The normal distribution is the probability distribution where the mean, median, and mode are equal.
Understanding the Normal Distribution
The normal distribution, also known as the Gaussian distribution or bell curve, is a fundamental concept in statistics. It describes how many natural phenomena are distributed around a central value.
A key characteristic of the normal distribution is its perfect symmetry around its center. This symmetry is precisely why the mean, median, and mode all coincide at the same point.
- Mean: The arithmetic average of all values in the distribution.
- Median: The middle value when all values are arranged in order. In a symmetric distribution, this point divides the data set into two equal halves.
- Mode: The value that appears most frequently in the distribution. In a perfectly symmetric, unimodal distribution like the normal distribution, the highest point (peak) represents the mode.
As stated in the reference, "Mean, Mode and Median of normal distribution are equal." This is a well-known property.
Why Are They Equal in a Normal Distribution?
The symmetry of the normal distribution is the key factor:
- Mode: The peak of the bell curve is exactly in the center, where the highest frequency occurs. This peak is the mode.
- Median: Because the distribution is symmetric, 50% of the data falls to the left of the center, and 50% falls to the right. The point that splits the data exactly in half is the median, which is the center.
- Mean: In a symmetric distribution, the mean is not pulled towards extreme values on either side. It is located exactly at the center, balancing the distribution.
Therefore, for a normal distribution, the single point at the center of the bell curve represents the mean, median, and mode simultaneously.
This property makes the normal distribution particularly easy to work with and interpret compared to skewed distributions where these three measures of central tendency can be quite different.
