A testing service has 1000 raw scores. It wants to transform the distribution so that the mean =10 and the standard deviation = 1. To do so ?
What was the mean and standard deviation of the original distribution of raw scores?
To transform the distribution of the raw scores so that the mean becomes 10 and the standard deviation becomes 1, you need to follow these steps:
1. Calculate the mean and standard deviation of the given set of raw scores.
2. Apply a linear transformation to each raw score using the following formula:
**Transformed Score = (Raw Score - Mean) / Standard Deviation**
This formula is known as the z-score formula, which transforms the values in a distribution to have a mean of 0 and a standard deviation of 1 (standardizing the distribution).
In this case, since you want the mean to be 10 and the standard deviation to be 1, you need to multiply the transformed score by the desired standard deviation (1) and then add the desired mean (10):
**Transformed Score = (Raw Score - Mean) / Standard Deviation * Desired Standard Deviation + Desired Mean**
Therefore, the formula becomes:
**Transformed Score = (Raw Score - Mean) / Standard Deviation + Desired Mean**
3. Apply this transformation to each raw score to obtain the transformed scores.
For example, let's suppose one of the raw scores is 15 and you want to find its transformed score. Assuming the given mean is 10 and standard deviation is 1, you can calculate it as follows:
Transformed Score = (15 - 10) / 1 + 10 = 5 + 10 = 15
In this case, since the raw score is already above the desired mean, the transformed score remains the same.
Repeat this process for all the raw scores to obtain the transformed distribution with a mean of 10 and a standard deviation of 1.