Question

from unit 1, lesson 14
the data represents the number of hours 10 students slept on sunday night.
6 6 7 7 7 8 8 8 8 9
are there any outliers? explain your reasoning.

Explanation:

Step1: Find the median

First, find the median of the data set. The data set has 10 values. The median is the average of the 5th and 6th values. The 5th value is 7 and the 6th value is 8. So, the median \(Q_2=\frac{7 + 8}{2}=7.5\).

Step2: Find \(Q_1\) and \(Q_3\)

The lower half of the data is \(6,6,7,7,7\). The median of the lower half \(Q_1 = 7\). The upper half of the data is \(8,8,8,8,9\). The median of the upper half \(Q_3=8\).

Step3: Calculate the inter - quartile range (IQR)

The inter - quartile range \(IQR=Q_3 - Q_1\). So, \(IQR = 8-7=1\).

Step4: Find the lower and upper bounds

The lower bound is \(Q_1-1.5\times IQR\). \(Q_1-1.5\times IQR=7 - 1.5\times1=5.5\). The upper bound is \(Q_3 + 1.5\times IQR\). \(Q_3+1.5\times IQR=8 + 1.5\times1=9.5\).

Step5: Check for outliers

All the data values \(6,6,7,7,7,8,8,8,8,9\) are within the range \([5.5,9.5]\).

Answer:

No, there are no outliers. All data values are within the range \(Q_1 - 1.5\times IQR=5.5\) and \(Q_3+1.5\times IQR = 9.5\).