What is the best way to factor out an unknown value from a percentage?

Maybe I didn't word the question the best... Here's my deal. I have 10 customer surveys returned to me with an average of 90.00% (satisfaction.) I wish to find out what my average would be if I threw out one survey. Here's the problem: I cannot see the individual scores on the surveys. Please help! Thanks!

What is the best way to factor out an unknown value from a percentage?
The info you've presented is insufficient to determine this. Along with the mean (average) of 90%, you need one other piece of data: the standard deviation, with the assumption that the data are normally distributed.





The average doesn't tell you about the spread of the data. That is, you may have 10 surveys all with 90% satisfaction, or 9 with 100% and one with 0%.





If you knew you had a standard deviation of +/-5% for example, you'd know that if you picked a survey at random, there'd be a ~68% chance that it would be between 85 and 95%.





You've stated that you now know that you have a standard deviation of 1.2. This means that if you picked one survey at random, it would have a 68% chance that it was between 88.8 and 91.2. Throwing that one away would change the average from 90% to the range (89.87 - 90.12). Stated again: if you threw away one survey at random, there's a 68% chance that your new average would be between 89.87% and 90.12%.





You're not going to get a single number out of this since the answer is the probability that it is a certain number. That is, you can say with 68% confidence that the new average will be within the range above. You can similarly determine what the average would be with 95 or 99% confidence by using the 2nd or 3rd standard deviatoins, respectively.





With your newest edit, you've completely change the problem.


You now have restrictions on all of the scores and know one of them. I'm taking all of your info literally, and you've asked what's the probability that the one score you pick is %26lt;4 OR %26gt;4. This is different from ,%26lt;=4 and %26gt;=4.





You already know that one of the scores is less than 4, specifically 3, 2, or 1. You've said it's a 5-point scale, so I am assuming that there is no 0. Regardless of that this smallest score is, the probability of choosing it is 1/10. This is the probability that the score is %26lt;4.





It's tougher to find out the probability that the score is %26gt;4, namely whether it is 5. You have to figure out how many 5s you would need if the smallest score is 3, 2, or 1. For example, let's do the instance where the smallest score is 3.





The total of the scores has to be 42 since the mean is 4.2 and there are 10 scores. If the smallest is 3, then the sum of the rest has to be 39. You would need x scores that are 4 points each and y scores that are 5 points each, meaning that 4*x+5*y = 39. The additional constraint is that there are 9 total scores, so x + y = 9. Solving for x and y, you get x = 6 and y = 3. So in this case, three scores are 5 and thus %26gt;4, so there's a 3/10 chance of picking a score %26gt;4.





If you figure it out for the rest, you'll get 4/10 and 5/10 chances. So the average is 4/10.





For your original question, %26lt;4 OR %26gt;4, the answer is 1/10 + 4/10 = 5/10. There's a 50% chance that in choosing one of the scores, you'll choose one that is %26lt;4 or %26gt;4 but not = 4.





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I didn't realize that the standard dev. constraint is still there. In that case, the only data set that gives you all of the conditions is where the smallest score is 1, and there are 4 4s and 5 5s. In that case, it's even easier, since 6 of the 10 are %26lt;4 or %26gt;4, so the answer in that case is 60%.

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