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Statistics question: Rescaling data?
Hi I have a question about my statistics homework. Here's the problem:
"A convenient rule of thumb is SAT = 40 X ACT +150; that is, multiply an ACT score by 40 and add 150 points to estimate the equivalent SAT score. An admissions officer reported the following statistics about the ACT scores of 2355 who applied to her college one year. Find the summaries of equivalent SAT scores.
Lowest score: 19
Mean: 27
Standard Deviation: 3
Q3: 30
Median: 28
IQR: 6"
What I'm having trouble with is the Standard Deviation. If I plug in the SD = 3 into the SAT = 40 X ACT +150, I get SAT = 270. But the answer key said that the answer is supposed to be 120. I later figured out it could be that you're supposed to only do multiplication or division with measures of spread? That would make sense because SAT = 40 X 3 = 120, but I'm not too sure if what I'm doing is right or not.
Can I have confirmation for this? My teacher didn't teach us this because we didn't have time. She gave us a powerpoint, but it was very vague, so I'm kind of confused.
Thanks!
2 Answers
- FlashRubinoLv 68 years agoFavourite answer
Hi Melon,
That's a very good question. In case you haven't gotten the answer yet, here's my take on it.
For background, it's useful to know that if you have a normally distributed random variable X with mean μ and standard deviation σ, then a random variable Y = X + k will have a mean of μ + k and an unchanged standard deviation,σ. Makes sense, doesn't it? How on earth could adding a CONSTANT change the VARIATION in a data set? If you have another random variable U, equal to c*X, then your mean wil be c*μ. AND your sd will be c*σ, because you've spread out the spread, so to speak.
So when you have SAT = 40ACT + 150, it's just as you've done. First, you converted X into U, so your sd and mean both increased 40-fold. Then you added a constant, so your mean went up, to 40μ + 150, but the standard deviation remained 40σ, because adding a constant added 0 variation.
Just a note, sometimes you'll need to find the variance for k*X, and it will be the variance of X times k^2.
Your teacher should not rush you off into a homework problem, although in your case it worked out well. You can ALWAYS assign it for the next lesson and give a review reading assignment with 5 minute quiz in its stead if you run out of time.
From the questions here, though, a lot of the time it's like sending troops out without their weapons. messy and bloody.
- 4 years ago
the two are splendid, 2 million new jobs have been created, a million.9 million new workers have entered the artwork rigidity So in different words we've extra extra or less a one hundred,000 job earnings over the process 15 months(no longer very sturdy)
