Prove that for a normal distribution, the mean, median, and mode are all equal.

A normal distribution is characterized by its bell-shaped curve, which is perfectly symmetrical around its central point. The probability density function of a normal distribution is given by:

where $\mu$ is the mean and $\sigma$ is the standard deviation.

Due to the perfect symmetry of the normal distribution, the data points are evenly distributed on both sides of the central value. The median is the value that divides the distribution into two equal halves, meaning 50% of the data falls below it and 50% falls above it. In a symmetrical distribution, this central point of symmetry is exactly where the mean is located. Therefore, for a normal distribution, the mean and median coincide.

The mode is the value that appears most frequently in a data set, or in a continuous distribution, it is the point where the probability density function reaches its maximum. For a normal distribution, the bell-shaped curve has a single peak. This peak occurs exactly at the center of the distribution, which is also the point of symmetry. Since the mean is at the center of symmetry, and the peak (mode) is also at the center, the mode is equal to the mean.

Since the normal distribution is perfectly symmetrical and unimodal (has a single peak), the mean, median, and mode all coincide at the central point of the distribution.
Therefore, for a normal distribution: