e. what percentage of the measurements in the data set lie to the right of the​ median?
In this explainer, we will learn how to construct and clarify data from box-and-whisker plots.
When we accept a numerical data set, a skilful style of showing how the information is spread out from the middle is with a box-and-whisker plot. Recollect that a numerical information set up is ane in which the values are measurements, like meridian, weight, or historic period. We say that the variable (the matter that nosotros are measuring, e.thou., 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 set of data is the smallest value in the data gear up.
The maximum of a gear up of data is the largest value in the data ready.
The range of a information set up is the maximum value minus the minimum value.
The starting time quartile, or Q1, is the value in a information set up beneath which 25% of the data lie.
The median, or second quartile (Q2), of a fix of data is the middle value of the information gear up upwardly. 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 set is given past 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.
At present, let usa 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 data, using v numbers from the data set: the maximum, the minimum, the kickoff quartile (Q1), the median, and the third quartile (Q3).
Let us list the features of the boxplot:
The horizontal axis covers all possible data values.
The box role 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 upwards 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 middle of the data. 50% of the information values prevarication above the median and 50% lie below the median.
Outliers, or farthermost values, in a information set are usually indicated on a box-and-whisker plot past the "star" symbol. If there is one or more than than outliers in a data gear up, for the purpose of drawing a box-and-whisker plot, we accept the minimum and maximum to exist the minimum and maximum values of the information prepare 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 information gear up about the ages of people present on a Saturday morning time in a swimming pool:
lowest value: vii
lower quartile: 10
median: fifteen
upper quartile: 22
highest value: 31
Draw a box-and-whisker plot using the information Adam has calculated from the information ready.
What is the overall historic period range of the Sat morn swimmers?
What percentage of Saturday morn swimmers were betwixt vii and 22 years ane-time?
Summate and interpret the percentage of Sat morning time swimmers covered by the box.
Answer
Part 1
To draw the box-and-whisker plot, our outset footstep is to depict and label an advisable 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 tin at nowadays marker the values Adam has calculated on our axis.
At nowadays, we can begin to plot our box and whiskers. Permit u.s. start by drawing the box. For the left side of the box, nosotros draw a vertical line above Q1. And for the right, nosotros draw a vertical line in a college place Q3. We besides include a line above the median.
Using the lines above Q1 and Q3 every bit the curt sides, we tin tin can course a rectangle, which is our box.
Note that we are non also concerned with how far to a higher identify the axis nosotros draw our box, although information technology should be shut enough to the centrality that nosotros can read which values the features of the box sit above.
The final pace is to depict the whiskers. Mark to a higher place where each of the minimum and the maximum values sits on the axis, in line with the heart of the short sides of the box, we and then join these marks to the box with horizontal lines.
This completes our box-and-whisker plot for the ages of Sat forenoon swimmers.
Part two
To find the range of the ages of the Saturday morning time swimmers, we subtract the minimum value (the lowest celebrated period) from the maximum value (the highest historic period). The highest age was 31 years and the lowest was vii years, then the range is
That is, the range of the ages of Sabbatum morning swimmers was 24 years.
Function 3
To find the pct of Sabbatum forenoon swimmers betwixt vii and 22 years former, we can utilise the data provided past Adam and shown in the box-and-whisker plot. Nosotros know that the youngest swimmer was 7 and that . We know as well that, by definition, 75% of the information lies below Q3, that is, betwixt the minimum information value and Q3.
So, nosotros tin say that 75% of the Saturday morning swimmers were betwixt vii and 22 years erstwhile.
Part 4
To summate the per centum of Saturday forenoon swimmers covered past the box, we can again use the information nosotros take 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 ten and that Q3, corresponding to the right-hand side of the box, is 22. We too know that, past definition, l% of the data prevarication betwixt Q1 and Q3. Then, since our box stretches from Q1 to Q3, the box must cover 50% of the information.
We tin can interpret this equally follows: 50% of the Sat morning swimmers were betwixt x and 22 years erstwhile.
