Box Plots
Emma is choosing between two phones. Both ads say "about 20 hours of battery." In reviewers' tests, Phone A almost always lasts between 18 and 22 hours. Phone B sometimes lasts 10 hours and sometimes 30. The averages can match, yet the phones behave very differently. A box plot (also called a box-and-whisker plot) shows this at a glance: not just the center of the data, but also how spread out it is and what shape it has. Which phone would you trust? We'll answer at the end.
The Five-Number Summary
The median splits ordered data into two halves. Quartiles split it into four parts with about the same number of values in each.
With an even number of values, there is no middle value to leave out. The median is the average of the two middle values, and the halves split the list exactly in two.
Try your own numbers in the tool below and watch the quartiles move.
The grades of a group of $9$ students in a math exam are: $3, 4, 5, 6, 7, 8, 8, 9, 10$. Determine the minimum value, the lower quartile ($Q_1$), the median ($Me$), the…
The grades of a group of $9$ students in a math exam are: $3, 4, 5, 6, 7, 8, 8, 9, 10$.
Determine the minimum value, the lower quartile ($Q_1$), the median ($Me$), the upper quartile ($Q_3$) and the maximum value to construct a box plot.
The prices (in dollars) of $11$ books on a shelf are: $12, 15, 15, 18, 20, 21, 24, 25, 28, 30, 35$. Calculate the five values needed to draw their box plot.
The prices (in dollars) of $11$ books on a shelf are: $12, 15, 15, 18, 20, 21, 24, 25, 28, 30, 35$.
Calculate the five values needed to draw their box plot.
A basketball team has scored the following points in its last $15$ games: $65, 72, 78, 81, 82, 85, 88, 90, 92, 95, 98, 101, 105, 108, 112$. Find the five-number summary…
A basketball team has scored the following points in its last $15$ games:
$65, 72, 78, 81, 82, 85, 88, 90, 92, 95, 98, 101, 105, 108, 112$.
Find the five-number summary and calculate the interquartile range.
Quartiles from Tables and Grouped Data
Sometimes you get a frequency table instead of a list. Don't rewrite hundreds of values: just count positions.
When data is grouped into intervals, you can't know the exact values, but you can estimate each quartile. Assume the values are spread evenly inside each interval.
The following frequency table shows the number of siblings of the students in a class: $0$ siblings: $6$ students $1$ sibling: $10$ students $2$ siblings: $8$ students…
The following frequency table shows the number of siblings of the students in a class:
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$0$ siblings: $6$ students
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$1$ sibling: $10$ students
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$2$ siblings: $8$ students
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$3$ siblings: $4$ students
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$4$ siblings: $2$ students
Determine the minimum, $Q_1$, $Me$, $Q_3$ and the maximum to be able to construct the corresponding box plot.
The salaries (in thousands of dollars) of $200$ workers are grouped into the following intervals: $[10, 20)$: $40$ workers $[20, 30)$: $80$ workers $[30, 40)$: $60$…
The salaries (in thousands of dollars) of $200$ workers are grouped into the following intervals:
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$[10, 20)$: $40$ workers
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$[20, 30)$: $80$ workers
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$[30, 40)$: $60$ workers
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$[40, 50)$: $20$ workers
Estimate the values of $Q_1$, $Me$ and $Q_3$ using linear interpolation within the intervals to be able to sketch their box plot.
Drawing and Reading a Box Plot
A box plot is drawn above a number line. The box goes from $Q_1$ to $Q_3$, with a line inside it at the median. Two whiskers reach out to the minimum and the maximum.
To read a box plot, look at where each vertical line meets the number line. To draw one: mark the five numbers on a scale, draw the box from $Q_1$ to $Q_3$, add the median line, then draw whiskers to the minimum and maximum.
For a perfectly symmetric data set, the median sits exactly halfway between $Q_1$ and $Q_3$, and also halfway between the minimum and the maximum. That lets you work backward from the range and the IQR.
The following box plot represents the ages of the visitors to a museum during a morning. From the graph, indicate the minimum age, the lower quartile, the median, the…
The following box plot represents the ages of the visitors to a museum during a morning.
From the graph, indicate the minimum age, the lower quartile, the median, the upper quartile and the maximum age of the visitors.
The five-number summary of the weights (in pounds) of the dogs treated in a veterinary clinic is: Minimum = $5$, $Q_1 = 12$, Median = $18$, $Q_3 = 24$, Maximum = $35$.…
The five-number summary of the weights (in pounds) of the dogs treated in a veterinary clinic is: Minimum = $5$, $Q_1 = 12$, Median = $18$, $Q_3 = 24$, Maximum = $35$.
Calculate the range of the distribution and the interquartile range (IQR).
A data distribution is perfectly symmetrical. It is known that its range is $40$ units, its interquartile range (IQR) is $16$ units and the minimum value is $10$.…
A data distribution is perfectly symmetrical. It is known that its range is $40$ units, its interquartile range (IQR) is $16$ units and the minimum value is $10$.
