eduKateSengkang · Family learning
Find your next learning step
Choose the route closest to your question. Read one worked example, try a fresh case and use the explanation to check the decision behind your answer.
Route 3 · Chapters 10–15
Distinguish centre, spread, whiskers and overlap.
Full chapter index · Start the diagnostic · Continue at the learning hub
A box plot can show whether two sets of results differ in their typical values or in how much they vary. It cannot tell you who recorded each result, what the mean was, or why one group differed from another. This guide starts with a few original lists you can sort at home, then shows how to draw a box plot, compare two plots on one scale and write conclusions that the evidence supports.
Cumulative frequency for families · Secondary 3 formal guide to box plots and other displays
Choose a chapter
Sort and locate the middle · Chapters 1–4
Build the box plot · Chapters 5–9
Compare what the plot shows · Chapters 10–15
Check conclusions · Chapters 16–20
Test and explain · Chapters 21–24
CHAPTER 1 OF 24 · Sort and locate the middle
1. Start with the question a box plot answers
Imagine two families time how long it takes to set out a board game on eight occasions. The full lists are useful when somebody asks which attempt was slowest. A box plot answers a different question: where are the lower, middle and upper parts of the ordered results? It keeps five markers rather than drawing a mark for every attempt. That makes two batches easier to compare, especially when the lists get longer.
Suppose the first batch was recorded as 11, 7, 9, 15, 8, 8, 10 and 12 minutes. Write the times on separate slips and sort them from least to greatest. Keep both slips marked 8. Two attempts took the same time, so both count. Removing one would change the size of the sample and can change the middle and quartiles. Sorting changes the order in which you read the results, but not the values recorded.
Ask the learner to distinguish the observation from the unit: one observation is one attempt, and the unit is minutes. “The median is 9.5 minutes” describes a duration. “There are eight observations” counts attempts. Keeping those sentences separate helps later when several attempts sit near one marker on the scale. It also stops a learner from taking the box to represent eight minutes just because the list has eight entries.
For the setup times, the ordered list is 7, 8, 8, 9, 10, 11, 12, 15. A learner who reads the centre of the original list sees 15 and 8, but those were merely in the fourth and fifth positions in the recording order. They do not tell us where the centre is by size. Use the ordered list to find that centre, and retain the original list if sequence is also useful.
One quick sorting method is to copy the data into a column, cross off each value as it is used, and put a mark beside each crossed-off entry. When there are equal values, cross off one occurrence at a time. Compare the length of the sorted list with the original length. An error check is to read the sorted list back and ask whether any value falls below the one immediately before it.
Suppose the last number was copied as 51 instead of 15. The list could still be perfectly sorted. Sorting does not check whether the measurements were copied accurately. Compare back with the record before arguing that the extreme value is suspicious. A box plot sometimes draws attention to an error, but it does not diagnose that error. One purpose of a table beside a practice plot is to show exactly what was plotted.
CHAPTER 3 OF 24 · Sort and locate the middle
3. Find the median when the list has an even length
With eight ordered values there are two central positions, fourth and fifth. In the setup example those hold 9 and 10. Their average is 9.5 minutes, which is the median under the usual convention for an even list. Nobody has to have taken exactly 9.5 minutes. The marker describes the position halfway between the two central observations.
A common first attempt is to add all eight values and divide by eight. That gives the mean, a different quantity. Here the total is 80, so the mean is 10 minutes. Compare what happens if 15 is replaced by 23 while the other seven values remain the same. The mean becomes 11 minutes; the two central values are still 9 and 10, so the median stays at 9.5. This is a reason to name the measure when reporting what is typical.
For a transfer check, give the learner 3, 4, 6, 9, 10, 12. Ask which positions determine the median before calculating it. They are the third and fourth, with values 6 and 9, giving 7.5. Averaging the first and last, 3 and 12, also gives 7.5 here by coincidence. Replace 12 with 16: the correct median stays 7.5 while the midpoint of the extremes moves to 9.5.
