In a perfectly normal distribution of scores, which of the following statements is true?

General
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in a perfectly normal distribution of scores, which of the following statements is true?

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In a perfectly normal distribution of scores, which of the following statements is true?

Answer:
In a perfectly normal distribution, which is also known as a Gaussian distribution, the following statements are true:

  1. The mean, median, and mode are all equal.
  2. The distribution is symmetric about the mean.
  3. Approximately 68% of the data falls within one standard deviation of the mean.
  4. Approximately 95% of the data falls within two standard deviations of the mean.
  5. Approximately 99.7% of the data falls within three standard deviations of the mean.

Let’s elaborate on each point to fully understand why these statements are true:

1. Mean, Median, and Mode:
In a normal distribution, the mean (average), median (middle value), and mode (most frequent value) are all located at the center of the distribution. This central point is the peak of the bell curve. Since the distribution is symmetric, these three measures of central tendency all coincide at the same point.

2. Symmetry:
A normal distribution is perfectly symmetric around its mean. This means that if you draw a line vertically through the mean, the left and right halves of the graph are mirror images of each other. This symmetry implies that half of the data values lie to the left of the mean, and half lie to the right.

3. 68-95-99.7 Rule:
This rule (also known as the empirical rule) describes how data is distributed in a normal distribution:

  • 68% within one standard deviation: If you move one standard deviation (σ) away from the mean (μ) in both directions (μ - σ to μ + σ), you will encompass about 68% of the total data.
  • 95% within two standard deviations: If you extend this to two standard deviations (μ - 2σ to μ + 2σ), you capture about 95% of the data.
  • 99.7% within three standard deviations: Extending it to three standard deviations (μ - 3σ to μ + 3σ) covers about 99.7% of the data.

These properties are fundamental characteristics of the normal distribution and are critical in statistical analysis.

Final Answer:
The true statements in a perfectly normal distribution of scores include:

  1. The mean, median, and mode are all equal.
  2. The distribution is symmetric about the mean.
  3. Approximately 68% of the data falls within one standard deviation of the mean.
  4. Approximately 95% of the data falls within two standard deviations of the mean.
  5. Approximately 99.7% of the data falls within three standard deviations of the mean.