Which Of The Following Box-and-whisker Plots Correctly Displays The Data Set? 12 11 15 12 19 20 19

Box-and-Whisker Plots

To understand box-and-whisker plots, you have to sympathize medians and quartiles of a data set.

The median is the heart number of a prepare of data, or the average of the two heart numbers (if at that place are an even number of data points).

The median ( $Q_{2}$ ) divides the information set into two parts, the upper set and the lower fix. The lower quartile ( $Q_{1}$ ) is the median of the lower half, and the upper quartile ( $Q_{iii}$ ) is the median of the upper half.

Example:

Find $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ for the following data fix, and draw a box-and-whisker plot.

${2, vi, seven, viii, viii, 11, 12, 13, fourteen, fifteen, 22, 23}$

There are $12$ data points. The middle ii are $11$ and $12$ . And so the median, $Q_{2}$ , is $11.5$ .

The "lower one-half" of the data set is the set ${2, 6, vii, 8, viii, 11}$ . The median here is $7.5$ . So $Q_{one} = 7.5$ .

The "upper half" of the information prepare is the gear up ${12, 13, xiv, xv, 22, 23}$ . The median here is $14.5$ . So $Q_{3} = 14.5$ .

A box-and-whisker plot displays the values $Q_{1}$ , $Q_{2}$ , and $Q_{iii}$ , along with the extreme values of the information set ( $two$ and $23$ , in this case):

A box & whisker plot shows a "box" with left border at $Q_{i}$ , right edge at $Q_{3}$ , the "center" of the box at $Q_{2}$ (the median) and the maximum and minimum equally "whiskers".

Notation that the plot divides the data into $four$ equal parts. The left whisker represents the bottom $25 %$ of the data, the left half of the box represents the second $25 %$ , the right half of the box represents the third $25 %$ , and the right whisker represents the summit $25 %$ .

Outliers

If a data value is very far abroad from the quartiles (either much less than $Q_{1}$ or much greater than $Q_{three}$ ), information technology is sometimes designated an outlier . Instead of beingness shown using the whiskers of the box-and-whisker plot, outliers are usually shown every bit separately plotted points.

The standard definition for an outlier is a number which is less than $Q_{1}$ or greater than $Q_{3}$ past more than than $ane.five$ times the interquartile range ( $IQR = Q_{3} - Q_{1}$ ). That is, an outlier is whatever number less than $Q_{1} - (1.5 \times IQR)$ or greater than $Q_{3} + (ane.5 \times IQR)$ .

Instance:

Find $Q_{1}$ , $Q_{2}$ , and $Q_{iii}$ for the following data set. Identify any outliers, and depict a box-and-whisker plot.

${five, 40, 42, 46, 48, 49, 50, 50, 52, 53, 55, 56, 58, 75, 102}$

There are $15$ values, arranged in increasing guild. So, $Q_{2}$ is the $8^{th}$ information point, $50$ .

$Q_{1}$ is the $4^{th}$ data signal, $46$ , and $Q_{three}$ is the $12^{thursday}$ data point, $56$ .

The interquartile range $IQR$ is $Q_{3} - Q_{1}$ or $56 - 47 = 10$ .

At present nosotros need to find whether there are values less than $Q_{1} - (i.v \times IQR)$ or greater than $Q_{3} + (1.5 \times IQR)$ .

$Q_{1} - (1.5 \times IQR) = 46 - 15 = 31$

$Q_{iii} + (1.5 \times IQR) = 56 + 15 = 71$

Since $v$ is less than $31$ and $75$ and $102$ are greater than $71$ , there are $3$ outliers.

The box-and-whisker plot is as shown. Note that $40$ and $58$ are shown as the ends of the whiskers, with the outliers plotted separately.