How To Find The Median In A Box Plot

Box-and-Whisker Plots

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

The median is the middle number of a set up of data, or the average of the two middle numbers (if there are an even number of data points).

The median ( $Q_{\mathrm{two}}$ ) divides the information set into two parts, the upper fix and the lower set. The lower quartile ( $Q_{\mathrm{ane}}$ ) is the median of the lower half, and the upper quartile ( $Q_3$ ) is the median of the upper half.

Instance:

Notice $Q_{\mathrm{one}}$ , $Q_2$ , and $Q_3$ for the following data set, and draw a box-and-whisker plot.

$\left\{\mathrm{two},6,7,8,8,\mathrm{xi},12,13,14,15,22,23\right\}$

There are $12$ data points. The centre two are $\mathrm{eleven}$ and $12$ . So the median, $Q_{\mathrm{ii}}$ , is $\mathrm{xi.5}$ .

The "lower half" of the data set up is the set $\left\{2,6,7,8,8,\mathrm{xi}\right\}$ . The median hither is $\mathrm{7.v}$ . And so $Q_{\mathrm{one}}=\mathrm{seven.5}$ .

The "upper half" of the data set is the prepare $\left\{12,13,14,15,22,23\right\}$ . The median here is $14.5$ . So $Q_3=14.5$ .

A box-and-whisker plot displays the values $Q_{\mathrm{one}}$ , $Q_2$ , and $Q_{\mathrm{iii}}$ , forth with the farthermost values of the information set ( $\mathrm{ii}$ and $23$ , in this case):

A box & whisker plot shows a "box" with left edge at $Q_1$ , right border at $Q_3$ , the "eye" of the box at $Q_{\mathrm{two}}$ (the median) and the maximum and minimum as "whiskers".

Note that the plot divides the data into $\mathrm{iv}$ equal parts. The left whisker represents the lesser $25\%$ of the data, the left half of the box represents the second $25\%$ , the right half of the box represents the 3rd $25\%$ , and the correct whisker represents the top $25\%$ .

Outliers

If a data value is very far away from the quartiles (either much less than $Q_1$ or much greater than $Q_3$ ), it is sometimes designated an outlier . Instead of existence shown using the whiskers of the box-and-whisker plot, outliers are ordinarily shown as separately plotted points.

The standard definition for an outlier is a number which is less than $Q_1$ or greater than $Q_{\mathrm{three}}$ by more than $1.5$ times the interquartile range ( $\text{IQR}=Q_{\mathrm{iii}}-Q_1$ ). That is, an outlier is any number less than $Q_{\mathrm{one}}-\left(\mathrm{one.5}\times \text{IQR}\right)$ or greater than $Q_{\mathrm{three}}+\left(\mathrm{1.five}\times \text{IQR}\right)$ .

Example:

Find $Q_1$ , $Q_2$ , and $Q_{\mathrm{iii}}$ for the post-obit data fix. Identify any outliers, and draw a box-and-whisker plot.

$\left\{5,40,42,46,48,49,50,\mathrm{fifty},52,53,55,56,58,75,102\right\}$

At that place are $\mathrm{fifteen}$ values, arranged in increasing gild. And so, $Q_2$ is the $8^{\text{th}}$ information point, $\mathrm{fifty}$ .

$Q_{\mathrm{i}}$ is the $4^{\text{th}}$ data point, $46$ , and $Q_3$ is the ${12}^{\text{th}}$ data point, $56$ .

The interquartile range $\text{IQR}$ is $Q_3-Q_1$ or $56-47=\mathrm{ten}$ .

Now we need to find whether at that place are values less than $Q_1-\left(\mathrm{one.five}\times \text{IQR}\right)$ or greater than $Q_3+\left(\mathrm{1.five}\times \text{IQR}\right)$ .

$Q_1-\left(\mathrm{1.v}\times \text{IQR}\right)=46-15=31$

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

Since $5$ 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.

Source: https://www.varsitytutors.com/hotmath/hotmath_help/topics/box-and-whisker-plots

Posted by: keenanmaked1947.blogspot.com

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