Understanding Normal Probability Curve Areas

The question asks us to match specific ranges defined by standard deviations ($\sigma$) on a Normal Probability Curve (NPC) with the percentage of the total area under the curve that falls within those ranges. The Normal Probability Curve, also known as the Gaussian distribution or bell curve, is symmetrical around its mean, which corresponds to the 0 standard deviation mark.

The total area under the NPC is 100%. The percentage of area between the mean (0) and a certain number of standard deviations is a key property. Due to symmetry, the area from 0 to $+z\sigma$ is the same as the area from 0 to $-z\sigma$.

Here are some common areas from the mean (0):

• Area from 0 to $+1\sigma$ or 0 to $-1\sigma$ is approximately $34.13\%$.
• Area from 0 to $+2\sigma$ or 0 to $-2\sigma$ is approximately $47.72\%$.
• Area from 0 to $+3\sigma$ or 0 to $-3\sigma$ is approximately $49.86\%$.

Calculating Areas for the Given Ranges

Let's calculate the percentage of area for each limit given in List I based on these properties:

(a) –2$\sigma$ to –1$\sigma$

This range is entirely to the left of the mean. We can find this area by taking the area from –2$\sigma$ to 0 and subtracting the area from –1$\sigma$ to 0.

• Area from –2$\sigma$ to 0 is the same as area from 0 to $+2\sigma$, which is $47.72\%$.
• Area from –1$\sigma$ to 0 is the same as area from 0 to $+1\sigma$, which is $34.13\%$.

Area (–2$\sigma$ to –1$\sigma$) = Area (–2$\sigma$ to 0) – Area (–1$\sigma$ to 0)

Area (–2$\sigma$ to –1$\sigma$) = $47.72\% - 34.13\% = 13.59\%$.

This matches option (II) in List II.

(b) 0 to +2$\sigma$

This is a standard range from the mean to $+2\sigma$.

Area (0 to +2$\sigma$) = $47.72\%$.

This matches option (V) in List II.

(c) –1$\sigma$ to +2$\sigma$

This range spans across the mean. We can find this area by adding the area from –1$\sigma$ to 0 and the area from 0 to +2$\sigma$.

• Area from –1$\sigma$ to 0 is the same as area from 0 to $+1\sigma$, which is $34.13\%$.
• Area from 0 to +2$\sigma$ is $47.72\%$.

Area (–1$\sigma$ to +2$\sigma$) = Area (–1$\sigma$ to 0) + Area (0 to +2$\sigma$)

Area (–1$\sigma$ to +2$\sigma$) = $34.13\% + 47.72\% = 81.85\%$.

This matches option (IV) in List II.

(d) –2$\sigma$ to +2$\sigma$

This range is symmetrical around the mean. We can find this area by taking twice the area from 0 to +2$\sigma$ (or 0 to –2$\sigma$).

Area (0 to +2$\sigma$) = $47.72\%$.

Area (–2$\sigma$ to +2$\sigma$) = $2 \times \text{Area (0 to +2}\sigma\text{)}$

Area (–2$\sigma$ to +2$\sigma$) = $2 \times 47.72\% = 95.44\%$.

This matches option (I) in List II.

Summary of Matching

Based on the calculations, the correct matching is:

• (a) –2$\sigma$ to –1$\sigma$ corresponds to (II) $13.59\%$
• (b) 0 to +2$\sigma$ corresponds to (V) $47.72\%$
• (c) –1$\sigma$ to +2$\sigma$ corresponds to (IV) $81.85\%$
• (d) –2$\sigma$ to +2$\sigma$ corresponds to (I) $95.44\%$

The correct matching is (a)-(II), (b)-(V), (c)-(IV), (d)-(I).

Revision Table: Normal Probability Curve Areas

Additional Information on Normal Distribution

The Normal Distribution is fundamental in statistics and probability. It's often used to model real-world data that clusters around a central value with symmetric tails. Key characteristics include:

• Symmetry: The distribution is symmetrical around its mean ($\mu$). The mean, median, and mode are all equal and located at the center.
• Bell Shape: The graph of the distribution is a bell shape, highest at the mean and tapering off towards the tails.
• Parameters: A normal distribution is completely defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The mean determines the center of the distribution, and the standard deviation determines its spread. A larger standard deviation means the data is more spread out.
• The Empirical Rule (68-95-99.7 Rule): This rule provides a quick estimate of the area within 1, 2, and 3 standard deviations of the mean. Approximately $68\%$ of the data falls within $\pm 1\sigma$, $95\%$ within $\pm 2\sigma$, and $99.7\%$ within $\pm 3\sigma$. The values $68.26\%$, $95.44\%$, and $99.72\%$ are more precise percentages often used in exams, as seen in this question.
• Z-scores: Any value ($x$) from a normal distribution can be converted into a Z-score using the formula: $Z = \frac{x - \mu}{\sigma}$. A Z-score represents how many standard deviations a value is away from the mean. For example, a Z-score of +2 means the value is 2 standard deviations above the mean. The question uses standard deviation units directly (–2$\sigma$, +1$\sigma$, etc.), which are equivalent to Z-scores.

Understanding these areas under the NPC is crucial for probability calculations, confidence intervals, and hypothesis testing in statistics.