Symmetric distribution and quartiles

A symmetric distribution is a type of distribution where the two halves of the distribution are mirror images of each other. In other words, the mean, median, and mode of the distribution are all equal.

Quartiles are values that divide a dataset into four equal parts. The first quartile (Q1) is the value that separates the lowest 25% of the data from the rest, while the third quartile (Q3) is the value that separates the highest 25% of the data from the rest.

The relationship between Q1, Q3, and the halved sum

If we add the third quartile (Q3) and the first quartile (Q1) of a symmetric distribution, we get the interquartile range (IQR), which is a measure of the spread of the data.

If we halve the sum of Q3 and Q1, we get the median (M) of the distribution. In a symmetric distribution, M is equal to Q2, which is the middle value of the dataset.

Since a symmetric distribution has equal halves, we can say that Q1 is equal to M - IQR/2 and Q3 is equal to M + IQR/2.

If we substitute these values in the expression Q3 + Q1/2, we get:

Q3 + Q1/2 = M + IQR/2 + M - IQR/2 / 2
= M

In other words, when we halve the sum of Q3 and Q1 of a symmetric distribution, we get the median of the distribution.

Conclusion

In a symmetric distribution, halving the sum of Q3 and Q1 gives us the median of the distribution. This is because the median is the value that separates the dataset into two equal halves, and in a symmetric distribution, Q1 and Q3 are equidistant from the median.