In this explainer, we will learn how to construct and analyze data from box-and-whisker plots.
When we have a numerical data set, a good way of showing how the data is spread out from the eye is with a box-and-whisker plot. Remember that a numerical data set up is 1 in which the values are measurements, like height, weight, or age. Nosotros say that the variable (the affair that nosotros are measuring, e.g., pinnacle) is numerical.
It volition exist helpful to remind ourselves of some useful terms earlier looking at examples of box-and-whisker plots and how to use them.
Some Useful Terms:
The minimum of a prepare of data is the smallest value in the information ready.
The maximum of a set of data is the largest value in the data ready.
The range of a information set is the maximum value minus the minimum value.
The first quartile, or Q1, is the value in a data set up beneath which 25% of the data prevarication.
The median, or second quartile (Q2), of a fix of information is the centre value of the information set up. Then, 50% of the information lie beneath the median.
The tertiary quartile, or Q3, is the value in a data prepare beneath which 75% of the data lie.
The interquartile range (IQR) of a data ready is given by and represents 50% of the data. That is, fifty% of the data lie betwixt Q1 and Q3.
An outlier is a value that is much smaller or much larger than most of the other values in a data prepare.
Now, let us look at the different components and features of a box-and-whisker plot.
Definition
A box-and-whisker plot (or boxplot) is a graph that illustrates the spread of a fix of numerical information, using five numbers from the data set: the maximum, the minimum, the get-go quartile (Q1), the median, and the tertiary quartile (Q3).
Let us list the features of the boxplot:
The horizontal axis covers all possible data values.
The box part of a box-and-whisker plot covers the middle l% of the values in the data set.
The whiskers each cover 25% of the information values.
The lower whisker covers all the data values from the minimum value up to Q1, that is, the lowest 25% of data values.
The upper whisker covers all the data values between Q3 and the maximum value, that is, the highest 25% of data values.
The median sits within the box and represents the center of the data. fifty% of the data values lie above the median and 50% lie below the median.
Outliers, or extreme values, in a information gear up are usually indicated on a box-and-whisker plot past the "star" symbol. If there is one or more than outliers in a data gear up, for the purpose of drawing a box-and-whisker plot, nosotros have the minimum and maximum to exist the minimum and maximum values of the information set excluding the outliers.
In our first example, we will draw a box-and-whisker plot for a specific data ready and interpret some of the features of this plot.
Instance 1: The Components of a Box-and-Whisker Plot
Adam has calculated the following data from a data set about the ages of people present on a Sat morning in a swimming puddle:
lowest value: seven
lower quartile: 10
median: fifteen
upper quartile: 22
highest value: 31
Depict a box-and-whisker plot using the information Adam has calculated from the information set up.
What is the overall age range of the Sat morning swimmers?
What percentage of Saturday morn swimmers were between 7 and 22 years one-time?
Summate and interpret the percentage of Saturday morn swimmers covered by the box.
Answer
Function 1
To depict the box-and-whisker plot, our outset pace is to depict and label an appropriate horizontal axis.
Since our least, or minimum, value is 7 and the highest, or maximum, value is 31, we tin first our axis at v and finish at 35. These are round numbers that encompass the whole range of our data. Nosotros can at present marker the values Adam has calculated on our axis.
At present, we can begin to plot our box and whiskers. Permit u.s. start by drawing the box. For the left side of the box, we draw a vertical line above Q1. And for the right, nosotros draw a vertical line in a higher place Q3. Nosotros besides include a line above the median.
Using the lines above Q1 and Q3 as the curt sides, we tin can form a rectangle, which is our box.
Note that nosotros are not too concerned with how far to a higher place the axis nosotros draw our box, although it should be close enough to the centrality that we can read which values the features of the box sit above.
The last pace is to draw the whiskers. Mark above where each of the minimum and the maximum values sits on the axis, in line with the eye of the short sides of the box, nosotros then join these marks to the box with horizontal lines.
This completes our box-and-whisker plot for the ages of Saturday morning swimmers.
Part 2
To find the range of the ages of the Sabbatum morning swimmers, we decrease the minimum value (the lowest historic period) from the maximum value (the highest age). The highest age was 31 years and the lowest was vii years, so the range is
That is, the range of the ages of Sabbatum morning swimmers was 24 years.
Function 3
To observe the pct of Sabbatum forenoon swimmers betwixt vii and 22 years sometime, we can use the information provided by Adam and shown in the box-and-whisker plot. Nosotros know that the youngest swimmer was 7 and that . We know also that, by definition, 75% of the data lies below Q3, that is, between the minimum data value and Q3.
So, we can say that 75% of the Saturday morning swimmers were between seven and 22 years old.
Part 4
To summate the percentage of Saturday forenoon swimmers covered past the box, we can again use the data we have from Adam and displayed in the box-and-whisker plot.
We know that Q1, which corresponds to the left-hand side of the box, is 10 and that Q3, corresponding to the right-hand side of the box, is 22. We also know that, by definition, 50% of the data lie between Q1 and Q3. And so, since our box stretches from Q1 to Q3, the box must cover 50% of the information.
We can interpret this as follows: 50% of the Sat forenoon swimmers were between x and 22 years quondam.
In our next instance, we use a box-and-whisker plot to determine percentages of a information gear up.
Case ii: Percentages from Box-and-Whisker Plots
The exam scores for a physics exam are displayed in the post-obit box-and-whisker plot. Make up one's mind the percent of students who had scores between 85 and 120.
Answer
We will apply the knowledge we accept well-nigh box-and-whisker plots to decide the percentage of students who had scores between 85 and 120 in their physics test.
We know that the left-paw edge of the box sits above the showtime quartile, Q1, of a data gear up and that the maximum data value sits below the signal at the right end of the right-hand whisker.
