When sampling is done without replacement then standard error of mean is:

Question

Prepp Team · Accepted Answer

Understanding Standard Error with and Without Replacement

The standard error of the mean ($SE_{\bar{X}}$) is a measure of the variability of sample means around the true population mean. It essentially tells us how much the sample mean is likely to vary from the population mean if we were to take many samples of the same size from the same population.

The formula for the standard error depends on whether sampling is done with replacement or without replacement, and whether the population is considered infinite or finite.

Standard Error Formula: Sampling With Replacement

When sampling is done with replacement, or from an infinite population, the selection of one item does not affect the probability of selecting other items. In this case, the standard error of the mean is given by the formula:

$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}$

Where:

$\sigma$ is the population standard deviation.
$n$ is the sample size.

Standard Error Formula: Sampling Without Replacement (Finite Population)

When sampling is done without replacement from a finite population (a population with a known, limited size $N$), the probability of selecting an item changes with each selection. This is because once an item is selected, it is not returned to the population pool.

In this scenario, the formula for the standard error of the mean includes a correction factor called the Finite Population Correction (FPC) factor. The formula is:

$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}} \times \sqrt{\dfrac{N-n}{N-1}}$

Where:

$\sigma$ is the population standard deviation.
$n$ is the sample size.
$N$ is the population size.
$\sqrt{\dfrac{N-n}{N-1}}$ is the Finite Population Correction (FPC) factor.

The FPC factor accounts for the fact that as you sample a larger fraction of the finite population, the variability of the sample mean decreases. If the sample size $n$ is very small compared to the population size $N$ (typically when $n/N < 0.05$), the FPC factor is close to 1, and the formula approximates the standard error for sampling with replacement or from an infinite population.

Analyzing the Options for Standard Error Without Replacement

Let's look at the given options in the context of sampling without replacement from a finite population:

$SE_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}$
This is the formula for sampling with replacement or from an infinite population. It does not include the FPC factor needed for sampling without replacement from a finite population.
$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-n}{N-1}}$
This formula includes the term $\dfrac{\sigma}{\sqrt{n}}$ multiplied by the Finite Population Correction (FPC) factor $\sqrt{\dfrac{N-n}{N-1}}$. This is the correct formula for the standard error of the mean when sampling is done without replacement from a finite population.
$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-1}{N-n}}$
This formula includes the reciprocal of the correct FPC factor inside the square root. This is not the standard formula for the standard error without replacement.
$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}\sqrt{1-\dfrac{n}{N}}$
This formula is similar to the correct one, but the term inside the square root is $\left(1-\dfrac{n}{N}\right)$. While related to the finite population correction, the standard FPC factor is $\sqrt{\dfrac{N-n}{N-1}}$. For large N, $\dfrac{N-n}{N-1}$ is approximately equal to $\dfrac{N-n}{N} = 1-\dfrac{n}{N}$, so this is an approximation often used when N is large. However, the exact formula uses $\dfrac{N-n}{N-1}$.

Based on the standard statistical definition, the formula for the standard error of the mean when sampling is done without replacement from a finite population of size $N$ is $SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-n}{N-1}}$.

Conclusion on Standard Error

When sampling is performed without replacement from a finite population, the standard error of the mean is calculated using the formula that incorporates the Finite Population Correction (FPC) factor.

Sampling Method	Population Type	Standard Error Formula ($SE_{\bar{X}}$)
With Replacement	Finite or Infinite	$\dfrac{\sigma}{\sqrt{n}}$
Without Replacement	Infinite	$\dfrac{\sigma}{\sqrt{n}}$
Without Replacement	Finite (size $N$)	$\dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-n}{N-1}}$

Revision Table: Standard Error Formulas

Let's quickly summarize the key formulas for standard error based on sampling methods.

Formula	Applicable Scenario
$\dfrac{\sigma}{\sqrt{n}}$	Sampling with replacement, or from an infinite population (standard standard error).
$\dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-n}{N-1}}$	Sampling without replacement from a finite population of size $N$. Includes the Finite Population Correction factor.

Additional Information: Finite Population Correction (FPC)

The Finite Population Correction factor, $\sqrt{\dfrac{N-n}{N-1}}$, is always less than or equal to 1 (for $n > 1$). It reduces the standard error compared to the infinite population formula. This makes intuitive sense: if you sample a large portion of a finite population without replacement, your sample mean is likely to be closer to the true population mean because you have information from a substantial part of the population. As the sample size $n$ approaches the population size $N$, the term $\dfrac{N-n}{N-1}$ approaches 0, and thus the standard error approaches 0. This also makes sense, as sampling the entire population ($n=N$) means the sample mean is the population mean, with zero variability.

The FPC factor is typically ignored if the sample size $n$ is small relative to the population size $N$, often if $n/N < 0.05$. In such cases, $\sqrt{\dfrac{N-n}{N-1}} \approx 1$, and the simpler formula $\dfrac{\sigma}{\sqrt{n}}$ is used.

Prepp Team · Answer

$SE_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}$

Prepp Team · Answer

$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}} \sqrt{\dfrac{N-1}{N-n}}$

Prepp Team · Answer

$SE_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}\sqrt{1-\dfrac{n}{N}}$