Calculate the exact values of $Q_1$, $Me$, $Q_3$ and the maximum value.
Each Quarter Holds 25% of the Data
The five numbers cut the data into four parts. Each part has about $25\%$ of the values, no matter how long it looks on the plot.
So the box holds the middle $50\%$ of the data. Above $Q_3$ is $25\%$, and below $Q_1$ is $25\%$. Half the data is above the median.
In a box plot representing the time (in minutes) that students take to get to school, the lower quartile is $15$ and the upper quartile is $30$. What exact percentage of…
In a box plot representing the time (in minutes) that students take to get to school, the lower quartile is $15$ and the upper quartile is $30$.
What exact percentage of the students takes between $15$ and $30$ minutes to get to school?
A box plot summarizes the heights of $120$ athletes. If the upper quartile ($Q_3$) is $185$ inches and the maximum value is $205$ inches, approximately how many athletes…
A box plot summarizes the heights of $120$ athletes. If the upper quartile ($Q_3$) is $185$ inches and the maximum value is $205$ inches, approximately how many athletes have a height greater than $185$ inches? Assume there are no repeated values that coincide exactly with the quartiles.
In an engineering company, the $25\%$ of the highest-earning employees have a monthly salary greater than $\$3500…
In an engineering company, the $25\%$ of the highest-earning employees have a monthly salary greater than $\$3500$. In addition, the lowest salary in the company is $\$1200$ and $50\%$ of the workforce earns less than $\$2100$.
Identify which specific parameters of the box plot (Minimum, $Q_1$, $Me$, $Q_3$, Maximum) the values $\$3500$, $\$1200$ and $\$2100$ correspond to.
Spread and Skew in the Plot
Where the median line sits inside the box, and how long each whisker is, tell you the shape of the data.
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Skewed right (a tail toward high values): the box sits to the left, the median is near $Q_1$, and the right whisker is long.
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Skewed left (a tail toward low values): the opposite.
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Roughly symmetric: the median is near the middle of the box and the whiskers have similar lengths.
A short whisker means that quarter of the data is packed into a small interval. A long whisker means that quarter is spread over a wide interval. Likewise, if the distance from $Q_1$ to the median is small and the distance from the median to $Q_3$ is large, then the second quarter is concentrated and the third quarter is spread out. If you drew a histogram of right-skewed data, it would have a long tail on the right.
A box plot shows that the distance between the lower quartile and the median is $2$ units, while the distance between the median and the upper quartile is $10$ units.…
A box plot shows that the distance between the lower quartile and the median is $2$ units, while the distance between the median and the upper quartile is $10$ units.
What does this difference indicate about the distribution and concentration of the data in the central part?
In a box plot, the left whisker (from the minimum to $Q_1$) is very short, while the right whisker (from $Q_3$ to the maximum) is extremely long. What does this mean…
In a box plot, the left whisker (from the minimum to $Q_1$) is very short, while the right whisker (from $Q_3$ to the maximum) is extremely long.
What does this mean regarding the dispersion of the data at the extremes of the distribution?
A box plot has a very narrow box shifted towards the left extreme, with a very long right whisker. If you had to represent this same data in a histogram, what shape…
A box plot has a very narrow box shifted towards the left extreme, with a very long right whisker. If you had to represent this same data in a histogram, what shape would the distribution have? (Symmetrical, right-skewed or left-skewed). Briefly explain why.
The Median Line Is Not the Mean
The mean and the median react differently to extreme values. Look at five salaries (in thousands of dollars): $30, 32, 35, 38, 90$. The median is $35$. The mean is $\dfrac{225}{5} = 45$. The one large value pulls the mean up, but not the median.
So for right-skewed data, the mean is usually greater than the median. For left-skewed data, the mean is usually less than the median. This is a pattern, not a guarantee: a box plot alone does not give you enough information to calculate the mean.
State whether the following statement is true or false and briefly justify your answer: "In any box plot, the arithmetic mean always coincides exactly with the inner…
State whether the following statement is true or false and briefly justify your answer:
"In any box plot, the arithmetic mean always coincides exactly with the inner vertical line that divides the box."
In a box plot of the income of a population, it is observed that the distance from $Q_1$ to the median is $\$500$, while from the median to $Q3$ is $\$2500$. In…
In a box plot of the income of a population, it is observed that the distance from $Q_1$ to the median is $\$500$, while from the median to $Q3$ is $\$2500$. In addition, the upper whisker is much longer than the lower one.
What relationship do you expect there to be between the arithmetic mean and the median (which will be greater)? Reason your answer based on the skewness shown by the box.
Comparing Distributions
The real power of box plots is drawing several on the same number line. Always compare the same three things.
The following graph shows the box plots of the battery life (in hours) of two brands of cell phones, Alpha and Beta. a) Which brand has a higher median for its battery…
The following graph shows the box plots of the battery life (in hours) of two brands of cell phones, Alpha and Beta.
- a) Which brand has a higher median for its battery life?
- b) Which of the two brands shows greater variability in the central $50\%$ of its data (greater interquartile range)?
