Question

The Interquartile Range (IQR) for a set of normally distributed data with mean µ = 80 is 12. What is the approximate value of σ ?

Answer 1

IQR holds 50% of the data. So in your case between 74 and 86.
A difference of 6, 25% off-mean

Between `mu-sigma` and `mu+sigma` there will be 64% of your data.
So `mu` will be (a bit) larger than 6, as it represents 32% off-mean.

Since the bell curve is already reasonably steep at the 25%-off points, I would - without calculating - estimate `mu` to be around 7.

Calculator:
If you own a TI-84 or something like that, you can set up two functions:
Y1= normalcdf(74,86,80,X) -- (left, right, mean, unknown std)
Y2=0.50 -- fraction of data in this part (50%)
And then use Intersect. The X-value will be `sigma`

Tip:
Always make a sketch of the curve and draw in the things that you know. This helps to visualise the problem.