For a statistical sample with a 95% confidence level, explain what occurs that other 5% of the time. Is that sample more likely to be close (but not within) the margin or error than to be far away?

Question

For a statistical sample with a 95% confidence level, explain what occurs that other 5% of the time. Is that sample more likely to be close (but not within) the margin or error than to be far away?

In: Mathematics

Answer 1

In theory, in the general case, it can be either way — **it depends entirely on the probabilty distribution**. However, for most common distributions (like Gaussian), it’s acutally the case that, as you say, “the 5% is more likely to be close (but not within) the margin or error than to be far away”.

See it like this: to say “95% confidence level” is like saying: “95% of the sand inside this box is on the left half of it”.

“What about the other 5% of the sand?” you ask “I know it is on the right half of the box, but is it more likely to be near the center, or farther on the right?”.

The reality is, you have been told literally nothing about that 5% of the sand (except that it’s not on the left); it *could* be piled all on the far right side, and it could be shaped as a duck sand statuette. There’s nothing against this. But, *if* you assume that the sand in the box is just in a big conic pile (which is a common way for sand to be distributed), *then*, by knowing that 95% of it is on the left half, you also know that most of the 5% on the right is closer to the center than to the right-end side of the box.

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Edit: **for example, compare**:

Case 1: 95% of the people in my class is male (with 95% confidence, a random person from my class will be male). Are the other 5%, the not male ones, likely to be more masculine and not completely feminine? Ansewer: no, because males/females tend to be **distributed** in two separate clusters, with statistically very little in between.

Case 2: 95% of swedish men are taller then 180cm. Does it mean that the other 5%, shorter than 180cm, are likely not to be much shorter than that? Answer: yes, because people’s height tends to be **distributed** as a gaussian dustribution (a bit like the pile of sand in the box).

Conclusion: **it depends on the distribution.**