Answer :
To understand the ranges for the 68-95-99.7 distribution, also known as the empirical rule, let’s break it down step-by-step:
1. Understanding the Empirical Rule:
- The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
2. Mean and Standard Deviation:
- For a standard normal distribution, the mean (μ) is 0 and the standard deviation (σ) is 1.
3. Ranges According to the Empirical Rule:
- 68% Interval:
- The range is from (mean - σ) to (mean + σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 1 = -1
- Upper bound: 0 + 1 = 1
- So, the 68% interval is: (-1, 1).
- 95% Interval:
- The range is from (mean - 2σ) to (mean + 2σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 2 = -2
- Upper bound: 0 + 2 = 2
- So, the 95% interval is: (-2, 2).
- 99.7% Interval:
- The range is from (mean - 3σ) to (mean + 3σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 3 = -3
- Upper bound: 0 + 3 = 3
- So, the 99.7% interval is: (-3, 3).
So, the ranges for the 68-95-99.7 distribution are:
- 68% interval: (-1, 1)
- 95% interval: (-2, 2)
- 99.7% interval: (-3, 3)
1. Understanding the Empirical Rule:
- The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
2. Mean and Standard Deviation:
- For a standard normal distribution, the mean (μ) is 0 and the standard deviation (σ) is 1.
3. Ranges According to the Empirical Rule:
- 68% Interval:
- The range is from (mean - σ) to (mean + σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 1 = -1
- Upper bound: 0 + 1 = 1
- So, the 68% interval is: (-1, 1).
- 95% Interval:
- The range is from (mean - 2σ) to (mean + 2σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 2 = -2
- Upper bound: 0 + 2 = 2
- So, the 95% interval is: (-2, 2).
- 99.7% Interval:
- The range is from (mean - 3σ) to (mean + 3σ).
- With the mean being 0 and the standard deviation being 1:
- Lower bound: 0 - 3 = -3
- Upper bound: 0 + 3 = 3
- So, the 99.7% interval is: (-3, 3).
So, the ranges for the 68-95-99.7 distribution are:
- 68% interval: (-1, 1)
- 95% interval: (-2, 2)
- 99.7% interval: (-3, 3)
