Five Number Summary: Complete R, Python, SPSS and Excel Guide

Specific interpretation: The G3 final grade boxplot shows a minimum of 0, Q1 of 10, median of 12, Q3 of 14 and maximum of 19. The box begins at Q1 and ends at Q3, so the middle 50% of final grades are between 10 and 14. The median line at 12 sits inside the box and shows that half of students scored 12 or below while the other half scored 12 or above.

The lower tail is important. Even though most final grades cluster between 10 and 14, some scores fall at or near 0. These very low values appear as unusual observations compared with the central grade pattern. The upper end reaches 19, showing that high-performing students exist, but the upper tail is not as visually extreme as the low-score points.

Why this chart matters: This chart gives a complete distribution summary in one visual. It is more informative than reporting only the mean because it shows center, spread and possible outliers at the same time.

Reporting decision: Report G3 as median = 12, Q1 = 10, Q3 = 14, IQR = 4, minimum = 0 and maximum = 19. Also mention that a small number of very low scores create lower-end unusual values.

Specific interpretation: The G1, G2 and G3 boxplots show that the three grade variables have broadly similar distributions, but G3 has a slightly higher median and a wider middle spread. G1 and G2 both have median values around 11, while G3 has a median of 12. This suggests that the final grade distribution is slightly shifted upward compared with earlier grade periods.

The boxes for G1 and G2 are similar, with Q1 around 10 and Q3 around 13. G3 has Q1 around 10 and Q3 around 14, producing a wider IQR. This means final grades show slightly more spread in the middle 50% of students. The presence of low outliers near 0 across the grade variables indicates that a small number of students performed very poorly or had zero recorded outcomes.

Why this chart matters: It compares three related grade variables at once. Instead of looking at separate tables, the boxplot shows that G3 is not only centered slightly higher but also has more middle spread than G1 and G2.

Reporting decision: State that G3 has the highest median among the three grade variables and a slightly larger IQR, while all three variables show some low-end unusual values.

Specific interpretation: This chart separates the five values for G1, G2 and G3 into clear panels. For G1, the values are minimum 0, Q1 10, median 11, Q3 13 and maximum 19. G2 shows the same five-number pattern: minimum 0, Q1 10, median 11, Q3 13 and maximum 19. G3 changes slightly, with minimum 0, Q1 10, median 12, Q3 14 and maximum 19.

The chart makes the grade progression easy to see. The minimum and maximum stay the same across G1, G2 and G3, but the median and Q3 increase in G3. This means the final grade distribution is not simply wider; its central location also moves upward. The increase from median 11 to median 12 is small but meaningful in a student-performance context because it represents a shift in the middle student outcome.

Why this chart matters: Boxplots are useful, but some readers find the exact five values difficult to read from a boxplot alone. This chart labels each value directly, making it ideal for reports, teaching and interpretation.

Reporting decision: Report that G1 and G2 share the same five-number structure, while G3 improves at the median and upper quartile level, with median 12 and Q3 14.

Specific interpretation: The IQR comparison chart shows how much the middle 50% of each numeric variable varies. Absences has the largest IQR, with an IQR of 6. This means the central spread of absences is much wider than most other numeric variables in the dataset. G3 has an IQR of 4, while G1 and G2 each have an IQR of 3.

Several variables, such as failures, traveltime, studytime, famrel, freetime and Dalc, have small IQR values. A small IQR does not automatically mean the variable is unimportant; it means that most students are concentrated in a narrower middle range. For variables measured on small ordinal scales, a small IQR is common because the available response categories are limited.

The grade variables show moderate central spread. G3 is more variable than G1 and G2 in the middle 50%, while absences is the most spread-out variable. This supports the earlier descriptive-statistics finding that absences is highly variable and likely skewed.

Why this chart matters: IQR comparison helps identify which variables have the most robust spread without being overly influenced by extreme values. It is especially useful before choosing transformations, outlier handling or nonparametric summaries.

