In a standard normal distribution, 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. $99.7\%$
• B. $50\%$
• C. $68\%$
• D. $95\%$

Answer: (C)

Verified step by step guidance

1. Recall that a standard normal distribution is a normal distribution with a mean \(\mu = 0\) and a standard deviation \(\sigma = 1\).
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 distribution function (CDF) values for \(z = 1\) and \(z = -1\) and then subtracting: \(P(-1 < Z < 1) = \Phi(1) - \Phi(-1)\), where \(\Phi(z)\) is the CDF of the standard normal distribution.
5. Since \(\Phi(-1) = 1 - \Phi(1)\) due to symmetry, this calculation confirms that about 68% of the data falls within one standard deviation of the mean.