In more detail
For a mean it is mean ± critical value × SD / √n. The term SD / √n is the standard error. With large samples the critical value for 95% is 1.96; with small samples it comes from the t distribution with n − 1 degrees of freedom and is larger. A mean of 103 with SD 15 and n = 100 has a 95% interval of about 100.06 to 105.94. Because the width shrinks with √n, halving it takes four times the data.
Why it matters
A single estimate hides how precise it is; the interval shows it. Strictly, the 95% describes the method: across many samples, 95% of intervals built this way contain the true value, while any one interval either contains it or not. Using 1.96 on a small sample gives an interval that is too narrow and overconfident.
Example
The same mean (103) and SD (15) from only n = 25 people:
standard error = 15 / √25 = 3
t critical (24 df, 95%) = 2.064
interval = 103 ± 2.064 × 3 = 103 ± 6.19 → 96.81 to 109.19
Using 1.96 instead would give 97.12 to 108.88, narrower than the data justify.
Common confusion
A confidence interval is not the range that holds 95% of individual values. That range is about two standard deviations either side of the mean and does not shrink as n grows; the confidence interval uses the standard error and does.