Question

assume this is a positive ordered set of data. which statistics show values that would most likely describe the following set of data? x, 9, 11, 13, y, 20
range = 23
range = 17
mean = 12
median = 10
interquartile range = 7
interquartile range = 12
describe the data
do not describe the data

Explanation:

Step1: Recall the formula for range

Range = maximum value - minimum value. Given the data set \(X,9,11,13,Y,20\) and range \(= 23\). If we assume \(Y\) is the maximum value, then \(Y - X=23\). If we assume \(20\) is the maximum value, then \(20 - X = 23\) (but \(X\) would be negative, which is against the "positive ordered set" condition). So \(Y\) is the maximum. Also, for range \( = 17\), if \(Y\) is the maximum, \(Y - 9=17\) (since \(9\) is the minimum in some cases of wrong calculation). But for a valid positive ordered set, when calculating range correctly (with \(Y\) as max and \(X\) as min), range formula is key.

Step2: Recall the formula for inter - quartile range

Inter - quartile range (IQR) is the difference between the third quartile (\(Q_3\)) and the first quartile (\(Q_1\)), \(IQR=Q_3 - Q_1\). For a set of data \(n\) values (here \(n = 6\) values \(X,9,11,13,Y,20\)). The median of the lower half (\(Q_1\)) and upper half (\(Q_3\)) is calculated. If \(IQR = 7\), for a small set, it's a measure of spread of the middle \(50\%\) of the data. But if \(IQR=12\), it's a larger spread. The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{X + 9+11+13+Y + 20}{6}=12\), so \(X + Y=39\). The median of an even - numbered set ( \(n = 6\)) is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. If \(X\leq9\) and \(Y\geq13\), median \(=\frac{11 + 13}{2}=12
eq10\).

Answer:

Describe the data: Range \(=23\), Inter - quartile range \(=7\), Mean \(=12\)
Do not describe the data: Range \(=17\), Inter - quartile range \(=12\), Median \(=10\)