Explain Normal distribution curve.

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The normal distribution curve, often referred to as the bell curve due to its bell-shaped appearance, is a fundamental concept in statistics and probability theory. It represents a continuous probability distribution characterized by its symmetric shape and defined by two parameters: the mean (Î¼) and the standard deviation (Ïƒ). The mean determines the center of the distribution, while the standard deviation controls the spread or width of the curve.

A key feature of the normal distribution is that it is perfectly symmetrical around the mean. This symmetry implies that the mean, median, and mode of the distribution are equal. The area under the curve represents the total probability and sums up to 1. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations, a principle known as the empirical rule or the 68-95-99.7 rule.

The normal distribution is crucial in statistics because of the Central Limit Theorem, which states that the distribution of sample means approximates a normal distribution as the sample size becomes large, regardless of the shape of the population distribution. This property makes it a cornerstone in statistical methods, including hypothesis testing and confidence intervals, and it's widely applicable in various fields like psychology, finance, natural sciences, and social sciences. The normal distribution also serves as a foundation for other distributions, such as the t-distribution and chi-squared distribution.