The Empirical Rule

In most cases with normal distributions, about 68.27% of the data will be found within one standard deviation above or below the average. Approximately 95.45% of the data falls within two times the standard deviation range, and around 99.73% falls within three times the standard deviation range. This is often called the “Empirical Rule,” which helps us understand where the majority of data in a normal distribution is located.

In other words, μ ± 1σ covers 68.27% of the area, μ ± 2σ covers 95.45% of the area, and μ ± 3σ represents 99.73% of the area.

Normal Distribution in Business Statistics

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What is Normal Distribution?

Normal Distribution is defined as the probability distribution that tends to be symmetric about the mean; i.e., data near the mean occurs more as compared to the data far away from the mean. The two parameters of normal distribution are mean (μ) and standard deviation (σ). Hence, the notation of the normal distribution is...

Probability Density Function (PDF) of Normal Distribution

The Probability Density Function (PDF) of a normal distribution, often denoted as f(x), describes the likelihood of a random variable taking on a specific value within the distribution. In simpler terms, it tells the probability of getting a particular result. For a normal distribution, the PDF is represented by the well-known bell-shaped curve. This curve is centered at the mean (average) value, and its shape is determined by the standard deviation, which measures how spread out the data is. The PDF shows that values near the mean are more probable, while values farther from the mean are less likely....

Standard Normal Distribution

As different combinations of μ and σ lead to different normal distributions. The Standard normal distribution is defined as the value of normal distribution at μ = 0 and σ = 1. This is known as z-transformation....

Properties of Normal Distribution

Equality of Mean, Median, and Mode: In a normal distribution, the average (mean), middle value (median), and most frequent value (mode) are all the same.Positive Value: For any value of x, f(x) will have a positive value.Defined by Mean and Standard Deviation: A normal distribution is uniquely determined by two parameters, the average (mean) and the spread or variability (standard deviation). These parameters describe its uni-modal, bell-shaped, and symmetrical curve.Symmetry at the Center: A normal distribution curve is symmetrical. If you fold it in half at the center, both sides look the same. This means that the values on one side of the mean are like a mirror image of the other side.Total Area under the Curve: The entire area under the normal distribution curve adds up to 1. In simpler terms, if you add up all the probabilities of all possible values, it equals 100%.Half of Values on each side of the Center: Exactly half of the values are to the right of the center, and the other half is to the left of the center. This reflects the even distribution of data.One Peak: The normal distribution curve has only one hump or peak. It does not have multiple peaks. This is called a uni-modal distribution....

The Empirical Rule

In most cases with normal distributions, about 68.27% of the data will be found within one standard deviation above or below the average. Approximately 95.45% of the data falls within two times the standard deviation range, and around 99.73% falls within three times the standard deviation range. This is often called the “Empirical Rule,” which helps us understand where the majority of data in a normal distribution is located....

Parameters of Normal Distribution

The parameters of the Normal Distribution, also known as the Gaussian Distribution, are the mean and the standard deviation. The mean represents the average or central value of the data, while the standard deviation measures the spread or variability of the data points. Together, these parameters describe the shape and characteristics of the bell-shaped curve that is typical of the Normal Distribution. The mean determines the center of the curve, and the standard deviation controls how wide or narrow it is. These parameters are essential for understanding and analysing data in various fields, including statistics, science, and finance....

Curve of Normal Distribution

The curve of a normal distribution, often referred to as a bell curve, is a symmetrical, smooth, and continuous graph that depicts the distribution of data. It has the following characteristics:...

Examples of Normal Distribution

Example 1:...

Applications of Normal Distribution in Business Statistics

Normal Distribution is used in business statistics in the following ways:...

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