From the story. The Rocket Boys numbered their rockets, and by May 1960 they had reached Auk XXXI. With many flights to compare, telling a real improvement from luck takes statistics. (More in October Sky.)
Fly the same rocket five times and you get five heights, because wind, launch angle and the motor vary a little. The mean is the typical value. The range (highest minus lowest) is a quick measure of spread. The standard deviation is a better one: roughly the typical distance of a flight from the mean.
Key idea
$\bar x = \dfrac{\sum x_i}{n}$, median = middle value in order, range = max − min, $s = \sqrt{\dfrac{\sum (x_i - \bar x)^2}{n - 1}}$. A change in design is believable when the new mean differs from the old by more than the spread.
Worked example: Five flights
Heights: $310, 330, 300, 340, 320$ m. Find the mean and standard deviation.
- Mean: $1{,}600/5 = 320\ \text{m}$.
- Differences: $-10, 10, -20, 20, 0$; squares add to $1{,}000$.
- $s = \sqrt{1{,}000/4} = \sqrt{250} \approx 15.8\ \text{m}$.
Answer: $\bar x = 320\ \text{m}$, $s \approx 15.8\ \text{m}$
Common mistake
Dividing by $n$ instead of $n - 1$ for a sample, or reporting the range as if it were the standard deviation. The range depends only on the two most extreme flights.