In our next instance, we use a box-and-whisker plot to make up one's mind percentages of a information gear upwardly.
Example ii: Percentages from Box-and-Whisker Plots
The exam scores for a physics exam are displayed in the post-obit box-and-whisker plot. Brand up one's heed the percent of students who had scores betwixt 85 and 120.
Respond
We will apply the cognition nosotros accept well-about box-and-whisker plots to decide the pct of students who had scores between 85 and 120 in their physics exam.
Nosotros know that the left-mitt edge of the box sits in a higher place the showtime quartile, Q1, of a data gear up and that the maximum data value sits below the point at the correct cease of the correct-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 below Q1.
If 25% of the data in a data gear up prevarication below Q1, then the rest of the data prepare must lie above Q1 (i.east., the remaining 75%), that is, that 75% of the data ready lies between Q1 and the maximum data 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, nosotros tin can say that the median of the data is 56. Nosotros know that the median is the center value of the data fix, splitting the data set up in ii. This ways that 50% of the data lie beneath the median.
We can therefore answer as follows: yes, one-half of the information is in the interval 36 to 56.
Information technology is worth noting that in this example we accept an outlier in the data fix (a value of fourscore), which is much college than the bulk of the data values.
Technically, this is the maximum value in the information fix, 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 lxxx, ane-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 made a error.
The style is 49.
Respond
To decide which of the reasons given is correct, for why the line inside the box is further to the right, allow united states of america marking the quantities that we know on our box-and-whisker plot.
Nosotros can can at present address each possibility in plough.
Nosotros can can encounter from our plot that the line higher up the median within the box is closer to the right-hand border than the left-mitt border of the box. And nosotros know that the correct-hand edge of the box sits above Q3, whereas the left-mitt edge sits to a college place Q1. And and so "A" must exist correct: the median is closer to Q3 than to Q1.
It is non possible to tell what the mean of a data ready is from a box-and-whisker plot. We can but tell what the median is, and in this instance data technology looks every bit 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 exist off-centre within the box.
A box-and-whisker plot can just tell the states the value of the median, not the style of a information gear up. So we cannot say that the manner is 49.
We can conclude that choice "A" is correct.
In our last example, nosotros will encounter how to proceeds and interpret data from a box-and-whisker plot.
Instance 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 per centum of days was the temperature between and ?
On roughly what pct of days was the temperature greater than ?
Reply
In social order to respond the questions (A)–(I), let us mark the quantities we know on our box-and-whisker plot.
In a box-and-whisker plot, the median of the information fix 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 up sits beneath the correct-paw cease of the correct-mitt (or upper) whisker. Equally we 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 gear up sits beneath the left-paw cease of the left-manus (or lower) whisker. Equally we tin come across from our plot, the minimum value here is 18. And then the minimum temperature recorded was .
In a box-and-whisker plot, the lower quartile (Q1) of a data gear upward 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 information set up is the value that sits beneath the correct-paw border 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 is the distance between the lower and upper quartiles, which is given past . 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 case, nosotros can see from our box-and-whisker plot that the maximum value in the data set is 31 and the minimum value is 18. So the range is The range of temperatures was therefore .
To notice what pct of days the temperature was between and , nosotros notation in one case once more that the lower quartile, Q1, is and that the median is . We know as well that fifty% of the data prevarication below the median and that 25% of the data lie beneath Q1.
And 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 notice on what per centum of days the temperature was greater than , nosotros notation once more that the median was and that l% of the data lie above the median.
Then, the temperature was greater than on l% of the days.
Let united states conclude past reminding ourselves of the main features of a box-and-whisker plot.
Fundamental Points
The box role of a box-and-whisker plot covers the centre 50% of the values in the data prepare.
The whiskers each cover 25% of the information values.
The lower whisker covers all the information values from the minimum value upwards to Q1, that is, the lowest 25% of information 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 eye of the data. 50% of the data values lie above the median and fifty% lie beneath the median.
Outliers, or extreme values, in a data set are ordinarily indicated on a box-and-whisker plot by the "star" symbol.
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