A normal distribution has a mean of 20 and a standard deviation of 2. If you convert every score in the distribution to Z scores, which of the following statements would be true of the resulting distribution?
What are you choices?
To determine what would be true of the resulting distribution when converting every score in a normal distribution to Z scores, we need to understand what a Z score represents.
A Z score, also known as a standard score, measures the number of standard deviations a particular data point is away from the mean. It tells us how far from the mean a data point is in relation to the standard deviation.
In this case, the given normal distribution has a mean of 20 and a standard deviation of 2. To convert a score X to a Z score, we use the formula:
Z = (X - μ) / σ
Where:
Z is the Z score,
X is the data point (score),
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Let's consider the statements and analyze if they would be true for the resulting distribution:
1. The mean of the resulting distribution would be 0.
When we convert every score in a normal distribution to Z scores, the mean of the resulting distribution will always be 0. This is because the Z score formula subtracts the mean from each data point.
2. The standard deviation of the resulting distribution would be 1.
The standard deviation of the resulting distribution will also always be 1. This is because the Z score formula divides each data point by the standard deviation, effectively scaling the distribution to a standard deviation of 1.
3. The resulting distribution would have the same shape as the original distribution.
Converting scores to Z scores does not change the shape of the distribution. It only transforms the data points and scales them accordingly. Therefore, the resulting distribution will have the same shape as the original distribution, which, in this case, is a normal distribution.
Therefore, the statements that would be true of the resulting distribution when converting every score in a normal distribution with a mean of 20 and a standard deviation of 2 to Z scores are:
1. The mean of the resulting distribution would be 0.
2. The standard deviation of the resulting distribution would be 1.