CHAPTER 4 OF 24 · Sort and locate the middle
4. Find quartiles by stating the convention
In this guide, when finding quartiles from a small list, take the median of the lower half for Q1 and the median of the upper half for Q3. For an odd number of observations, leave out the overall middle observation from those halves. School questions, software and textbooks can use different conventions; follow the one given. Being explicit is especially useful in a small data set, where different conventions can make noticeably different markers.
Split the eight setup times into 7, 8, 8, 9 and 10, 11, 12, 15. The median of the lower four is 8. The median of the upper four is 11.5. The overall median, 9.5, is not placed into either half as an extra observation. It came from the middle positions, so adding it would alter the number of observations used for each quartile.
Now sort an odd list: 4, 5, 6, 8, 9, 10, 14. The overall median is 8. The lower half 4, 5, 6 gives Q1 = 5. The upper half 9, 10, 14 gives Q3 = 10. Some methods include the 8 in both halves or interpolate using prescribed positions. Before disagreeing with a calculator, compare the method rather than assuming that one person misread the list.
In a standard box plot with whiskers at the observed extremes, the five-number summary runs minimum, Q1, median, Q3 and maximum. The setup data give 7, 8, 9.5, 11.5 and 15 minutes. Each label corresponds to a marker on the same measurement axis. The first and last are recorded observations. A quartile or median can fall between observations because of how the method works.
If a learner writes 7, 9.5, 8, 11.5, 15, ask whether the median can sit outside the interval from Q1 to Q3. The ordering check catches the swap without having to recalculate the whole list. An equal pair is possible: if all four values in the lower half are 8, then the minimum and Q1 are both 8. The rule is nondecreasing order, so successive numbers can coincide.
Practise labelling the five positions for the odd list in the previous chapter: minimum 4, Q1 5, median 8, Q3 10, maximum 14. Now ask what changed if only the maximum becomes 20. It is the same list apart from its last observation, so with this quartile method the interior markers remain the same. That distinction is a preview of how changing one extreme can affect one measure of spread without affecting another.
For the setup data, choose a number line from 6 to 16 minutes with equally spaced one-minute ticks. Mark 7 and 15 for the endpoints. Mark 8 and 11.5 for the box edges and 9.5 for the line within the box. Put each half-minute value halfway between adjacent ticks. If you mark all five values equally far apart just because the labels follow an order, the plot gives false information about how the values are spaced.
Use squared paper if possible. Decide how many grid squares represent one minute, then keep that rule throughout the plot. If two squares represent one minute, the right box edge at 11.5 belongs one square after 11. The physical length of a gap on the paper is useful only because the units are consistent. Label the axis “Setup time (minutes)” rather than only “time” or “attempts.”
Try making a second axis beginning at 0. Both plots can correctly show the same gaps if the scales are uniform, but the unused space looks different. A displayed zero is often useful for context; it is not required to find the five marker values. If you compare two data sets, plot them against the same axis, so a one-minute gap is the same physical width for both. Do not use a separate scale for each row.
CHAPTER 7 OF 24 · Build the box plot
7. Build the box, the median line and the whiskers
Draw the box from Q1 to Q3 above the axis. Its vertical height is a drawing choice. The horizontal extent records data values. Draw a line across the box at the median, not halfway across by default. For the setup example the midpoint of the box is 9.75, whereas the median is 9.5. A small difference may look negligible in a rough sketch but matters if you label exact values.
Draw the lower whisker from Q1 back to the minimum and the upper whisker from Q3 out to the maximum. Add end marks if your drawing convention uses them. The whiskers need not have the same length. Here the lower whisker covers 8 − 7 = 1 minute and the upper covers 15 − 11.5 = 3.5 minutes. The numbers tell you how far the measurements extend, not how many attempts are on each bit of line.
Ask the learner to create a plot from 2, 5, 6, 8, 8, 8 and 12. The median and Q3 coincide. Mark both labels at the right edge of the box. Here the lower half has median 5 and the upper half has median 8. There is no requirement to draw a second line offset from the box merely to make it visible. The geometry follows the measurements; where labels need room, use a small annotation rather than moving the marker.
CHAPTER 8 OF 24 · Build the box plot
8. Read values off the axis and explain the units
To read a printed plot, point to a marker and project it to the measurement axis. Read the tick labels before estimating a value between them. If the axis marks 0, 5, 10 and 15 with equal spacing, a median halfway between 5 and 10 is 7.5. It is not the sixth value merely because it lies close to a sixth visible grid mark. The number line gives measurements, not list positions.
A plot with thick lines or widely spaced ticks may only support a rounded reading. If the median seems just beyond 7.5, writing “about 8 minutes” can be more defensible than “7.83 minutes.” Use an exact value when supplied by the question or clearly indicated on the scale. Give the same care to the units in both cases.
Compare two printed summaries: one labels a time in seconds and the other in minutes. A median of 48 seconds is 0.8 minutes; it is not longer than a median of 2 minutes. Re-express each marker on the same unit before plotting them together. This applies to spreads too. A gap of 30 seconds represents half a minute no matter where its endpoints lie. A worksheet that mixes units can test whether the learner is reading the quantities rather than comparing the printed digits.
CHAPTER 9 OF 24 · Build the box plot
9. Distinguish range from the interquartile range
The range is maximum minus minimum. For the setup data, it is 15 − 7 = 8 minutes. The interquartile range, or IQR, is Q3 minus Q1. It is 11.5 − 8 = 3.5 minutes. A box plot makes both differences visible: the range spans the end markers in this convention, and the IQR is the width of the box along the measurement axis.
Replace the maximum setup time 15 with 23. The range becomes 23 − 7 = 16 minutes. The IQR stays 3.5 minutes because the quartiles do not change in that example. This makes the two measures useful for different questions. The range tells how far apart the most extreme measurements are. The IQR describes the width between the quartiles, so one large extreme may leave it unchanged.
Ask what a notice could safely report if two groups have identical IQRs of 3 minutes. They have the same spread between their own first and third quartiles. That does not mean the groups have the same median, full range or maximum. A useful conclusion names the statistic and its interpretation: “The middle spread was the same in both groups, but their typical setup times differed.” Check the medians before including that last clause.
Suppose a simple plot has a lower whisker from 2 to 4, a box from 4 to 11 and an upper whisker from 11 to 12. The lower whisker covers more units than the upper whisker. You cannot conclude that the lower whisker contains exactly twice as many observations. The lengths measure how far apart the endpoints are along the measurement axis.
Quartiles divide ordered data by position, so people often speak of four roughly equal parts. Exact counts above, below or within a marker depend on sample size, the rule used to find the quartiles and ties at the markers. If Q1 is 4 and five observations are all equal to 4, a question about values strictly less than 4 is different from a question about values at most 4. The plot does not identify all the observations at 4.
Make two lists of eight observations that include the same extremes but differently spaced values. Ask where the first four positions lie on each axis. In one list, the first four can span two units; in the other, they can span six. The number of positions is the same while the range of their values differs. This separates a visual comparison of widths from a counting question that needs the original list.
CHAPTER 11 OF 24 · Compare what the plot shows
11. A long whisker suggests an extension, not a complete shape
Consider summaries A = 2, 4, 5, 6, 13 and B = 2, 4, 5, 6, 8. They share a box and a lower endpoint. A extends farther above Q3 than B. Say “A has a longer upper whisker and a larger range.” If the context is measurement duration, that means A includes a slower extreme. The summaries alone do not show how many values are close to 13 or where any gap lies.
Some courses use words such as right-skewed when the longer tail extends toward higher values. Treat a long upper whisker as a clue, especially when the box and median also support the description. It is not a substitute for seeing the raw observations or another display. Two very different distributions can share a five-number summary.
Have the learner imagine that one batch with a maximum of 13 had a single slow attempt, while another had several slow attempts. Ask for a display that could help distinguish those possibilities. A dot plot of the raw values would make repeated measurements visible. A box plot is useful because it compresses information; the same compression explains why it cannot provide every detail. Choose a display based on the question you need to answer.
