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I know that t-value can be calculated by

$(sample mean-population mean)/sem$

and that the standard error (SE) means the t value is calculated according to the sample size.

What I do not understand with SE is how much error can we expect when we say sample mean represents the mean of the larger population (Statistics in plain English, page 59) is directly related to t-test.

How can we relate this to the relationship between standard deviation (SD) and Z-score: I always thought the reason why we use SD in Z-score is because it sets general intervals for calculating possibilities in a Normal distribution. Is this a wrong approach as well?

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  • $\begingroup$ Standard error is the standard deviation of sample estimate. t becomes z when the sample size is large, i.e. tends to infinity(theoretically speaking) $\endgroup$
    – DevD
    Jul 5 at 6:54

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I think you have two related concepts here: #1 is how standard error (SE) relates to sampling variation, or how precisely we can estimate a population mean based on sample data; and #2 is about the difference between a standard error and a standard deviation.

Some of this confusion arises because we use similar-looking tools in different settings (explained more below).

1. Standard error and estimating the population mean

The short version here is that standard error [of the mean] is an estimate of the extent to which sampling variability might be impacting our estimate of the population mean, based on our sample data.

We often use the t-distribution to think about this hypothetical distribution of sample means, as in smaller samples this is a better option than using the Normal distribution (which tends to lead to underestimation of statistical inferences i.e. too liberal). As noted in a comment to your question, as sample size gets larger the value of $t$ from the t-distribution (which depends on sample size) converges towards the value of $Z$ from the Normal distribution.

2. Standard error and standard deviation

As above: when we're talking about standard error of the mean, we're talking about the expected/estimated variation in repeated sample means drawn from the same population.

When we're looking at standard deviation, we're considering variation in individual observations around the mean: typically variation of individual's values around the sample mean (e.g. if we're estimating mean height from a sample of students).

This can sometimes be re-scaled into a Z-score, which is the difference between a single observation and the mean divided by the standard deviation (see formula below). The Z-score is then the number of units of standard deviation a particular observation falls from the mean in that sample.

$$\frac{X_i - \bar{X}}{SD}$$


Footnote 1. Z-scores can be calculated for a given observation relative to a reference population (e.g. one real-world example is a child's body mass index [BMI] expressed as a Z-score relative to the mean and standard deviation of a reference population of children of the same age/sex)

Footnote 2. Z-statistics are sometimes calculated in hypothesis testing too, but I'll ignore that here.

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