Students' Test Scores: 95, 92, 90, 90, 83, 83, 8

Given The Following Students Test Scores 95 92 90 90 83 83 83

Given the following students' test scores (95, 92, 90, 90, 83, 83, 83, 74, 60, and 50), identify the mean, median, mode, range, variance, and standard deviation for the sample. Write a word summary and analysis discussing the results of your calculations.

1. State your results for the sample: the mean, median, mode, range, variance, and standard deviation.

2. Explain which method is best for this data set. Why?

3. Conduct a one-sample t-test and interpret the results (use a population mean of 70).

4. In what situations would this information be useful?

Prepare this assignment according to the guidelines found in the APA Style Guide.

Paper For Above instruction

Analysis of Student Test Scores and Statistical Testing

The analysis of the given set of student test scores (95, 92, 90, 90, 83, 83, 83, 74, 60, and 50) reveals several important statistical measures that describe the distribution and central tendencies of the data. These measures include the mean, median, mode, range, variance, and standard deviation. Additionally, conducting a one-sample t-test against a population mean of 70 provides insight into whether this sample significantly differs from the assumed population standard. This report will discuss each of these results, justify the most appropriate descriptive method for this data set, and explore potential practical applications of such statistical analysis.

Descriptive Statistics

Mean

The mean, or average, is calculated by summing all the data points and dividing by the total number of observations. Adding the scores: 95 + 92 + 90 + 90 + 83 + 83 + 83 + 74 + 60 + 50 = 810. Dividing by 10, the total number of scores, yields a mean of 81.0.

Median

The median is the middle value when data are ordered. Ordering the scores: 50, 60, 74, 83, 83, 83, 90, 90, 92, 95. With an even number of observations, the median is the average of the fifth and sixth scores: (83 + 83) / 2 = 83.

Mode

The mode signifies the most frequently occurring score. Here, 83 appears three times, more than any other score, thus the mode is 83.

Range

The range measures the spread of the data: maximum score minus minimum score. 95 - 50 = 45.

Variance and Standard Deviation

The variance indicates how much the scores vary around the mean. Calculating variance involves taking each score's deviation from the mean, squaring it, summing these squared deviations, and dividing by n-1 for a sample.

Calculations show the variance is approximately 290.27, and the standard deviation, the square root of the variance, is approximately 17.04. These metrics suggest moderate spread around the mean score of 81.

Analysis and Best Method for Data Representation

Among the descriptive measures, the median and mode effectively summarize the typical performance, especially given the presence of extreme scores (50, 60) that skew the mean downward if the dataset were larger. The mean provides an overall average but may be less representative due to outliers. Given the symmetry and the skewness observed, the median is particularly useful in this context, as it is unaffected by the extreme low score (50). The mode highlights the most common achievement level, which is 83, offering insight into the most typical performance.

Therefore, for this specific data set, reporting the median and mode alongside the mean gives a comprehensive understanding of the scores' distribution, capturing central tendency and most common outcomes more reliably than the mean alone.

Conducting a One-Sample t-Test

To determine if the mean score significantly differs from the population mean of 70, a one-sample t-test was performed. Using the sample mean (81), standard deviation (17.04), and sample size (n=10), the t-statistic was computed as:

t = (sample mean - population mean) / (sample standard deviation / √n) = (81 - 70) / (17.04 / √10) ≈ 3.66

With 9 degrees of freedom, referencing t-distribution tables or software indicates a p-value less than 0.01. Thus, the test shows significant difference, suggesting that the sample scores are statistically higher than the population mean of 70 at a 5% significance level.

Practical Applications of this Analysis

The statistical insights derived from these scores are valuable in educational settings, such as assessing the effectiveness of teaching strategies or curriculum adjustments. Educators can use the mean and median to identify typical student performance levels, the mode to recognize the most common score, and the variance or standard deviation to understand score variability. The t-test provides evidence on whether students outperform a benchmark or baseline expectation, aiding decision-making regarding instructional methods or resource allocation. Such analyses are also crucial for identifying at-risk students or evaluating the impact of interventions across different classes or schools.

Conclusion

This comprehensive statistical evaluation of student test scores demonstrates the importance of multiple metrics in accurately portraying data characteristics. The use of descriptive statistics such as the median and mode offers practical advantages in understanding typical performance, especially with skewed data. The significance of the t-test confirms that the students' scores are considerably above the baseline score of 70, providing evidence for educational decision-making. Overall, applying these statistical tools enhances our ability to interpret educational data meaningfully and implement informed strategies that support student success.

References

• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
• Laerd Statistics. (2017). One Sample t-test Assumptions. Retrieved from https://statistics.laerd.com/statistical-guides/one-sample-t-test-statistics.php
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.