Look at the question a shape claim is meant to answer. A family wants to know whether most trips took less than ten minutes. Calling the plot right-skewed does not answer that. If the median is 11, at least about half the trips reach the middle level or higher, so “most took less than ten” is not supported. If the median is 8, the plot supports a lower central value, but the exact count under ten still needs the data. A named shape is a description of the display rather than permission to invent a frequency.
CHAPTER 12 OF 24 · Compare what the plot shows
12. Know which whisker convention is being used
This article uses minimum-to-maximum whiskers unless stated otherwise. Some modified box plots use whiskers that stop at the most extreme observations within specified limits and draw more distant values as separate points. A common rule uses fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. The whisker ends at an observed value, not automatically at the fence. Read the caption or the problem before interpreting the ends.
For example, suppose Q1 = 10 and Q3 = 14. The IQR is 4, so the upper fence under that rule is 20. If the observations above Q3 are 15, 18 and 23, the upper whisker would end at 18 and 23 would be drawn separately. Under the simple convention the upper whisker would end at 23. The two pictures differ even though the underlying data do not.
A flagged value is a reason to look at the record. It might be a valid observation, a measurement under different conditions or a copying mistake. Do not drop it just because the software marks it. If you remove a verified recording error, explain the reason and report which data were used. A family exercise can focus on reading the legend and distinguishing the fence, the whisker endpoint and the overall maximum.
CHAPTER 13 OF 24 · Compare what the plot shows
13. Compare medians on the same scale
Suppose setup method A has five-number summary 5, 7, 8, 10, 14 and method B has 4, 6, 7, 8, 10, in minutes. The median of B is one minute lower. A precise statement is “B had a lower median setup time in these observations, 7 minutes against 8 minutes.” That does not mean every use of B was faster than every use of A.
Test the stronger claim against the endpoints. B includes an attempt that took 10 minutes. A includes an attempt that took 5 minutes. Thus at least one B attempt was slower than at least one A attempt. The fact that B has a better median cannot override those examples. A learner who knows the exact values can reject the overclaim without guessing about everything between them.
Ask what would be needed to say that every attempt for B was faster than every attempt for A. In this simple whisker convention, the maximum for B would have to be strictly below the minimum for A. If those extremes were equal, “every B attempt was at most as long as every A attempt” would be possible, but “strictly faster” would fail for the equal pair. The wording of a comparison can turn on one boundary.
CHAPTER 14 OF 24 · Compare what the plot shows
14. Compare spread without confusing it with performance
In the A and B example, IQRs are 10 − 7 = 3 minutes and 8 − 6 = 2 minutes. B has a smaller spread in the middle of the ordered data. A range is 9 minutes and B range is 6 minutes, so the full observed spread is smaller for B as well. Say which measure you are using when describing one method as more consistent. Sometimes the IQR and range point in opposite directions.
Try C = 3, 7, 8, 9, 20 and D = 5, 6, 8, 10, 13. They share a median of 8. C has IQR 2 and range 17. D has IQR 4 and range 8. A person who says C is “more variable” is using the range; a person who says D is “more variable” may be using the IQR. Both can make a supported statement if they specify the statistic. An unqualified ranking hides the difference.
Return to a decision about planning setup time. If you want an estimate of a typical setup, use the median. If you need to allow for observed worst cases, check maxima and ask whether the conditions were comparable. The purpose of the decision helps determine which part of the plot deserves emphasis. A lower IQR by itself says nothing about which method is usually faster.
CHAPTER 15 OF 24 · Compare what the plot shows
15. Look at overlap without inventing percentages
For A = 5, 7, 8, 10, 14 and B = 4, 6, 7, 8, 10, the boxes overlap from 7 to 8 minutes. This shows that the ranges between the first and third quartiles have a shared interval. It does not tell us exactly how many attempts from each group took values in that interval. One group could have many observations at 7; another could have few there. A continuous box edge is not a record of every value in the space it covers.
