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What is the formula for sample standard deviation of a small sample size?

The formula for sample standard deviation is given by:
~~$$s = \sqrt{\frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N-1}}$$~~
**Am I right that when the sample size is small ($N<30$), the formula for sample standard deviation becomes:**
~~$$s = t_{N-1, confidence} \sqrt{\frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N-1}} \ \ \ \ \ \ \ \ ?$$~~
Here, $t_{N-1, confidence}$ is the Student's $t$ coefficient obtained from the tables (the tables are given, for instance, [here](https://www.scribbr.com/statistics/students-t-table/)).
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The reason I am asking is the following. I had no doubts whatsoever that I must multiply standard deviation by the Student's $t$ coefficient for a small sample size. And I have been doing it all the time. I used it in a draft for an article. When I was checking the draft, I decided to check this formula. First, I could not remember where I got it from. Second, I searched the books and the Internet only to find out that people use Student's $t$ coefficient to calculate confidence interval, not standard deviation. I tried to derive my formula from the formula for confidence interval and failed. I talked to a fellow student who also thinks standard deviation must be multiplied by Student's $t$ coefficient and also does not remember why.

The formula for sample standard deviation is given by:
$$s = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}}$$
**Am I right that when the sample size is small ($N<30$), the formula for sample standard deviation becomes:**
$$s = t_{N-1, confidence} \sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}} \ \ \ \ \ \ \ \ ?$$
Here, $t_{N-1, confidence}$ is the Student's $t$ coefficient obtained from the tables (the tables are given, for instance, [here](https://www.scribbr.com/statistics/students-t-table/)).
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The reason I am asking is the following. I had no doubts whatsoever that I must multiply standard deviation by the Student's $t$ coefficient for a small sample size. And I have been doing it all the time. I used it in a draft for an article. When I was checking the draft, I decided to check this formula. First, I could not remember where I got it from. Second, I searched the books and the Internet only to find out that people use Student's $t$ coefficient to calculate confidence interval, not standard deviation. I tried to derive my formula from the formula for confidence interval and failed. I talked to a fellow student who also thinks standard deviation must be multiplied by Student's $t$ coefficient and also does not remember why.

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