Normal Distribution and Quartiles

Normal distribution, also known as Gaussian distribution, is one of the most commonly used probability distributions in statistics. It is characterized by a bell-shaped curve that is symmetrical around the mean of the distribution. In a normal distribution, the quartiles divide the data into four equal parts, with each part containing 25% of the data.

Equidistant Quartiles from Mean

The mean, median, and mode are all measures of central tendency that describe the location of the center of the distribution. In a normal distribution, these measures are all equal and located at the center of the distribution. However, quartiles are not measures of central tendency but rather measures of dispersion that describe the spread of the data.

In a normal distribution, the quartiles are equidistant from the mean. This means that the distance between the mean and the first quartile (Q1) is the same as the distance between the mean and the third quartile (Q3). This property holds true regardless of the shape of the distribution, as long as it is normal.

Explanation

The reason why the quartiles are equidistant from the mean in a normal distribution is because of the way the distribution is constructed. A normal distribution is constructed so that the area under the curve between any two points is proportional to the probability of observing a value between those two points. This means that the area under the curve between the mean and each quartile is equal, and therefore the quartiles are equidistant from the mean.

Conclusion

In summary, in a normal distribution, the quartiles are equidistant from the mean. This property reflects the way a normal distribution is constructed, and holds true regardless of the shape of the distribution. Quartiles are measures of dispersion that describe the spread of the data, while the mean, median, and mode are measures of central tendency that describe the location of the center of the distribution.