Ardem Collected Data From A Class Survey. He Then Randomly Selected Samples Of Five Responses To Generate Four Samples.$[ \begin{array}{|c|c|c|c|c|c|} \hline \multicolumn{6}{|c|}{\text{Survey Data}} \ \hline \text{Sample 1} & 4 & 5 & 2 & 4 & 3
Introduction
In the world of statistics, collecting and analyzing data is a crucial step in understanding various phenomena. Ardem, a statistics enthusiast, conducted a class survey to gather information about a particular topic. He then randomly selected samples of five responses to generate four samples. In this article, we will delve into the world of statistics and explore the concept of sampling and data analysis using Ardem's survey data.
Understanding the Survey Data
The survey data collected by Ardem consists of five responses from a class survey. The data is presented in the following table:
| Sample 1 | Sample 2 | Sample 3 | Sample 4 |
|---|---|---|---|
| 4 | 5 | 2 | 4 |
| 5 | 2 | 4 | 3 |
| 2 | 4 | 3 | 5 |
| 4 | 3 | 5 | 2 |
| 3 | 5 | 2 | 4 |
Calculating the Sample Means
To calculate the sample means, we need to add up all the values in each sample and divide by the number of values. Let's calculate the sample means for each of the four samples.
Sample 1
| Value | |
|---|---|
| 4 | |
| 5 | |
| 2 | |
| 4 | |
| 3 |
Sample Mean = (4 + 5 + 2 + 4 + 3) / 5 = 18 / 5 = 3.6
Sample 2
| Value | |
|---|---|
| 5 | |
| 2 | |
| 4 | |
| 3 | |
| 5 |
Sample Mean = (5 + 2 + 4 + 3 + 5) / 5 = 19 / 5 = 3.8
Sample 3
| Value | |
|---|---|
| 2 | |
| 4 | |
| 3 | |
| 5 | |
| 2 |
Sample Mean = (2 + 4 + 3 + 5 + 2) / 5 = 16 / 5 = 3.2
Sample 4
| Value | |
|---|---|
| 4 | |
| 3 | |
| 5 | |
| 2 | |
| 4 |
Sample Mean = (4 + 3 + 5 + 2 + 4) / 5 = 18 / 5 = 3.6
Calculating the Population Mean
To calculate the population mean, we need to add up all the values in the survey data and divide by the total number of values.
| Value | |
|---|---|
| 4 | |
| 5 | |
| 2 | |
| 4 | |
| 3 | |
| 5 | |
| 2 | |
| 4 | |
| 3 | |
| 5 | |
| 2 | |
| 4 |
Population Mean = (4 + 5 + 2 + 4 + 3 + 5 + 2 + 4 + 3 + 5 + 2 + 4) / 12 = 43 / 12 = 3.5833
Calculating the Sample Standard Deviation
To calculate the sample standard deviation, we need to follow these steps:
- Calculate the sample mean.
- Calculate the deviations from the sample mean.
- Square each deviation.
- Calculate the average of the squared deviations.
- Take the square root of the average of the squared deviations.
Let's calculate the sample standard deviation for each of the four samples.
Sample 1
| Value | Deviation | Squared Deviation |
|---|---|---|
| 4 | -0.6 | 0.36 |
| 5 | -0.6 | 0.36 |
| 2 | 0.8 | 0.64 |
| 4 | -0.6 | 0.36 |
| 3 | 0.2 | 0.04 |
Sample Standard Deviation = sqrt((0.36 + 0.36 + 0.64 + 0.36 + 0.04) / 4) = sqrt(1.76 / 4) = sqrt(0.44) = 0.664
Sample 2
| Value | Deviation | Squared Deviation |
|---|---|---|
| 5 | -0.8 | 0.64 |
| 2 | 0.8 | 0.64 |
| 4 | -0.8 | 0.64 |
| 3 | 0.8 | 0.64 |
| 5 | -0.8 | 0.64 |
Sample Standard Deviation = sqrt((0.64 + 0.64 + 0.64 + 0.64 + 0.64) / 5) = sqrt(3.2 / 5) = sqrt(0.64) = 0.8
Sample 3
| Value | Deviation | Squared Deviation |
|---|---|---|
| 2 | 0.8 | 0.64 |
| 4 | -0.8 | 0.64 |
| 3 | 0.2 | 0.04 |
| 5 | -0.8 | 0.64 |
| 2 | 0.8 | 0.64 |
Sample Standard Deviation = sqrt((0.64 + 0.64 + 0.04 + 0.64 + 0.64) / 5) = sqrt(2.4 / 5) = sqrt(0.48) = 0.692
Sample 4
| Value | Deviation | Squared Deviation |
|---|---|---|
| 4 | -0.5833 | 0.3393 |
| 3 | 0.4167 | 0.1733 |
| 5 | -0.5833 | 0.3393 |
| 2 | 0.4167 | 0.1733 |
| 4 | -0.5833 | 0.3393 |
Sample Standard Deviation = sqrt((0.3393 + 0.1733 + 0.3393 + 0.1733 + 0.3393) / 5) = sqrt(1.3633 / 5) = sqrt(0.27266) = 0.522
Conclusion
Introduction
In our previous article, we explored the concept of sampling and data analysis using Ardem's survey data. We calculated the sample means, population mean, and sample standard deviations for each of the four samples. In this article, we will answer some frequently asked questions about the survey data and provide additional insights.
Q: What is the purpose of sampling in statistics?
A: Sampling is a method used in statistics to select a subset of data from a larger population. The purpose of sampling is to make inferences about the population based on the sample data. In Ardem's survey, the purpose of sampling was to select a subset of five responses from the class survey to generate four samples.
Q: How do you calculate the sample mean?
A: To calculate the sample mean, you need to add up all the values in the sample and divide by the number of values. For example, in Sample 1, the sample mean is calculated as follows:
Sample Mean = (4 + 5 + 2 + 4 + 3) / 5 = 18 / 5 = 3.6
Q: What is the difference between the sample mean and the population mean?
A: The sample mean is an estimate of the population mean, which is the true mean of the population. In Ardem's survey, the sample mean is close to the population mean, which suggests that the samples are representative of the population.
Q: How do you calculate the sample standard deviation?
A: To calculate the sample standard deviation, you need to follow these steps:
- Calculate the sample mean.
- Calculate the deviations from the sample mean.
- Square each deviation.
- Calculate the average of the squared deviations.
- Take the square root of the average of the squared deviations.
For example, in Sample 1, the sample standard deviation is calculated as follows:
Sample Standard Deviation = sqrt((0.36 + 0.36 + 0.64 + 0.36 + 0.04) / 4) = sqrt(1.76 / 4) = sqrt(0.44) = 0.664
Q: What is the purpose of calculating the sample standard deviation?
A: The sample standard deviation is a measure of the spread or dispersion of the data. It is used to calculate the standard error of the mean, which is a measure of the variability of the sample mean.
Q: How do you interpret the results of the survey data analysis?
A: The results of the survey data analysis provide insights into the characteristics of the population. In Ardem's survey, the results suggest that the sample means are close to the population mean, and the sample standard deviations are similar for each sample. This suggests that the samples are representative of the population and that the data is consistent.
Q: What are some limitations of the survey data analysis?
A: Some limitations of the survey data analysis include:
- The sample size is small, which may not be representative of the population.
- The survey data may not be representative of the population due to biases or errors.
- The analysis assumes that the data is normally distributed, which may not be the case.
Conclusion
In this article, we answered some frequently asked questions about the survey data and provided additional insights. We hope that this article has provided a better understanding of the concept of sampling and data analysis using Ardem's survey data.