Question 1131821: If one is only given a sample mean and sample standard deviation, is it possible to calculate the percentage of data that falls within one standard deviation of the mean? If it is possible, how or what formula is used?
I've looked all over, and cannot figure out how it can be done.
Any guidance would be very helpful.
Thank you
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the percentage of date that is within 1 standard deviation about the mean is the same, regardless of the data.
it's a fixed amount.
you use the z-score formula to derive this.
then using the z-score formula, the mean is always 0 and the standard deviation is always 1.
you can determine the percentages directly from the z-score formula.
when you want to be 1 standard deviation about the mean, you will need to be between a z-score of -1 and 1.
look up a z-score of -1 and look up a z-score of 1 and you will find that the area to the left of a z-score of -1 is equal to ... and the x-score to the left of a z-score of 1 is equal to ...
subtract the smaller area from the larger area and you get the area that is between those z-scores.
multiply that by 100 and you get the percentage of the area under the normal distribution curve that is 1 standard deviation about the mean.
you can use a normal distribution table or a normal distribution calculator.
i used the TI-84 plus and got the following:
area to the left of a z-score of -1 = .1586552596
area to the left of a z-score of 1 = .8413447404
the area in between is the larger area minus the smaller area = .6826894809.
visually, this looks like this.
it doesn't matter what the underlying data is.
the percentage will be the same.
to find, however, the underlying data, you need more information.
the information you need is:
sample mean
population or sample standard deviation
sample size
if you have the population standard deviation, use that.
if you don't, use the sample standard deviation.
you must have the sample size because the underlying data limits will be different based on that.
the percentage of the data between, however, will always be the same.
you would use the z-score formula to find the raw score associated with that z-score.
the z-score formula is:
z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error
the formula for standard error is s = standard deviation / square root of sample size
for example:
assume the sample mean is 100 and sample standard deviation is 10 and you want to find the raw date of the sample that is 1 standard deviation about the mean.
the standard error for a sample size of 50 is s = 10 / sqrt(50) = 1.414213562.
remember, 1 standard deviation about the mean is between z-score of -1 and 1.
if the sample size is 50, then:
z = (x - m) / s becomes -1 = (x - 100) / 1.414213562 for the low side.
solve for x to get x = -1 * 1.414213562 + 100 = 98.58578644.
z = (x - m) / s becomes 1 = (x - 100) / 1.414213562 for the high side.
solve for x to get x = 1 * 1.414213562 + 100 = 101.4142136.
the area between those 2 raw scores is equal to .6826894809
this can be seen visually below.
if the sample size is 100, your standard error becomes 10 / sqrt(100) = 1.
solving for x from z-score of -1 gets you x = -1 * 1 + 100 = 99
solving for x from a z-score of 1 gets you x = 1 * 1 + 100 = 101
the area between these 2 raw scores is equal to .6826894809.
this can be see visually below.
the area between is the same, regardless of the underlying data.
your question was:
If one is only given a sample mean and sample standard deviation, is it possible to calculate the percentage of data that falls within one standard deviation of the mean? If it is possible, how or what formula is used?
the percentage of data that lies within one standard deviation about the mean is the same, regardless of the underlying data.
the raw scores associated with this percentage, however, will be different, depending on the sample size.
to find the raw scores, you need to know the sample size.
the sample size determines the standard error which determines what those raw scores will be.
hopefully this answers your question in a manner that you can understand.
if you have further questions about this, let me know what they are and i'll try to answer as best i can.
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