From our box-and-whisker plot, nosotros can run across that Q1 corresponds to a score of 85 and that the maximum score was 120. Nosotros also know that 25% of the values in a information gear up prevarication beneath Q1.
If 25% of the data in a data set prevarication below Q1, then the balance of the data set must lie above Q1 (i.e., the remaining 75%), that is, that 75% of the data ready lies between Q1 and the maximum information value.
In our case, since Q1 corresponds to a score of 85, and the maximum score was 120, nosotros can say that 75 per centum of the students had scores between 85 and 120 in their physics examination.
Example 3: Percentages from a Box-and-Whisker Plot
Is half of the information in the interval 36 to 56?
Reply
Since the data value 36 sits beneath the end of the lower whisker in the box-and-whisker plot, we tin say that the minimum data value is 36. Similarly, since the information value 56 sits below the vertical bar within the box, we can say that the median of the information is 56. Nosotros know that the median is the center value of the data set, splitting the data set in two. This means that 50% of the data prevarication below the median.
We can therefore answer as follows: yes, half of the data is in the interval 36 to 56.
It is worth noting that in this example we have an outlier in the data fix (a value of 80), which is much college than the majority of the data values.
Technically, this is the maximum value in the information set, but if we had been asked if one-half of the data is in the interval 56 to 64, nosotros would answer as follows: yes, excluding the outlier at eighty, one-half of the data is in the interval 56 to 64.
Example 4: Interpreting a Box-and-Whisker Plot
Look at the box-and-whisker plot. Give a reason why the line within the box is further to the correct.
The median is closer to Q3 than Q1.
The mean is about 49.
The person who made the graph fabricated a mistake.
The way is 49.
Respond
To determine which of the reasons given is correct, for why the line inside the box is further to the right, allow us marking the quantities that we know on our box-and-whisker plot.
Nosotros tin can at present address each possibility in turn.
Nosotros tin can run into from our plot that the line above the median within the box is closer to the right-hand edge than the left-paw edge of the box. And we know that the correct-hand edge of the box sits above Q3, whereas the left-mitt edge sits to a higher place Q1. And then "A" must be correct: the median is closer to Q3 than to Q1.
It is non possible to tell what the mean of a data set is from a box-and-whisker plot. We can but tell what the median is, and in this example information technology looks as though the median is approximately 49.
We cannot tell whether the person who made the graph messed upward or not, but it is perfectly possible for the line above the median in a box-and-whisker plot to be off-centre within the box.
A box-and-whisker plot can merely tell us the value of the median, non the style of a data gear up. So we cannot say that the manner is 49.
We can conclude that pick "A" is correct.
In our final example, we will encounter how to gain and interpret data from a box-and-whisker plot.
Example v: Gaining and Interpreting Information from a Box-and-Whisker Plot
The boxplot shows the daily temperatures at a seaside resort during the calendar month of Baronial.
What was the median temperature?
What was the maximum temperature recorded?
What was the minimum temperature recorded?
What was the lower quartile of the temperatures?
What was the upper quartile of the temperatures?
What was the interquartile range of the temperatures?
What was the range of the temperatures?
On roughly what percentage of days was the temperature between and ?
On roughly what percent of days was the temperature greater than ?
Reply
In social club to respond the questions (A)–(I), let us marker the quantities nosotros know on our box-and-whisker plot.
In a box-and-whisker plot, the median of the information set is the value that sits beneath the vertical line inside the box. In our case, this value is 24. And so, the median temperature was .
In a box-and-whisker plot, the maximum value in the data set sits beneath the correct-mitt cease of the correct-paw (or upper) whisker. Equally nosotros can encounter from our plot, the maximum value here is 31. So the maximum temperature recorded was .
In a box-and-whisker plot, the minimum value in the data set sits below the left-paw cease of the left-manus (or lower) whisker. Equally we can come across from our plot, the minimum value hither is 18. So the minimum temperature recorded was .
In a box-and-whisker plot, the lower quartile (Q1) of a data gear up is the value that sits beneath the left-manus edge of the box. In our case, this value is 22. So, the lower quartile of the temperatures was .
In a box-and-whisker plot, the upper quartile (Q3) of a data set is the value that sits beneath the correct-paw edge of the box. In our case, this value is 28. Then the upper quartile of the temperatures was .
The interquartile range (IQR) of a data set up is the distance between the lower and upper quartiles, which is given by . In this case, So, the interquartile range of the temperatures was .
The range of a data set is given by the maximum value minus the minimum value. In our example, nosotros can see from our box-and-whisker plot that the maximum value in the information set is 31 and the minimum value is 18. So the range is The range of temperatures was therefore .
To notice what percentage of days the temperature was between and , nosotros note once again that the lower quartile, Q1, is and that the median is . We know also that fifty% of the data lie below the median and that 25% of the data lie below Q1.
And so, between Q1 and the median, there must exist of the data. Therefore, on 25% of the days the temperature was between (which is Q1) and (which is the median).
To observe on what per centum of days the temperature was greater than , we annotation once more that the median was and that l% of the data lie above the median.
So, the temperature was greater than on fifty% of the days.
Let us conclude past reminding ourselves of the master features of a box-and-whisker plot.
Central Points
The box part of a box-and-whisker plot covers the centre 50% of the values in the data set.
The whiskers each embrace 25% of the information values.
The lower whisker covers all the data values from the minimum value up to Q1, that is, the lowest 25% of data values.
The upper whisker covers all the information values between Q3 and the maximum value, that is, the highest 25% of data values.
The median sits within the box and represents the center of the data. 50% of the data values lie above the median and l% lie below the median.
Outliers, or extreme values, in a data set are ordinarily indicated on a box-and-whisker plot past the "star" symbol.
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