"Using the empirical rule to identify values and percentages of a normal distribution"

Understanding Normal Distribution

• Normal Distribution: Often referred to as a bell curve, is a probability distribution that is symmetric about the mean.

  • The mean is located at the highest point of the curve (the peak).

  • The left and right sides of the curve are mirror images of each other.

• Mean: The central value of a normal distribution.

• Standard Deviation (stdv): A measure of the amount of variation or dispersion in a set of values.

Empirical Rule

• The empirical rule is a foundational concept in statistics concerning the distribution of data within a normal distribution. It states:

  1. Approximately 68% of the data falls within 1 standard deviation of the mean.

  2. Approximately 95% of the data falls within 2 standard deviations of the mean.

  3. Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Illustration of Empirical Rule

• 1 Std Dev from Mean: Represents about 68% of total area.

• 2 Std Devs from Mean: Represents about 95% of total area.

• 3 Std Devs from Mean: Represents about 99.7% of total area.

Application Example

• Sample Problem Overview: The lengths of movie files that are streamed follow a normal distribution.

  • Mean: 170.8 min

  • Standard Deviation: 19.7 min

Calculating Boundaries using the Empirical Rule

• Calculations for Values:

  • Above Mean:

  • Mean + 1 Std Dev: $170.8 + 1 imes 19.7 = 190.5$ min

  • Mean + 2 Std Devs: $170.8 + 2 imes 19.7 = 210.2$ min

  • Mean + 3 Std Devs: $170.8 + 3 imes 19.7 = 229.9$ min

  • Below Mean:

  • Mean - 1 Std Dev: $170.8 - 1 imes 19.7 = 151.1$ min

  • Mean - 2 Std Devs: $170.8 - 2 imes 19.7 = 131.4$ min

  • Mean - 3 Std Devs: $170.8 - 3 imes 19.7 = 111.7$ min

Choosing Shaded Area Percentage

• Final Thought: To find the percentage of the total area shaded under the curve, based on the calculations: 68% of the area lies between 1 standard deviation above and below the mean is a likely conclusion for the example presented.

• Summary of Values:

  • V = Mean - 1 Std Dev = 151.1 min

  • U = Mean + 1 Std Dev = 190.5 min

  • W = Mean + 2 Std Devs = 210.2 min

Conclusion

Understanding the empirical rule allows for better comprehension of how data is concentrated around the mean in a normal distribution, providing foundational skills for statistical analysis.