In a standard normal distribution, approximately what percentage of the data falls within one standard deviation of the mean (that is, between $-1$ and $1$ on the z-score scale)?

• A. About $95\%$
• B. About $99.7\%$
• C. About $50\%$
• D. About $68\%$

Answer: (D)

Verified step by step guidance

1. Recall that a standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
2. Understand that the question asks for the percentage of data within one standard deviation from the mean, which corresponds to the interval between z = -1 and z = 1 on the z-score scale.
3. Use the empirical rule (68-95-99.7 rule) which states that approximately 68% of the data in a normal distribution lies within one standard deviation of the mean.
4. Alternatively, you can find this percentage by calculating the cumulative probability from z = -1 to z = 1 using the standard normal distribution table or a calculator: compute \(P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)\).
5. Recognize that this probability corresponds to about 68%, confirming that approximately 68% of the data falls within one standard deviation of the mean in a standard normal distribution.