Ask Uncle Colin: A Stretch, Indeed

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

Dear Uncle Colin,

My research has determined that female adults have a mean overhead reach of 208.5cm, with a standard deviation of 8.6cm, and follows a normal distribution.

I wanted to know the probability that the mean overhead reach of 50 female adults would lie between 180cm and 200cm and got a crazy small answer of about $2 \times 10^{-13}$. What am I doing wrong?

-- Got Adults Under Stretching, Somehow


That looks about right, actually! You would expect the mean of a sample of 50 stretchers to follow a normal distribution with a mean of 208.5cm and a standard deviation of $\frac{8.6}{\sqrt{50}}\text{cm} \approx 1.22$cm.

The z-score for a mean of 180cm is -23.4, and the z-score for 200cm is about -6.99. You would turn to your normal distribution tables and say "oh pooh. Those numbers aren't on my tables." Instead, you turn to Wolfram|Alpha or (better) Abramovitz and Stegun, and find that the associated probabilities are $9.8 \times 10^{-122}$ and $1.4 \times 10^{-12}$. The difference between those is, well, about $1.4 \times 10^{-12}$ -- the first number is minuscule in comparison.

Does this number look about right? Well, to have an observed mean of 50 observations a full standard deviation below the expected mean is (very rough ballpark) about the same as the probability of all 50 being below the mean. That would be a probability of $2^{-50} \approx 10^{-15}$, which is a similar magnitude.

Hope that helps!

-- Uncle Colin


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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