In some contexts when denoting sample standard deviation, I notice a capital $S$ and sometimes a small $s$. I also notice this in the same standard textbook. Do they mean different things in context or just the same?
Context:
$F$-distribution calculation concerning two variances:
$F = \frac{S_2^2}{S_1^2}$
These variables were substituted as the following
$s_1^2 = 15,750 \qquad s_2^2 = 10,920$
Both were clearly stated as sample variances. This was also noticed for many other formulas in the book. The capital was used in the formula while the small letters denote the value. Some other sites use only the small $s$ for all cases. Why not use the small $s$ for the formula in the first place?
I also noticed that capital $S$ is general test statistics in hypothesis testing, while the Smith-Satterthwaite test formula is composed only of small $s$'s. What is the significance (if any)?
[Book: Miller & Freund's Probability and Statistics for Engineers – 8th ed.]
Best Answer
On page 82 of your book, second last paragraph they say:
$S^2$ is used for sample variance (as a random variable) in that sense on p189 and p190 (in the second case with subscripts) for example.
Lower case $s$'s would then go with the numbers from a sample (being a specific value taken by the random variable, as they said).