The following graph shows the execution times (in milliseconds) of three different computer algorithms processing the same amount of data. If you need to choose an…
The following graph shows the execution times (in milliseconds) of three different computer algorithms processing the same amount of data.
If you need to choose an algorithm for a critical real-time system where it is essential that the execution time never exceeds $60$ ms, but at the same time you want the overall performance (median) to be as fast as possible, which algorithm would you choose and why? Justify your answer by ruling out the other two.
Outliers and the 1.5 × IQR Rule
A very long whisker may be hiding one odd value. That value is an outlier, and it is worth plotting separately instead of stretching the whisker to it.
Sometimes an unknown value decides whether there is an outlier. Then you solve an inequality.
In a study on the number of heartbeats per minute of a group of athletes at rest, the following summary was obtained: $Q_1 = 48$, $Me = 52$, $Q_3 = 58$. One of the…
In a study on the number of heartbeats per minute of a group of athletes at rest, the following summary was obtained: $Q_1 = 48$, $Me = 52$, $Q_3 = 58$.
One of the athletes recorded $75$ heartbeats per minute. Using the mathematical criterion of the interquartile range (a value is an outlier if it is greater than $Q_3 + 1,5 \cdot RIC$), determine if this value should be represented as an outlier in the box plot.
Given the following ordered set of data on the reaction time (in milliseconds) of an experiment: $120, 135, 140, 145, 150, 160, 170, x$. Find the minimum value that the…
Given the following ordered set of data on the reaction time (in milliseconds) of an experiment: $120, 135, 140, 145, 150, 160, 170, x$.
Find the minimum value that the unknown $x$ must have to be considered an upper outlier in the box plot. (Remember that for this calculation $N=8$).
What Happens When the Data Changes
Suppose every value in a data set is changed in the same way, like adding a curve to every score, or converting units.
A box plot has been constructed with the maximum temperatures of a city in July, obtaining $Q_1 = 28^\circ \text{C}$, $Me = 31^\circ \text{C}$ and…
A box plot has been constructed with the maximum temperatures of a city in July, obtaining $Q_1 = 28^\circ \text{C}$, $Me = 31^\circ \text{C}$ and $Q_3 = 35^\circ \text{C}$.
If due to a heatwave, the following year all the recorded maximum temperatures increase by exactly $3^\circ \text{C}$, what will be the new values of $Q_1$, $Me$ and $Q_3$? Will the value of the interquartile range change?
A set of data has the following five-number summary: Minimum = $10$, $Q_1 = 20$, $Me = 25$, $Q_3 = 35$, Maximum = $50$. The linear transformation $y = -2x + 100$ is…
A set of data has the following five-number summary: Minimum = $10$, $Q_1 = 20$, $Me = 25$, $Q_3 = 35$, Maximum = $50$.
The linear transformation $y = -2x + 100$ is applied to all the data.
Calculate the new five-number summary for the variable $y$ and determine its new interquartile range. Be especially careful with the order of the quartiles when multiplying by a negative number.
Building Data from a Summary, and What a Box Hides
You can also work backward: given a five-number summary, build a data set that matches it. Use the positions of the quartiles in an ordered list.
For other data sizes, the quartile positions change. With $6$ values, for example, the median is the average of $a_3$ and $a_4$, and each half has three values. Write down which positions give each number before you choose values.
A box plot hides the individual values. Two very different data sets can have exactly the same five numbers.
Set A is $0, 2, 4, 4, 6, 6, 8, 10$ and Set B is $0, 3, 3, 4, 6, 7, 7, 10$. Both have the summary $0, 3, 5, 7, 10$. A has values spread fairly evenly, but B has values bunched around $3$ and $7$. The box plot cannot tell them apart, so when the shape matters, also look at a dot plot or a histogram.
Make up a set of data consisting of exactly $6$ positive integers whose box plot has the following parameters: Minimum = $2$, $Q_1 = 5$, Median = $8$, $Q_3 = 12$,…
Make up a set of data consisting of exactly $6$ positive integers whose box plot has the following parameters:
Minimum = $2$, $Q_1 = 5$, Median = $8$, $Q_3 = 12$, Maximum = $15$.
(Note: Use the standard interpolation rules for the median of the lower and upper halves to calculate the quartiles).
A box plot summarizes the center and dispersion using quartiles, but it can hide the true shape of the distribution. Propose two different sets of data of $8$ elements…
A box plot summarizes the center and dispersion using quartiles, but it can hide the true shape of the distribution.
Propose two different sets of data of $8$ elements each: the first must be a uniform distribution (equally spaced values) and the second a strongly bimodal distribution (values concentrated at the extremes), such that BOTH sets have exactly the same Minimum, $Q_1$, Median, $Q_3$ and Maximum.
Back to the start: Phone A, with batteries lasting 18 to 22 hours, has a box plot with an IQR of only about $4$ hours and short whiskers. Phone B, which swings between 10 and 30 hours, has a wide box and long whiskers. If the averages are the same, the box plots still show that Phone A is far more predictable.