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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.]

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  • $\begingroup$ 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? $\endgroup$ – Matthew Drury Oct 18 '15 at 4:47
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    $\begingroup$ 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. $\endgroup$ – Glen_b Oct 18 '15 at 4:54
  • $\begingroup$ @Glen_b and Mathew: Edit confirmed. Kindly look at the context. $\endgroup$ – Ébe Isaac Oct 18 '15 at 5:06
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    $\begingroup$ 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? $\endgroup$ – Glen_b Oct 18 '15 at 5:20
  • $\begingroup$ What pages are the part you quote from? $\endgroup$ – Glen_b Oct 18 '15 at 5:25
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On page 82 of your book, second last paragraph they say:

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).

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  • $\begingroup$ Great, I can understand the need for caps in the formula now. Is it fine to include the small letters directly in the formula? $\endgroup$ – Ébe Isaac Oct 18 '15 at 9:26
  • $\begingroup$ If the formula is describing the relationship between random variables, you'd have capitals on both sides. If it's relating observed sample quantities to observed sample values (i.e. if you are writing it in terms of specific values taken by the variables) then you'd have lower case letters ("small letters") on both sides. What you would avoid (if you're using the convention in that text) is mixing capital and lower case letters, because you'd be mixing together variables with the specific values taken by them. $\endgroup$ – Glen_b Oct 18 '15 at 9:43

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