The median, which can also be known as the second quartile (Q_2), separates the data into upper and lower halves.
The first quartile (Q_1) is the median of the lower half of the data.
The third quartile (Q_3) is the median of the upper half of the data.
We begin by listing the data in numerical order. Since we have an even number of values, the median is the mean of the two middle values.
1, 2, 4, 6, 7, 7, 8, 8, 9, 12, 13, 15, 15,
22, 37, 39 ⇓ Q_2 = 8+ 9/2=8.5
Next, we find the quartiles Q_1 and Q_3. Since each half of data contains 8 values (which is an even number) we will find the mean of the two middle values of each half of data.
1, 2, 4, 6, 7, 7, 8, 8, | 9, 12, 13, 15, 15, 22, 37, 39 ⇓
lQ_1=6+ 7/2=6.5 Q_3=15+ 15/2=15
The box-and-whisker plot consists of three parts: a box and two whiskers.
The left whisker extends from the minimum to the first quartile, 1 to 6.5.
The box extends from the first to the third quartile (6.5 to 15) and has a vertical line through the median, which is 8.5.
The right whisker extends from the third quartile to the maximum, which in our case is 15 to 39.
Let's make our box-and-whisker plot!
b To identify any outliers, we start by calculating the interquartile range (IQR). This is the difference between the upper and lower quartiles.
Q_3- Q_1 = IQR ⇔ 15- 6.5= 8.5
Next, we need to determine the maximum and minimum values for data to be considered outlier. Outliers are more than 1.5 times the IQR away from the upper and lower quartiles.
l Q_1-1.5* IQR ⇔ 6.5-1.5( 8.5)=- 6.25 Q_3+1.5* IQR ⇔ 8.5+1.5( 8.5)=21.25
This tells us that any values lower than -6.25 or greater than 21.25 are outliers. Let's identify the outliers from the given data set!
1, 2, 4, 6, 7, 7, 8, 8, 9, 12, 13, 15, 15,
22, 37, 39
We found that 22, 37, and 39 are outliers. We will remove these from the data set and make a revised box-and-whisker plot. Since we removed 3 values,we now have an odd number of values. This means that the median is the middle value of the data set.
1, 2, 4, 6, 7, 7, 8, 8, 9, 12, 13, 15, 15 ⇓
Q_2= 8
Since each half of data has 6 values, which is an even number, we can find the quartiles like we did in Part A.
1, 2, 4, 6, 7, 7, | 8 | 8, 9, 12, 13, 15, 15 ⇓
lQ_1=4+ 6/2=5 Q_3=12+ 13/2=12.5
Let's make our revised box-and-whisker plot using these values!
c Let's compare both box-and-whisker plots.
We can see that the whiskers of our plot get smaller as we remove the outliers. The box also becomes smaller and the median approaches the center of the box.
