When both normal-distribution cutoffs are above the mean, both standardised values are positive. Convert any upper-tail probability to a left-tail probability before using inverse normal, then subtract the two standardisation equations to find the standard deviation.
The 2027 A-Level H2 Mathematics syllabus (9758) includes finding relationships between a cutoff, the mean and the standard deviation from a given probability. The original example below shows why the midpoint shortcut fails when the cutoffs are on the same side.
- Rewrite both probabilities as areas to the left.
- Check the signs of the two standard normal values.
- Subtract the equations and substitute back for the mean.
Worked example | A lower-tail and an upper-tail statement
A normal random variable X has P(X ≤ 58) = 0.75 and P(X > 66) = 0.10. Find its mean μ and standard deviation σ. Since X is continuous, the probability at a single cutoff is zero. The second statement gives P(X ≤ 66) = 0.90.
The standard normal quantiles are approximately 0.67449 and 1.28155. So 58 = μ + 0.67449σ and 66 = μ + 1.28155σ. Both cutoffs are above the mean because both left-tail areas exceed 0.5.
Subtract the first equation from the second. Then 8 ≈ 0.60706σ, giving σ ≈ 13.18. Substituting into the first equation gives μ ≈ 49.11. Keep the quantiles at calculator precision until the final step.
Why the midpoint is too large
The midpoint of 58 and 66 is 62. If 62 were the mean, 58 would be below it and its left-tail probability would be less than 0.5. The question gives 0.75. That is enough to reject the midpoint before doing algebra.
The midpoint can give the mean when the cutoffs are equally far to either side, as with the 25th and 75th percentiles of a normal distribution. In this example the cutoffs are the 75th and 90th percentiles.
Try two lower-side cutoffs
Suppose the 10th percentile is 34 and the 25th percentile is 42. Use −1.28155 and −0.67449 respectively. Subtracting again gives σ ≈ 13.18. Substitution gives μ ≈ 50.89, above both cutoffs.
A study partner can sketch where each cutoff sits before using inverse normal. If an upper-tail area of 0.10 leads to a negative standard normal value for its boundary, check whether you used 0.10 as the left-tail area by mistake.
Frequently asked questions
Can I input an upper tail directly?
Use the calculator’s documented tail setting. With a left-tail input, enter one minus the upper-tail probability.
Can a standard deviation be negative?
No. A negative result signals an error such as mismatched cutoffs and quantiles or subtraction in inconsistent orders.
Official information
SEAB’s 2027 A-Level H2 Mathematics syllabus (9758), Section 6.3 Normal Distribution. Checked 9 November 2026.