In the ordinary box-plot convention used here, the box extends from first quartile Q₁ to third quartile Q₃, and the line inside the box marks the median Q₂. Read the numerical axis to obtain its value. Its physical position inside the box need not be halfway across the box.

The median is a positional statistic based on ordered observations. The mean is total divided by count. They can coincide, but they are not interchangeable. Some software overlays a separate mean marker; if a legend explicitly identifies such a marker, read it accordingly. Do not assume every extra symbol is the median.

Both lists are ordered and contain seven values. The fourth value is five, so both medians are five. Excluding that middle value, the lower halves are 0, 2, 3 and 0, 2, 4, whose medians are both two. The upper halves have medians eight.

The minimum is zero and maximum ten in both lists. Thus each ordinary min-to-max box plot has minimum zero, Q₁ two, median five, Q₃ eight and maximum ten. This worked example uses a stated convention because quartile algorithms differ. Follow the convention specified in a question rather than assuming every calculator produces identical quartiles.

The first list sums to 34 and the second to 36. Each contains seven observations, so the means are 34/7 ≈ 4.86 and 36/7 ≈ 5.14. Some interior values changed without moving the summary positions. The two box plots are identical under the chosen convention, but the means are different.

This is a direct demonstration of missing information. A box plot compresses a data set into selected summaries. It does not preserve the total. You cannot recover an exact mean merely by averaging the five displayed summary numbers or by taking the middle of the box. Those calculations create other numbers without reconstructing the observations.

For this example, Q₁ = 2, median = 5 and Q₃ = 8. The interquartile range is 8 − 2 = 6. Under an ordinary min-to-max plot, the range is 10 − 0 = 10. These quantities are available directly from the specified summary.

The box describes the central spread, conventionally associated with the middle half of the distribution. With finite samples, ties and different quartile definitions, do not turn every segment into an exact count without checking the data and convention. A long segment shows a wider value interval, not automatically more observations.

Not always. An ordinary min-to-max box plot uses extrema as whisker endpoints. A modified box plot may draw whiskers to the most extreme observations within a rule and plot outliers separately. A common rule uses fences Q₁ − 1.5 IQR and Q₃ + 1.5 IQR.

The fences are thresholds, not necessarily the actual whisker positions. A whisker usually ends at a qualifying observed value under that modified convention. Read the task or legend before calling a whisker endpoint the maximum. Our original seven-value example explicitly uses the ordinary min-to-max form, so that ambiguity does not affect its comparison.

Compare corresponding statistics on a common numerical scale. A higher median indicates a higher middle observation level under the given interpretation. A smaller IQR indicates a narrower central spread. Neither statement alone establishes which data set has the larger mean.

Also check what the measurements mean. For completion times, smaller may be faster; for a score, larger may indicate more points. A statistically correct comparison should name the quantity instead of saying one plot is simply better. Avoid claiming that every individual in one group exceeds every individual in another because their medians differ.

Question 1: a box plot labels Q₁ = 12, median = 18 and Q₃ = 25. Find the median and IQR. Question 2: can its mean be read exactly from those three labels alone? Question 3: if an ordinary plot also has minimum 4 and maximum 40, find the range. Question 4: must a longer box contain more observations than another group’s shorter box?

Answers: 1) Median eighteen and IQR thirteen. 2) No; the total and observations are not determined by those labels. 3) Range thirty-six. 4) No; box length measures value spread, while counts also depend on sample sizes and conventions. Ask your child to explain what information the graph supplies, not merely to calculate a subtraction.

Bring the plot and the sentence your child wrote about it. A note such as ‘calls the median line the mean’ identifies a terminology and interpretation issue. In a small-group 3-pax lesson, students can construct the two original data sets above and see why the same plot does not fix the total.

For parents considering Secondary 4 Mathematics tuition in Punggol, the next-question routine is to read the axis, identify the plotting convention, name the statistic and state only what the summary supports. If the mean is not shown or otherwise determined, saying so is a sound mathematical conclusion rather than an incomplete attempt.