STK110 Empirical Rule and Bell-Shaped Distribution Analysis

Course Code: STK110
Topic: Empirical Rule and Normal Distribution Application.
Contextual Metadata: * Slide Number: Slide 60 of 76. * Timestamp: 16 April at 15:00. * Interface Details: The presentation software displays a zoom level of $95\%$ . * Reference Codes: A partial identifier "*** IN 19970" is visible on the interface. * *Key Statistical Benchmarks: A sequence of values is displayed on the interface representing standard deviation increments from the mean: $24$ * $36$ * $48$ * $60$ * $72$ * $84$ * $96$

Distribution Type: The marks are described as "bell-shaped." In statistics, this confirms that the data follows a Normal (Gaussian) Distribution. Common properties include: * Symmetry: The distribution is symmetric about the mean. * Central Tendency: The mean, median, and mode are essentially equal at the center of the distribution. * Asymptotic Nature: The tails of the curve approach the horizontal axis but never touch it.
Mean ( $\mu$ ): The average semester mark is established as $60\%$ .
Standard Deviation ( $\sigma$ ): The spread of the data or average distance from the mean is $12\%$ .

The provided material utilizes the Empirical Rule, which describes the percentage of data falling within specific standard deviations from the mean in a bell-shaped distribution: * One Standard Deviation ( $\mu \pm 1\sigma$ ): Approximately $68\%$ of the data falls within this range. Based on the data, this range is $60 - 12$ to $60 + 12$ , or $[48, 72]$ . * Two Standard Deviations ( $\mu \pm 2\sigma$ ): Approximately $95\%$ of the data falls within this range. For this dataset, that is $60 - 2(12)$ to $60 + 2(12)$ , or $[36, 84]$ . * Three Standard Deviations ( $\mu \pm 3\sigma$ ): Approximately $99.7\%$ of the data falls within this range. For this dataset, that is $60 - 3(12)$ to $60 + 3(12)$ , or $[24, 96]$ .

Task: Calculate the percentage of students' marks below $48\%$ .
Step 1: Determine the Z-score/Standard Deviation Distance: * The value $48\%$ is exactly one standard deviation below the mean: $\frac{48 - 60}{12} = -1\sigma$ .
Step 2: Apply the Empirical Rule: * The rule states that $68\%$ of observations are between $-1\sigma$ ( $48\%$ ) and $+1\sigma$ ( $72\%$ ). * The total area under the curve is $100\%$ . * The area outside of the center range is $100\% - 68\% = 32\%$ .
Step 3: Account for Symmetry: * Because the bell curve is symmetric, the $32\%$ area outside the center is split equally between the two tails. * The tail below $-1\sigma$ ( $48\%$ ) is $\frac{32\%}{2} = 16\%$ .
Final Result: $16\%$ of students' marks are below $48\%$ .

Task: Calculate the mark representing the 97.5th percentile ( $P_{97.5}$ ).
Definition: A percentile is the value below which a given percentage of observations fall. For $P_{97.5}$ , $97.5\%$ of the area is to the left, and $2.5\%$ is to the right.
Relating to Standard Deviations: * We know from the Empirical Rule that $95\%$ of data falls between $\mu \pm 2\sigma$ . * The total area remaining in both tails is $100\% - 95\% = 5\%$ . * Because the distribution is symmetric, each tail contains $\frac{5\%}{2} = 2.5\%$ . * Therefore, the area to the left of $\mu + 2\sigma$ includes the main $95\%$ center plus the bottom $2.5\%$ tail ( $95\% + 2.5\% = 97.5\%$ ).
Execution: * $P_{97.5}$ corresponds to $\mu + 2\sigma$ . * Calculation: $60 + 2(12) = 60 + 24 = 84$ .
Final Result: The 97.5th percentile mark is $84\%$ .

Task: Determine the thresholds below or above which a mark is considered an outlier.
Statistical Criterion: In the context of the Empirical Rule and bell-shaped distributions, values that lie more than three standard deviations ( $3\sigma$ ) from the mean are typically categorized as outliers.
Calculating the Upper Threshold: * Formula: $\mu + 3\sigma$ * Calculation: $60 + 3(12) = 60 + 36 = 96$
Calculating the Lower Threshold: * Formula: $\mu - 3\sigma$ * Calculation: $60 - 3(12) = 60 - 36 = 24$
Final Result: A semester mark below $24\%$ or above $96\%$ will be considered an outlier.