Calculating the Standard Error of the Mean

Oct 13, 2025 by leolavanderia.com.br 56 views

Introduction

In statistics, the standard error of the mean (SEM) is a measure of the variability or dispersion of a sample mean. It represents how much the sample mean is likely to deviate from the true population mean. In this article, we will calculate the standard error of the mean for a set of three adult males' weights.

Calculating the Standard Error of the Mean

The standard error of the mean is given by the formula:

S_{E} = \frac{S_x}{\sqrt{n}}

where $S_x$ is the sample standard deviation and $n$ is the sample size.

Step 1: Calculate the Sample Standard Deviation

To calculate the sample standard deviation, we first need to calculate the sample variance. The sample variance is given by:

s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}

where $x_i$ is the individual data point, $\bar{x}$ is the sample mean, and $n$ is the sample size.

For the three adult males' weights, we have:

Weight (pounds)	160	215	195
Mean	190

First, we need to calculate the sample mean:

\bar{x} = \frac{160 + 215 + 195}{3} = 190

Now, we can calculate the sample variance:

s^2 = \frac{(160-190)^2 + (215-190)^2 + (195-190)^2}{3-1}

s^2 = \frac{(-30)^2 + 25^2 + 5^2}{2}

s^2 = \frac{900 + 625 + 25}{2}

s^2 = \frac{1550}{2}

s^2 = 775

Now, we can calculate the sample standard deviation:

s = \sqrt{s^2} = \sqrt{775}

s = 27.84

Step 2: Calculate the Standard Error of the Mean

Now that we have the sample standard deviation, we can calculate the standard error of the mean:

S_{E} = \frac{S_x}{\sqrt{n}}

S_{E} = \frac{27.84}{\sqrt{3}}

S_{E} = \frac{27.84}{1.732}

S_{E} = 16.07

Conclusion

In this article, we calculated the standard error of the mean for a set of three adult males' weights. We first calculated the sample standard deviation using the sample variance formula, and then used the sample standard deviation to calculate the standard error of the mean. The standard error of the mean is a measure of the variability or dispersion of a sample mean, and it represents how much the sample mean is likely to deviate from the true population mean.

Answer

The correct answer is:

D) 16.07

Q&A: Calculating the Standard Error of the Mean

Q: What is the standard error of the mean?

A: The standard error of the mean (SEM) is a measure of the variability or dispersion of a sample mean. It represents how much the sample mean is likely to deviate from the true population mean.

Q: What is the formula for calculating the standard error of the mean?

A: The formula for calculating the standard error of the mean is:

S_{E} = \frac{S_x}{\sqrt{n}}

where $S_x$ is the sample standard deviation and $n$ is the sample size.

Q: How do I calculate the sample standard deviation?

A: To calculate the sample standard deviation, you first need to calculate the sample variance. The sample variance is given by:

s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}

where $x_i$ is the individual data point, $\bar{x}$ is the sample mean, and $n$ is the sample size.

Q: What is the sample mean?

A: The sample mean is the average of the individual data points. It is calculated by summing up all the data points and dividing by the number of data points.

Q: How do I calculate the sample variance?

A: To calculate the sample variance, you need to subtract the sample mean from each individual data point, square the result, and then sum up the squared results. Finally, you divide the sum by the number of data points minus one.

Q: What is the sample standard deviation?

A: The sample standard deviation is the square root of the sample variance. It is a measure of the spread or dispersion of the individual data points.

Q: How do I calculate the standard error of the mean using the sample standard deviation?

A: To calculate the standard error of the mean using the sample standard deviation, you simply divide the sample standard deviation by the square root of the sample size.

Q: What is the correct answer for the standard error of the mean for the three adult males' weights?

A: The correct answer is:

D) 16.07

This is the standard error of the mean for the three adult males' weights.

Common Mistakes to Avoid

Not calculating the sample mean correctly: Make sure to sum up all the data points and divide by the number of data points to get the sample mean.
Not calculating the sample variance correctly: Make sure to subtract the sample mean from each individual data point, square the result, and then sum up the squared results.
Not calculating the sample standard deviation correctly: Make sure to take the square root of the sample variance to get the sample standard deviation.
Not calculating the standard error of the mean correctly: Make sure to divide the sample standard deviation by the square root of the sample size to get the standard error of the mean.

Conclusion

In this article, we provided a step-by-step guide on how to calculate the standard error of the mean for a set of three adult males' weights. We also answered some common questions related to calculating the standard error of the mean. By following the steps outlined in this article, you should be able to calculate the standard error of the mean for any given set of data.