Final answer:
If a measurement of 25 inches is far from a mean of 16 inches depends on the standard deviation. With a standard deviation of 3 inches, 25 inches is 3 standard deviations away, which is considered far. With a standard deviation of 7 inches, it is about 1.29 standard deviations away, which is less extreme.
Step-by-step explanation:
When determining if a measure of 25 inches is "far away" from a mean of 16 inches, it depends on the standard deviation of the underlying data.
(a) If the standard deviation is 3 inches, you calculate how many standard deviations 25 is from 16 by subtracting the mean from the measure and then dividing by the standard deviation: (25 - 16) / 3 = 9 / 3 = 3 standard deviations.
(b) With a standard deviation of 3 inches, 25 inches is indeed far away from a mean of 16 inches, as it is 3 standard deviations from the mean.
(c) If the standard deviation were 7 inches, then (25 - 16) / 7 = 9 / 7 = approximately 1.29 standard deviations. In this case, being approximately 1.29 standard deviations from the mean suggests that 25 inches is not as far from the mean as in the case with a smaller standard deviation.
The number of standard deviations away from the mean is a key factor in determining how unusual or extreme a data point is in relation to the overall data set. This is often referred to as a z-score in statistics.