Imagine you are in charge of reviewing applications for your academic institution. As a preliminary measure, you want to review only the top 25% of applicants on the basis of standardized test scores. How does calculating the z-score provide additional information regarding how each subject did overall? Provide specific examples of how you would use the information and interpret the scores.
If you want top 25%, you need everyone who had a z-score greater than 0.6745 –
Next, you can order the z-scores from highest to lowest to get the best of the best Take a set of four numbers 95,98,90,95, representing test score for different parts of an entrance exam. Then take the average and standard deviation of this set and calculate the z-score. Do this for every individual. Then compare the z-scores obtained for each to rank the students from best to worst, sorting the z-score from highest to lowest.
This is just an added to question to your answer.
Let's say that the standardized test had 100 possible points. If we wanted to identify the top 25%, why not just select those with raw scores above 75?
Why do we have to standardize the scores?
Calculating the z-score can provide additional information regarding how each applicant performed overall compared to the entire applicant pool. The z-score measures the number of standard deviations an individual's test score is from the mean score.
To use the z-score for reviewing applications, you would follow these steps:
1. Calculate the mean and standard deviation of the standardized test scores for all applicants.
2. For each applicant, calculate their z-score using the formula: z = (X - mean) / standard deviation, where X is the applicant's test score.
3. Rank the applicants based on their z-scores.
4. Set a threshold z-score to determine which applicants are in the top 25%.
For example, let's say the mean test score is 80 and the standard deviation is 10. If an applicant has a test score of 90, the z-score would be (90 - 80) / 10 = 1. Similarly, if another applicant has a score of 70, the z-score would be (70 - 80) / 10 = -1.
Interpreting the z-scores will help differentiate applicants and identify those in the top 25%. A positive z-score indicates an applicant performed above the mean, while a negative z-score indicates the applicant performed below the mean. The higher the z-score, the further above the mean the applicant's score is, indicating a stronger performance.
Using this information, you could review applications by focusing on those with the highest z-scores to identify the top 25% of applicants. This approach allows for a more objective evaluation, considering both the applicant's score and their relative performance compared to the entire applicant pool.
Calculating the z-score provides additional information regarding how each subject did overall by standardizing their test scores. The z-score helps to measure how far an individual's test score deviates from the mean of the entire population of test scores, and gives you a sense of their relative performance compared to others.
To calculate the z-score for a particular test score, you need to know the mean and standard deviation of the entire population of test scores. The formula for calculating the z-score is:
z = (x - μ) / σ
where:
- z is the z-score
- x is the individual test score
- μ is the mean of the population of test scores
- σ is the standard deviation of the population of test scores
Here's an example to illustrate how you can use this information:
Let's say you have a population of test scores with a mean (μ) of 70 and a standard deviation (σ) of 10. Now, suppose you receive an application with a test score of 80. To calculate the z-score for this applicant, you would use the formula:
z = (80 - 70) / 10 = 1
A z-score of 1 indicates that this applicant's test score is 1 standard deviation above the mean. This suggests that the applicant performed well compared to the rest of the population. You can then use this information to make decisions regarding their application.
To identify the top 25% of applicants based on standardized test scores, you can sort all the applicants' z-scores in descending order. The top 25% would be those applicants with the highest z-scores, indicating their performance well above the average.
By utilizing z-scores, you gain a more accurate understanding of how each applicant performed relative to the entire population of test scores, rather than just relying on their raw test scores. This allows for a fairer comparison between applicants, taking into account the variability in the distribution of test scores.