What percentage of scores falls within three standard deviations from the mean for the normal variate?
99.7%
Understanding the Normal Distribution and Standard Deviation
The question asks about the percentage of data within a certain range from the mean in a normal distribution. A normal distribution, often called a bell curve, is a common type of distribution where most data points cluster around the average (mean), and the spread decreases symmetrically away from the mean.
Standard deviation ($\sigma$) is a measure that tells us how spread out the numbers are in a data set. A small standard deviation means the data points are clustered close to the mean, while a large standard deviation means they are spread out over a wider range.
The Empirical Rule (68-95-99.7 Rule) for Normal Distributions
For a normal distribution, there is a useful rule called the empirical rule or the 68-95-99.7 rule. This rule states that:
- Approximately 68% of the data falls within one standard deviation ($\pm 1\sigma$) of the mean ($\mu$). This means between $\mu - 1\sigma$ and $\mu + 1\sigma$.
- Approximately 95% of the data falls within two standard deviations ($\pm 2\sigma$) of the mean ($\mu$). This means between $\mu - 2\sigma$ and $\mu + 2\sigma$.
- Approximately 99.7% of the data falls within three standard deviations ($\pm 3\sigma$) of the mean ($\mu$). This means between $\mu - 3\sigma$ and $\mu + 3\sigma$.
This rule is a key concept when working with normal distributions and understanding the spread of data relative to the mean using standard deviations.
Percentage within Three Standard Deviations
According to the empirical rule, the percentage of scores or data points that fall within three standard deviations ($\pm 3\sigma$) from the mean ($\mu$) in a normal distribution is approximately 99.7%.
This means that almost all data points in a normal distribution are expected to be found within this range.
| Range from the Mean | Percentage of Data (Approx.) |
|---|---|
| Within 1 standard deviation ($\mu \pm 1\sigma$) | 68% |
| Within 2 standard deviations ($\mu \pm 2\sigma$) | 95% |
| Within 3 standard deviations ($\mu \pm 3\sigma$) | 99.7% |
Therefore, the percentage of scores falling within three standard deviations from the mean for the normal variate is 99.7%.
Revision Table: Normal Distribution Key Percentages
| Concept | Description |
|---|---|
| Normal Distribution | Bell-shaped, symmetrical curve. Mean, median, and mode are equal. |
| Standard Deviation ($\sigma$) | Measures the spread or dispersion of data points around the mean. |
| Empirical Rule | States percentages of data within 1, 2, and 3 standard deviations of the mean for a normal distribution. |
| $\mu \pm 1\sigma$ | Approximately 68% of data. |
| $\mu \pm 2\sigma$ | Approximately 95% of data. |
| $\mu \pm 3\sigma$ | Approximately 99.7% of data. |
Additional Information: Properties of Normal Distribution
Beyond the empirical rule, the normal distribution has several important properties:
- It is perfectly symmetrical about the mean. This means the left side of the curve is a mirror image of the right side.
- The total area under the normal curve is equal to 1 (or 100%), representing the total probability or proportion of data.
- The curve extends infinitely in both directions but gets very close to the horizontal axis.
- The mean, median, and mode of a normal distribution are all located at the same point, the center of the distribution.
- Many natural phenomena (like height, weight, test scores) tend to follow a normal distribution.
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