If two boxes do not overlap, it is reasonable to say their middle ranges are separated. The whiskers can still overlap, so individual observations may cross. For example, summaries E = 1, 3, 4, 5, 9 and F = 4, 6, 7, 8, 11 have separated boxes but ranges that overlap from 4 to 9. E includes a value of 9, whereas F includes a value of 4. A statement about all individual comparisons would fail.
Ask the learner to draw both examples with a common number line. Use one colour to mark box intervals and a second to mark full ranges. Explain the difference in one sentence. Avoid asking what percentage of individual values is in the overlap unless raw data or specific counts are given.
CHAPTER 16 OF 24 · Check conclusions
16. Identical boxes can hide different samples
Take list X = 2, 4, 6, 7, 9, 10, 12, 15 and list Y = 2, 5, 5, 8, 8, 11, 11, 15. Under the convention used here, both have Q1 = 5, median = 8 and Q3 = 11, and both have extremes 2 and 15. In X, no observation is exactly 8; in Y, two observations are exactly 8. The plots are identical, and the raw lists still answer questions differently.
Count observations strictly above 10. For X they are 12 and 15; for Y they are 11, 11 and 15. The plot cannot reproduce those counts. If someone needs to know how many attempts exceeded a ten-minute time limit, keep or ask for the raw data. If all you need is a compact comparison of the centre and quartile spread, the box plot can still be the right choice.
The samples need not have the same size either. Repeat every observation of X twice to make sixteen observations. With this example the five marks stay the same under our convention. A plot of ordinary fixed box height would not show that change. Write the sample size beside each plot if it matters for the reader’s interpretation. Do not infer sample size from the height of an ordinary box.
One striking consequence is that the mean happens to be fixed by the five numbers in these two eight-value examples. Under this specific quartile rule, the interior observations form three pairs; their sums are twice Q1, twice the median and twice Q3. Add those sums to the minimum and maximum to recover the total. This shortcut fails as a general box-plot rule because other sample sizes leave observations outside the pairs that determine the markers. It is useful to explain the exception before offering a longer counterexample.
CHAPTER 17 OF 24 · Check conclusions
17. Compare fairly before making a recommendation
A family compares assembly times for two paper models. Model A was timed in a quiet room, and model B was timed at a busy table where tools were shared. A lower median for A is a valid descriptive claim about the observations. The setup does not isolate the effect of the model design. Quiet, access to tools and previous practice could also influence the times. A box plot cannot repair those differences.
A fairer small comparison might have the same people try both designs with the same supplies. If everyone tries A first, practice or tiredness might affect B. Alternating the order is one way to reduce that concern. Keep the question modest: “Which design took less time under these test conditions?” A family need not conduct a full study to recognise the difference between describing a result and explaining its cause.
Sometimes the data are paired: the same child tries both methods. Two box plots summarise the times separately and discard the pairings. A table of each child’s two times, or a plot of the difference for each pair, helps answer whether most children were faster with one method. Ask whether the conclusion is about whole groups or within-person changes before choosing a display.
A learner writes “The right whisker is longer, so more people took longer.” Ask which part is directly visible. The whisker covers a larger measurement interval. The number of people in any specified interval needs separate evidence. Give the student a pair of summaries with the same upper quartile and different maxima, and ask them to make a claim that mentions only those markers.
Another learner writes “A has a smaller box, so A is faster.” The smaller box shows a smaller IQR, not necessarily a lower median. Try A = 9, 10, 11, 12, 13 and B = 2, 4, 7, 8, 14. A has IQR 2, while B has IQR 4. A’s median is nevertheless higher, so its typical time is longer in this example. The first error was choosing a spread measure to answer a centre question.
