation of 𝑥 times the summation of 𝑦. All over the square root of 𝑛 times the summation of 𝑥 squared minus the summation of 𝑥 squared times 𝑛 times the summation of 𝑦 squared minus the summation of 𝑦 squared.
And while it’s a long formula, in this case, there’s something we can notice. Our 𝑛 equals five. And the summation of 𝑥 times 𝑦 equals 75. We need to subtract that from 15 times 25. This will all be over the square root of some value in the denominator.
However, let’s examine the numerator a little bit more closely. I can rewrite 75 as three times 25. Five times 75 equals five times three times 25. If we regroup one more time, five times three equals 15. We rewrite five times 75 as 15 times 25. And then we see that our numerator is 15 times 25 minus 15 times 25. Our numerator equals zero. And that means the linear correlation coefficient, 𝑟, equals zero over some value. And we don’t need to know what’s in the denominator. 𝑟 equals zero.