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\( \square \% \) of the area lies below the mean.

Ask by Carter Love. in the United States
Feb 05,2025

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50% of the area lies below the mean.

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In a **symmetric distribution**, such as the **normal (bell-shaped) distribution**, the mean and median are equal. This symmetry ensures that: - **50% of the area lies below the mean.** - **50% of the area lies above the mean.** So, for a normal distribution: \[ \boxed{50\%} \] This means that **50% of the data points fall below the mean value**. **Visual Representation:** Imagine the classic bell curve: ``` * * * * * * * ---------|--------- Mean * * * * * * * ``` Here, the vertical line represents the mean. The area under the curve to the left of this line is exactly half of the total area, confirming that **50% of the area lies below the mean**. **Note:** In **skewed distributions**, the percentage of area below the mean can differ from 50%, depending on the direction and degree of skewness. However, for symmetric distributions like the normal distribution, it's always 50%. Answer: 50 %

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Bonus Knowledge

In a normal distribution, about \(50\%\) of the area under the curve lies below the mean. This characteristic highlights the symmetry of the normal distribution, where the mean, median, and mode are all equal and centrally located. So, if you were to slice this bell-shaped curve right down the middle, you'd find that half of the data points fall below the mean! When dealing with data sets, understanding the area representation can help in making better predictions. For example, if you know your data follows a normal distribution, knowing that \(50\%\) lies below the mean can help you assess probabilities for various outcomes, making it easier to make informed decisions in fields like finance, engineering, or social sciences.

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