Flashcard quiz 14, Question 1
Is the algebraic solution below the only or the best way by far?
2n - 5 = n+5, then solve for n
2n=n+10
n=10
Pictures attached.
I can think of 1 more way,
The standard deviation of a list containing 2 distinct numbers must be half of its range.
To prove this,
For a list containing just two distinct numbers a and b:
I. The mean is x = (a + b)/2.
ii. The squared deviations from the mean are both ( (a−x)² = (b−x)² = ( (a−b)/2 )² ).
iii. Averaging these and taking the square root gives
\sigma = \sqrt{ \frac{ \left( \frac{a - b}{2} \right)^2 + \left( \frac{b - a}{2} \right)^2 }{2} } = \frac{ |a - b| }{ 2 }
Using this, we can do,
\frac{2n-n}{2} = 5
\frac{n}{2} = 5
n=10
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Oh wow. Yea there was probably no way I’d have thought of that haha! Thank you Joan!
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Would I be using an inconsistent method if I stuck with the route I found?
Uh, where’d you get your “method” from?
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I thought it up in review. Could be oversimplified. I thought the following:
For a list containing just two distinct numbers a and b, with mean m:
I. The distance from |a - m| = |b - m| = oversimplified σ
II. where a>b, a-σ = m = b+σ
so
a-σ = m = b+σ
2n - 5 = m = n + 5
2n - 5 = n + 5
maybe this always works when the list only contains two numbers.
Edit:
Oh this is just a rearrangement of the solution Greg used in the video. I just watched it again.
Yeah, that’s good
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Sweet! Thanks!