in a box plot the first quartile is x the median is y and the third quartile is 70. the median separates the box into 2 quartiles each with the same range
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To understand how to calculate the range, or spread, of the two quartiles in a box plot, it's important to have a clear understanding of what each component represents.
In a box plot, the box represents the interquartile range (IQR), which contains the middle 50% of the data. The median, also known as the second quartile (Q2), divides the IQR into two halves. The first quartile (Q1) represents the boundary that separates the first 25% of the data from the next 25%, and the third quartile (Q3) separates the next 25% from the final 25% of the data.
Based on the information given, we know that the value of the first quartile (Q1) is x, the median (Q2) is y, and the third quartile (Q3) is 70. This means that the range between Q1 and Q2 is y - x, and the range between Q2 and Q3 is 70 - y.
If we assume that the ranges of the two quartiles are equal, we can set up an equation:
y - x = 70 - y
To find the value of y, we need to solve this equation. Rearranging it, we have:
2y = 70 + x
y = (70 + x) / 2
By substituting this value back into the equation, we can find the range of each quartile.
Therefore, the range of the two quartiles would be:
First quartile (Q1) to median (Q2): y - x = [(70 + x) / 2] - x = 35 + (x/2) - x = 35 - (x/2)
Median (Q2) to third quartile (Q3): 70 - y = 70 - [(70 + x) / 2] = 70 - (70/2) - (x/2) = 35 - (x/2)
So, the first quartile (Q1) to median (Q2) has a range of 35 - (x/2), and the median (Q2) to third quartile (Q3) has a range of 35 - (x/2).