Normal Curve Areas Explained

The question asks to match specific ranges around the mean ($\mu$) of a normal distribution, defined by standard deviations ($\sigma$), to their corresponding percentage areas under the curve.

Matching Normal Distribution Areas

The standard percentages of data falling within certain standard deviations from the mean in a normal distribution are well-established:

• 1 Standard Deviation: Approximately 68.27% of the data falls within the range $\mu \pm 1\sigma$. From List II, 68.26% (IV) is the closest standard value.
• 2 Standard Deviations: Approximately 95.45% of the data falls within the range $\mu \pm 2\sigma$. From List II, 95.44% (III) is the closest standard value.
• 3 Standard Deviations: Approximately 99.73% of the data falls within the range $\mu \pm 3\sigma$. List II provides 99.73% (II) directly.
• 4 Standard Deviations: The range $\mu \pm 4\sigma$ encompasses almost all the data. The percentage is very close to 100%. From List II, 99.97% (V) represents this range.

Summary of Matches

Based on the standard properties of the normal distribution:

• (a) $\mu - \sigma$ to $\mu + \sigma$ matches with 68.26% (IV).
• (b) $\mu - 2\sigma$ to $\mu + 2\sigma$ matches with 95.44% (III).
• (c) $\mu - 3\sigma$ to $\mu + 3\sigma$ matches with 99.73% (II).
• (d) $\mu - 4\sigma$ to $\mu + 4\sigma$ matches with 99.97% (V).

Therefore, the correct matching is (a)-IV, (b)-III, (c)-II, (d)-V.