explain this---**Textual Information:** * **Title:** Normal distribution * **Mathematical Formula:** f(x) = (1 / (σ * sqrt(2π))) * e^(-(1/2) * ((x - μ) / σ)^2) * **Legend Data:** * mean: μ=0, variance: σ²=1.0 (Associated with a dark red/maroon curve) * mean: μ=0, variance: σ²=5.0 (Associated with a khaki/light brown curve) * mean: μ=-2, variance: σ²=0.5 (Associated with a green curve) **Chart/Diagram Description:** * **Type:** Line chart representing probability density functions of normal distributions. * **Main Elements:** * **Coordinate Axes:** * X-axis labeled 'x', with ticks and labels from -5 to 5 in increments of 1. * Y-axis labeled 'f(x)', with ticks and labels from 0.0 to 0.8 in increments of 0.2. * **Data Series (Curves):** Three distinct bell-shaped curves, each representing a normal distribution with specific parameters (mean μ and variance σ²). * Dark red/maroon curve: Centered at x=0, peaking near y=0.4, relatively broad base. * Khaki/light brown curve: Centered at x=0, peaking at a lower value (around y=0.18), much broader base. * Green curve: Centered at x=-2, peaking at the highest value (around y=0.79), narrow base. * **Legend:** A box showing the correspondence between curve colors and the parameters (mean and variance) of the normal distribution they represent. The legend is placed in the upper right portion of the chart area.

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The normal distribution is one of the most important probability distributions in statistics. It creates a symmetric, bell-shaped curve that appears frequently in nature and data analysis. This continuous distribution is characterized by its perfect symmetry around the mean, forming the classic bell shape we see here. Normal distributions are fundamental to many statistical methods and appear in countless real-world phenomena, from heights and weights to measurement errors and test scores. The normal distribution is one of the most important probability distributions in statistics. It describes many natural phenomena and is characterized by its distinctive bell-shaped curve. Also known as the Gaussian distribution after Carl Friedrich Gauss, it is completely defined by just two parameters: the mean, which determines the center, and the standard deviation, which controls the spread of the distribution. The normal distribution formula may look complex, but each component has a specific purpose. The mean μ determines where the curve is centered on the x-axis. The standard deviation σ controls how wide or narrow the curve appears. The normalization constant one over sigma root two pi ensures the total area under the curve equals one, making it a proper probability distribution. Finally, the exponential term creates the characteristic bell shape, with the negative exponent ensuring the curve peaks at the mean and decreases symmetrically on both sides. Here we see three normal distributions with different parameters. The dark red curve shows the standard normal distribution with mean zero and variance one. The khaki curve also has mean zero but with variance five, making it much wider and flatter. The green curve has mean negative two and variance zero point five, making it narrow and tall but shifted to the left. Notice how the mean controls the horizontal position while the variance controls the width and height of the curve. The normal distribution has several important properties that make it extremely useful. It is bell-shaped and perfectly symmetric around its mean. The total area under the curve always equals one, making it a valid probability distribution. Following the empirical rule, approximately 68 percent of data falls within one standard deviation of the mean, 95 percent within two standard deviations, and 99.7 percent within three standard deviations. The normal distribution appears everywhere in nature and science: human heights and weights, test scores, measurement errors, quality control in manufacturing, and financial modeling. It also forms the foundation of the Central Limit Theorem, one of the most important results in statistics. Let's explore how the parameters μ and σ affect the normal distribution. The mean μ controls the horizontal position of the curve - shifting it left or right without changing its shape. Watch as we move the mean from zero to negative two and back. The variance σ squared, or equivalently the standard deviation σ, controls the width and height of the curve. A larger standard deviation creates a wider, flatter curve, while a smaller standard deviation creates a narrower, taller curve. This inverse relationship between width and height ensures the area under the curve always remains equal to one. Now let's examine the three specific distributions from our chart. The dark red curve represents the standard normal distribution with mean zero and variance one - this is our reference distribution with moderate height and width. The khaki curve shares the same center but has variance five, making it much wider and flatter. Notice how the larger variance spreads the probability over a wider range, reducing the peak height. The green curve is shifted left to mean negative two and has variance zero point five. This small variance creates the narrowest and tallest curve, concentrating most of the probability near the mean. These examples perfectly demonstrate how mean controls position while variance controls the shape. This final comparison reveals the fundamental properties of normal distributions. Notice the inverse relationship between variance and peak height - the green curve with smallest variance has the highest peak, while the khaki curve with largest variance has the lowest peak. Despite their different shapes, all three curves have exactly the same total area under them, equal to one. The shaded regions demonstrate the empirical rule: 68 percent of the data falls within one standard deviation of the mean for each distribution. This rule applies universally to all normal distributions, making them incredibly useful for probability calculations and statistical inference in countless real-world applications.