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Answer:
**Normal Distribution and Its Properties:**
Normal distribution, also known as Gaussian distribution, is a fundamental concept in statistics and probability theory. It is characterized by a bell-shaped curve that is symmetric around the mean. Here are its key properties:
1. **Bell-shaped Curve:** The graph of a normal distribution is bell-shaped, with the highest point at the mean, which is also the median and mode of the distribution.
2. **Symmetry:** The distribution is symmetric around the mean, meaning that the probabilities on both sides of the mean are equal.
3. **Mean, Median, and Mode:** These three measures of central tendency are all located at the center of the distribution, which is the mean (\( \mu \)).
4. **Standard Deviation:** The spread of the distribution is determined by the standard deviation (\( \sigma \)). The larger the standard deviation, the wider the distribution.
5. **68-95-99.7 Rule:** This empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
6. **Probability Density Function:** The probability density function (pdf) of a normal distribution is given by:
\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
where \( x \) is a random variable, \( \mu \) is the mean, \( \sigma \) is the standard deviation, \( \pi \) is the mathematical constant pi, and \( e \) is the base of the natural logarithm.
**Real-life Phenomenon that Follows a Normal Distribution:**
One real-life phenomenon that often follows a normal distribution is human height. Here’s why:
- **Symmetry:** Heights of people tend to follow a symmetric distribution around the average height.
- **Central Tendency:** The average height (mean) of a population can be measured and is often reported.
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