Distribution of Mathematics Scores: A Normal Distribution Analysis

Understanding the distribution of scores is crucial in assessing the performance of a student population. In this article, we will analyze the distribution of mathematics scores for 300 students who have their scores normally distributed with a mean of 68 and a standard deviation of 3. We will use the z-score formula and the properties of the normal distribution to calculate the number of students achieving scores above 72 and those scoring 64 or below.

The Normal Distribution and Z-Score Calculation

The normal distribution, often referred to as the Gaussian distribution, is a continuous probability distribution that is widely used in statistical analyses due to its well-defined mathematical properties. The formula for calculating a z-score is given by:

Z-Score Formula: z (X - μ) / σ

Where: X is the raw score, μ is the mean, σ is the standard deviation.

A. Scores Greater Than 72

Firstly, let’s calculate the z-score for a score of 72:

Calculation: z (72 - 68) / 3 4 / 3 ≈ 1.33

The cumulative probability for a z-score of 1.33 in the standard normal distribution can be found using a z-table or a statistical calculator. The cumulative probability for Z 1.33 is approximately 0.9082. This indicates that about 90.82% of the students scored less than 72. Therefore, the percentage of students scoring more than 72 can be calculated as:

Probability of Students Scoring Greater Than 72: P(X > 72) 1 - P(X ≤ 72) 1 - 0.9082 ≈ 0.0918

The number of students scoring greater than 72 can be approximated as:

Number of Students 0.0918 * 300 ≈ 27.54

Rounding to the nearest whole number, we get approximately 28 students.

B. Scores Less Than or Equal to 64

Next, we need to calculate the z-score for a score of 64:

Calculation: z (64 - 68) / 3 -4 / 3 ≈ -1.33

The cumulative probability for a z-score of -1.33 in the standard normal distribution table is approximately 0.0918. This means about 9.18% of the students scored less than or equal to 64. The number of students can be calculated as:

Probability of Students Scoring Less Than or Equal to 64: P(X ≤ 64) 0.0918 * 300 ≈ 27.54

Rounding to the nearest whole number, we again get approximately 28 students.

Summary

Based on the above calculations, we can summarize the following findings:

Number of Students Scoring Greater Than 72: 28 students Number of Students Scoring Less Than or Equal to 64: 28 students

Accuracy and Rounding Considerations

It is important to note that while the z-score and probability calculations provide accurate estimates, they do involve rounding. In practice, it is not possible to have a fraction of a student. Hence, the final rounded values should be used for these types of statistical analyses.

Technical Aspects and Limitations

The marks represent a continuous distribution, and in reality, scores are discrete. Therefore, the normal distribution is an approximation that provides a useful and practical method for statistical analysis but may not perfectly reflect real-world data. Nonetheless, the normal distribution is a powerful tool for such analyses and is widely used in educational and scientific contexts.

Conclusion and Further Reading

This analysis demonstrates the application of the normal distribution and z-score calculations in determining the performance distribution of a student cohort. Understanding these concepts is essential for educational researchers, statisticians, and educators aiming to analyze and interpret data effectively.

For further reading, you may explore more statistical concepts and applications in:

Normal Distribution Online Statistics Courses Khan Academy - Statistics Probability