# Does $S$ and $s$ mean different things in statistics regarding standard deviation?

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.]

• This isn't answerable without more context. Can you perhaps find a short quote from this textbook that uses both notations and edit it into your question? Oct 18, 2015 at 4:47
• It really depends on whose notation you're looking at. In regression or univariate contexts I'd usually use $s$ for some kind of standard deviation and $S$ for some kind of sum of squares, but it's not universal. Please show the two uses you're comparing, preferably where the symbol is first defined. Oct 18, 2015 at 4:54
• @Glen_b and Mathew: Edit confirmed. Kindly look at the context. Oct 18, 2015 at 5:06
• The use of capital $S$ in that way is likely to indicate a random variable (and lower case $s$ an observed value) -- a common convention in statistics. Do they have a page near the start or end of the book where they discuss notation? Oct 18, 2015 at 5:20
• What pages are the part you quote from? Oct 18, 2015 at 5:25

Random variables are denoted by capital letters, $X$, $Y$ and so on, to distinguish them from their possible values given in lower case, $x$, $y$.
$S^2$ 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).