Use the table as a check after reading a conclusion. Point to the statistic, state the unit and name the claim it supports. Then identify one claim that would need more data. This helps the learner recover from a specific mistake rather than simply repeating the drawing. If a copied summary puts Q1 above the median, return to the ordered list and check which numbers were taken from which half.
| Question | Statistic or evidence | What it supports |
|---|---|---|
| What is the central time? | Median | A central position in the ordered times |
| How wide is the middle? | Q3 − Q1 | Spread between the quartiles |
| How far apart are the observed extremes? | Maximum − minimum | Full observed range |
| How many exceeded a threshold? | Raw list or threshold counts | An exact count under a stated boundary |
CHAPTER 19 OF 24 · Check conclusions
19. Independent practice: build and read a plot
1. Nine attempts to fold a card took 11, 6, 7, 9, 10, 5, 8, 12 and 15 seconds. Sort the values. Find the five-number summary using this guide’s convention. Mark the five values on one axis from 4 to 16 seconds. Then give the range and IQR with units.
2. In the sorted data from question 1, replace the maximum with 22. Find all five markers again. Which measure changes: range, IQR or both? Explain from the positions instead of listing two calculations alone.
3. A second card design has minimum 4, Q1 6, median 7, Q3 10 and maximum 13 seconds. Compare both original designs on the same axis. Which has the lower median? Which has the smaller IQR? Can you claim that all attempts for the second design were faster than all attempts for the first?
4. A classmate puts the lower and upper whisker ends equally far from the box because they think the plot must look symmetrical. Use question 1 to show how far each whisker should actually extend. Explain why visual balance is a poor drawing criterion here.
5. The first design’s Q1 is between two adjacent ticks on an axis labelled at whole seconds. Explain how to place it. What does its position say about whether somebody recorded that exact duration?
1. The ordered values are 5, 6, 7, 8, 9, 10, 11, 12, 15. Median 9 is omitted from the two halves. The first four give Q1 = (6 + 7) ÷ 2 = 6.5; the last four give Q3 = (11 + 12) ÷ 2 = 11.5. The box extends from 6.5 to 11.5, with the line at 9. Whiskers reach 5 and 15. Range is 10 seconds; IQR is 5 seconds.
2. The five values are 5, 6.5, 9, 11.5 and 22. The last observation changes, but neither of the two central positions of the upper half is the last position. The IQR stays 5 seconds. The new range is 22 − 5 = 17 seconds. On a shared plot choose a scale that includes the new extreme rather than squeezing it into the old range.
3. Design 2 has lower median, 7 rather than 9 seconds, and smaller IQR, 10 − 6 = 4 rather than 5 seconds. Its maximum of 13 exceeds design 1’s minimum of 5. Those two observed extremes show that not every design 2 attempt was faster than every design 1 attempt.
4. Lower whisker length is 6.5 − 5 = 1.5 seconds. Upper whisker length is 15 − 11.5 = 3.5 seconds. Symmetry would give a false picture. The two gaps must be drawn to scale regardless of whether the plot looks balanced.
5. Place Q1 halfway between 6 and 7. Q1 comes from averaging those two observations in this problem. None of the nine recorded times was 6.5, and a box plot does not promise that each summary marker is a measurement from the list.
1. Groups P and Q timed a fictional clean-up task. P has summary 3, 5, 8, 11, 13 minutes; Q has 2, 6, 8, 9, 18 minutes. Write two valid comparisons of spread and one valid comparison of centre. Is it accurate to say Q is less spread out without naming a measure?
2. A modified box plot uses the 1.5 × IQR rule. Its Q1 is 8 and Q3 is 12. The observed values above Q3 are 13, 16, 18 and 20. Find the upper fence. Identify the upper whisker endpoint and any separate point under the stated rule. Explain what remains to be checked about that point.
3. A learner claims that two samples with the same box plot must have the same mean. Compare these two ordered lists under the guide’s convention: 2, 4, 6, 7, 9, 10, 12, 15 and 2, 5, 5, 8, 8, 11, 11, 15. Check the claim. Then replace the second value 5 in the second list by 4 and its third value 5 by 6; instead of guessing, calculate whether that change affects the mean.
4. Try replacing the middle pair 7 and 9 in the first eight-item list by 8 and 8. Can that change the mean while keeping the median? Then use the longer list 1, 2, 4, 4, 6, 7, 8, 9, 10, 11, 11, 13, 14. Change one observation so the summary stays the same and the mean changes. Keep the list in ascending order.
