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I am trying to understand how, using only the standard deviation and mean, you are able to determine the first and third quartiles on a normal distribution.

I get that the area under the curve equals one; I understand that Q1 accounts for 25% of the area, and Q3 75%; but I do not understand how to leverage this information to calculate (with my TI-89) where Q1 and Q3 are...

I was able to reasonably estimate these values by looking at a z-score table for areas representing 25% and 75%, and I got z-scores of ±0.68. However, I need a greater degree of accuracy than manually estimating from a table, but I do not know how to go about this formulaically or through my TI-89.

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Is $\pm 0.67448975$ any more useful? – Henry Oct 16 '11 at 2:16
@Henry it is. I saw the number .68 and .67448 thrown around a few times, but I was never able to find a formula that explains how to actually get that number. – Moses Oct 17 '11 at 4:26
There isn't a closed-form formula in terms of simple functions, that's why people use tables or functions in statistical programs. Some calculators have the inverse normal CDF coded in, others can compute it if you tell them how, but many can't. If you know what you're doing you can get there (or pretty close) even with a very basic calculator, but it can be pretty tedious. – Glen_b Feb 26 '15 at 7:31
up vote 2 down vote accepted

The quantile or probit function, as you can see from the link (see "Computatuon"), is computed with inverse gaussian error function $erf^{-1}$ which I hope is downloadable for calculators like TI-89. Look here for instance.

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2nd Vars (Distr)>"InvNorm" next you subtract 1-% and enter this into your Inverse Norm along with your Mean and standard deviation.

Ex: Find the third quartile Q3 which is the IQ score separating the top 25% from the others. With a Mean of 100 and a Standard Deviation of 15.

1-.25=.75 in Inv Norm (.75,100,15)=110 My answer is 110

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