Expert: Sherry Wallin - 9/2/2010

Question
From a set of 1000 observations known to be normally distributed, the mean is 534 cm and SD is 13.5 cm.  How many observations are likely to exceed 561 cm? How many will be between 520.5 cm and 547.5 cm? Between what limits will the middle 50% of the observations lie?

Answer
Naresh~

One standard deviation will encompass the values between 520.5 and 547.5. Two standard deviations will encompass values between 507 and 561 and then three standard deviations will encompass between 493.5 and 574.5. Check out the empirical rule that tells you that 68% of the data falls within one standard deviation and 95% of the data will fall within two standard deviations and 99.7% of the data will fall within three standard deviations.

The number that is likely to exceed 561 is going to be half of the data outside two standard deviations. Since two standard deviations is where 561 is and that is 95% of the data then for greater than 561 it will be half of the difference between 100% and 95% or (100-95)/2 = 2.5%. So what is .025 * 1000? It is 25 observations.

How many will be between 520.5 cm and 547.5 cm? That is simply within one standard deviation so it will be .68 * 1000 or 680 observations.

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Since 50% is smaller than one standard deviation and by the empirical rule, 68% of the data is within one standard deviation, 50% of the data will lie within (50/68) = .735 of one standard deviation. That is the same as saying that you want .735(13.5) ~= 9.92. Now split 9.9 in half to see how many are to the left of the mean and how many are to the right of the mean. That is 534 + 4.46 = 538.46 and 534 - 4.46 = 529.54. So the middle 50% of observations will be in the interval [529.54,538.46].

Math Prof