In statistics, when describing a normal distribution, the mean, median, and mode play important roles and are often related in specific ways. Here’s a detailed breakdown of each:

### Normal Distribution

A normal distribution, often referred to as a Gaussian distribution, is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean (μ) and the standard deviation (σ). The mean defines the center of the distribution, while the standard deviation determines the width of the curve.

### Mean (μ)

**Definition**: The mean is the average of all data points in a dataset. In a normal distribution, it is the point around which the values are evenly distributed.**Properties**: In a normal distribution:- The mean is located at the center of the distribution.
- It is affected by extreme values (outliers).

**Mathematical Representation**: The mean can be mathematically represented as:[

\mu = \frac{\sum_{i=1}^{n} x_i}{n}

]where ( x_i ) represents each data point and ( n ) is the total number of points.

### Median

**Definition**: The median is the middle value of a dataset when it is ordered from least to greatest. For an even number of observations, it is the average of the two middle numbers.**Properties**: In a normal distribution:- The median is also located at the center of the distribution.
- Because the normal distribution is symmetric, the median will equal the mean.

**Mathematical Representation**:- For a dataset, it is often found as:

[

\text{Median} =

\begin{cases}

x*{(n/2)} & \text{if } n \text{ is even} \*{((n+1)/2)} & \text{if } n \text{ is odd}

x

\end{cases}

]

### Mode

**Definition**: The mode is the value that appears most frequently in a dataset. In a normal distribution, it is the peak point of the distribution curve.**Properties**: In a normal distribution:- The mode is also located at the center of the distribution.
- Like the mean and median, the mode in a normal distribution is equal to the mean and median.

**Mathematical Representation**: For a continuous distribution, the mode can be determined by finding the value of ( x ) that maximizes the probability density function (PDF):[

f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}

]

### Relationship Among Mean, Median, and Mode

In a normal distribution:

**Equality**: The mean, median, and mode are all equal to each other (mean = median = mode = μ).**Symmetry**: This equality is a direct result of the symmetry of the normal distribution. The peak is at the center, leading to data being evenly distributed around this point.

### Graphical Representation

On a graph of a normal distribution:

- The peak of the bell curve represents the mode.
- The center point of the curve is the mean and median.
- The area under the curve represents the total probability, which is equal to 1.

### Summary

In summary, in a normal distribution:

- Mean, median, and mode all coincide at the same point (the center of the distribution).
- They provide different perspectives on the central tendency of a dataset but yield the same value in the context of a normal distribution.
- Their equality is crucial in many statistical analyses and serves as a fundamental property of normal distributions.