Draw a Box Plot Online and Check

Typically, statisticians are going to use software to help them look at information using a box plot. All the same, when y'all are first learning about box plots, it tin exist helpful to learn how to sketch them past mitt. This way, you volition be very comfy with understanding the output from a computer or your calculator. In the following lesson, we will look at the steps needed to sketch boxplots from a given data set.

Example information

Retrieve, the goal of any graph is to summarize a information set. There are many possible graphs that ane can use to do this. One of the more common options is the histogram, but there are too dotplots, stem and leaf plots, and every bit we are reviewing here – boxplots (which are sometimes called box and whisker plots). Like a histogram, box plots ignore information about each individual data value and instead show the overall pattern.

To review the steps, nosotros will utilise the data prepare below. Allow's suppose this data set represents the salaries (in thousands) of a random sample of employees at a modest company.

vii
xiv
fourteen
xiv
16

18
xx
20
21
23

27
27
27
29
31

31
32
32
34
36

xl
40
40
40
40

42
51
56
60
65

Steps to Making Your Box plot

Pace ane: Calculate the v number summary for your data set

The five number summary consists of the minimum value, the first quartile, the median, the third quartile, and the maximum value. While these numbers can also be calculated past hand (here is how to calculate the median past mitt for case), they can quickly exist constitute on a TI83 or 84 calculator nether 1-varstats. The video beneath shows you how to become to that carte on the TI84:

For this information set, you volition get the post-obit output:
five-number-summary

Step 2: Place outliers

Other than "a unique value", there is non One definition across statistics that is used to find an outlier. As you written report statistics, yous volition see that dissimilar settings will use unlike techniques to flag or marking a potential outlier. With boxplots, this is done using something called "fences". The idea is that anything outside the fences is a potential outlier and shouldn't be included in the principal group that we graph. Instead information technology volition be marked with a asterisk or other symbol.

The lower fence

Whatsoever data value smaller than the lwoer fence will be considered an outlier. The lower contend is defined past the following formula:

$\text{lower contend} = Q_{1} – 1.5(IQR)$

This formula makes employ of the IQR, or interquartile range. This is defined as:

$\text{IQR} = Q_3 – Q_1$

Using the figurer output, we have for this data set $Q_1 = xx$ and $Q_3 = 40$. This gives the states:

$\begin{align} \text{IQR} &= Q_{3}-Q_{1}\\ &= 40 – 20\\ &= 20\end{align}$

and using this value:

$\begin{align} \text{lower fence} &= Q_{1} – 1.five(IQR) \\ &= 20 -1.5(20)\\ &= 20 – 30\\ &= -x\end{align}$

Since there are no values in the information set that are less than -x, at that place are no lower (small) outliers.

The upper fence

Like to the lower fence, anything data value larger than the upper fence volition be considered an outlier. This is defined by the following formula.

$\text{upper fence} = Q_{iii} + 1.5(IQR)$

Using the calculation to a higher place, we know that $\text{IQR} = xx$. Nosotros too had $Q_3 = 40$. Therefore:

$\begin{align}\text{upper fence} &= Q_{3} + ane.5(IQR)\\ &= 40 + 1.v(twenty) \\ &=twoscore + 30\\ &= 70\end{marshal}$

The largest value in the data prepare is 65, so this means in that location is no upper (large) outlier.

Since there were no small or large outliers in the set, we can conclude at that place are no outliers overall.

Step 3: Sketch the box plot using the model below

The main part of the box plot will be a line from the smallest number that is not an outlier to the largest number in our data prepare that is not an outlier. If a information set doesn't take any outliers (like this one), then this volition only be a line from the smallest value to the largest value. The remainder of the plot is made by drawing a box from $Q_{1}$ to $Q_{3}$ with a line in the middle for the median. As a general example:

general-box-plot

Additionally, if you are cartoon your box plot past paw you must call back of scale. In this data ready, the smallest is 7 and the largest is 65. And then starting the calibration at five and counting by v up to 65 or seventy would probably give a nice picture. So, since none of these are outliers, nosotros volition draw a line from vii, which is the smallest data value to 65, which is the largest data value. Finally, we will add a box from our quartiles ($Q_1 = 20$ and $Q_3 = 40$) and a line at the median of 31. All together nosotros have:

Of course, a software version will look quite a chip amend. Too notation that boxplots tin be drawn horizontally or vertically and y'all may run across either as you lot proceed your studies. As an instance, hither is the same boxplot done with R (a statistical software program) instead:

Summary

Remember – pay attending to how these box plots are put together in gild to do a better job at reading the information they provide. Since you now know that middle line is the median, you can merely look at the box plot and know that 50% of the salaries were less than $31,000 or so. As y'all can see, a box plot can not only prove you lot the overall pattern but also contains a lot of information about the information gear up. To see more most the data you tin gather from a boxplot, see: How to read a boxplot

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Source: https://www.mathbootcamps.com/how-to-make-a-boxplot-box-and-whiskers-plot-by-hand/

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