In a normal distribution with the mean being 500 and the standard deviation being 100, what raw score separates the top 40% of a distribution from the rest?
To find the raw score that separates the top 40% of a distribution from the rest in a normal distribution with a given mean and standard deviation, you can use the concept of z-scores.
A z-score is a measure of how many standard deviations a particular data point is from the mean of a distribution. By converting raw scores to z-scores, you can compare values from different normal distributions.
In this case, we want to determine the z-score that corresponds to the top 40% of the distribution. Since we are looking at the top portion, we are essentially finding the z-score that corresponds to the cumulative area under the normal curve of 1 - 0.40 = 0.60.
To find this z-score, we can use a standard normal distribution table (also known as the z-table) or a statistical calculator. The z-table provides the cumulative probability up to a specific z-score.
1. Using the z-table:
- Look up the cumulative probability (0.60) in the body of the z-table.
- Find the corresponding z-score in the margins of the table.
- For example, if the z-score is labeled as -0.25 in the table, it means that the top 40% of the data falls above the raw score that corresponds to the z-score of -0.25.
2. Using a statistical calculator:
- Input the cumulative probability (0.60) and the mean (500) and standard deviation (100) values into the calculator.
- The calculator will provide the z-score that corresponds to the top 40% of the distribution.
- For example, if the calculator returns a z-score of -0.25, it means that the top 40% of the data falls above the raw score that corresponds to the z-score of -0.25.
Once you have obtained the z-score, you can convert it back to a raw score using the formula:
raw score = mean + (z-score * standard deviation)
In this case:
raw score = 500 + (-0.25 * 100) = 475
Therefore, the raw score that separates the top 40% of the distribution from the rest is 475.