1. Both medians are 8 minutes. P’s IQR is 11 − 5 = 6 minutes and Q’s is 9 − 6 = 3 minutes. P’s range is 13 − 3 = 10 minutes and Q’s is 18 − 2 = 16 minutes. Q is less spread out between its quartiles but has a wider full observed range. The unqualified claim hides this split.
2. IQR is 12 − 8 = 4, so the upper fence is 12 + 6 = 18. The value 18 is on the fence and is eligible as the whisker endpoint under the usual inclusive rule. The 20 is plotted separately. Check the original recording and the conditions of that observation before treating it as an error.
3. Both lists sum to 65, so their means happen to match at 8.125. That agreement does not establish the claim. Replacing the pair 5 and 5 by 4 and 6 leaves its sum and the total unchanged. The midpoint of the pair, Q1, also stays unchanged. This particular attempt cannot produce different means.
4. The change to positions 4 and 5 keep their sum equal to 16, and so keep both median and mean. In an eight-item list using this method, the three marker pairs together with the endpoints actually account for every position. Use a longer list. For thirteen values with the overall median omitted, compare 1, 2, 4, 4, 6, 7, 8, 9, 10, 11, 11, 13, 14 with the same list except that the second value is 3. The quartile pairs and extremes do not change. The total rises by 1 and the mean by 1/13, so this pair disproves the general claim.
Choose a harmless measurement you can repeat, such as timing how long it takes to arrange six cards in order. Agree on when the timer starts and stops before making observations. Try one method several times and write every time down. Try another method under similar conditions. Label the two lists and keep their sizes visible. You can do the arithmetic on paper with a small sample.
After plotting, ask the learner for three sentences. First compare the median. Second compare the IQR. Third give one limitation. “Method B had a lower median in our trials, its middle spread was wider, and we did not vary who timed the attempts” is more useful than simply “B won.” If the observations are too few for a strong general conclusion, say that.
Then swap the task: show an unfamiliar pair of plots without the raw lists. Ask the learner to point to the markers behind each claim. When they ask a question about exact counts, mean or individual changes, ask what extra information would answer it. The goal is to move between raw observations, summaries and careful statements. Once that is comfortable, use the linked cumulative frequency guide to explore how a running total supports a different kind of question.
If the two learners disagree about an example, separate the calculation from the reporting decision. One can re-sort and recompute the five markers without seeing the other learner’s work. The other can inspect the labels, axis and wording without recalculating. Compare notes afterwards. A correct marker can still be drawn in the wrong place, and a correct plot can still be paired with a claim about causes that the trials never tested. Recording the specific error makes the next round of practice shorter and more useful.
Does the line in a box always mark the mean? No. The standard line inside the box marks the median. Some software can add a mean marker, so read the legend if you see a second mark. If you need the mean and only have a box plot, ask whether the raw values or the sum and sample size are available.
Can the box or whisker disappear? A box may have zero width if Q1 and Q3 are the same. A whisker may have zero length if its endpoint is at the corresponding box edge. A coincident marker can be hard to see without a label. Check the data and labels before deciding the drawing is missing information.
When should we move on? A learner is ready for a comparison question when they can label the five numbers, use one measurement axis and explain the difference between centre and spread. If the wrong quartiles keep appearing, revisit ordered positions. If claims overreach, ask for the raw value or extra evidence the claim needs.
A formal school guide can connect box plots with cumulative frequency and other displays. Use the stated convention for quartiles in each task. Bring one annotated plot and one comparison sentence to a lesson if help is needed; they show where the difficulty is, whether it lies in reading the markers, calculating a difference or making a fair claim.
References and further reading
The worked scenes and practice sets are original teaching examples. These references offer formal definitions or documented usage; the connected eduKate guides provide foundations and further study.
OpenStax: box plots · Cumulative frequency for families · Secondary 3 formal guide to box plots and other displays