Reporting decision: State that absences has the greatest middle-spread variability, while G3 has a larger IQR than G1 and G2. Use the IQR instead of only standard deviation when discussing skewed variables such as absences.

Specific interpretation: The G3 distribution chart places the five number summary directly on the histogram. The minimum line is at 0, Q1 is at 10, the median is at 12, Q3 is at 14 and the maximum is at 19. Most students are concentrated between 10 and 14, which matches the IQR result. The highest bars occur around the lower-middle final grade range, showing that many students scored close to the central grade values.

The chart also shows why the minimum value must be interpreted carefully. The minimum is far away from the main cluster of the data. A minimum of 0 is part of the five-number summary, but it does not represent the typical student. The typical student is better represented by the median and quartiles. This is exactly why five-number reporting is stronger than only reporting the range.

Compared with a simple boxplot, the histogram adds frequency information. It shows where the observations are densest, while the vertical lines show the exact summary boundaries. This combination helps readers understand both distribution shape and summary statistics.

Why this chart matters: It connects the numerical five-number summary to the actual distribution. Readers can see that Q1, median and Q3 fall inside the dense part of the distribution, while the minimum is separated from the main cluster.

Reporting decision: Report the five-number summary and add that the central mass of G3 scores lies between 10 and 14, with a small number of low-end unusual scores near 0.

Specific interpretation: The absences chart is a strong example of why the five number summary is useful for skewed data. Absences has Q1 = 0, median = 2 and Q3 = 6. This means many students have very few absences, and at least 25% of students have zero absences. The middle student has about 2 absences, while 75% of students have 6 or fewer absences.

The upper outlier fence is 15. This is calculated using the formula Q3 + 1.5 × IQR. Since Q3 is 6 and the IQR is 6, the upper fence is 6 + 1.5 × 6 = 15. Any absence value above 15 is considered unusually high by the 1.5 × IQR rule. The chart shows several values above this fence, including very high absence counts around the 20s and 30s.

This pattern indicates a right-skewed variable. Most students have low absences, but a small group has very high absence counts. The mean would be pulled upward by these high values, so the median and IQR are more reliable for summarizing the typical absence pattern.

Why this chart matters: It demonstrates practical outlier detection. The chart does not simply label high values as unusual; it shows the statistical fence used to identify them.

Reporting decision: Report absences using median and IQR rather than mean alone. A good statement is: absences were right-skewed, with median = 2, Q1 = 0, Q3 = 6, IQR = 6 and several high outliers above the upper fence of 15.

Specific interpretation: The school-level boxplot compares G3 final grade distributions for GP and MS. GP has a median around 13, while MS has a median around 11. This indicates that the typical final grade is higher for GP students than for MS students in this dataset. The difference in medians is easy to see because the median label is placed directly inside each box.

The GP box is positioned higher than the MS box, showing that not only the median but also much of the central distribution is shifted upward for GP. MS has a lower central location, and the box extends across a lower grade range. Both groups show some low values, but the overall grade pattern favors GP.

The chart also shows why a group comparison should not rely only on averages. A mean difference can be useful, but the boxplot shows the full distribution: median, quartiles, whiskers and unusual values. This helps determine whether the group difference is broad across the distribution or driven only by a few cases.

Why this chart matters: It provides a distribution-level comparison of final grades by school. It supports later group-comparison work such as effect size analysis, t tests or nonparametric comparison.

Reporting decision: State that GP students show a higher G3 median than MS students, with GP median = 13 and MS median = 11. This suggests better typical final-grade performance in GP for this sample.

Specific interpretation: The sex-based boxplot compares G3 final grades for female and male students. Female students have a median of 12, while male students have a median of 11. This shows a slight difference in typical final-grade performance, with female students having a somewhat higher central value.

The boxes overlap considerably, so the difference should be interpreted as modest rather than large. Female students show a central range that extends higher, while male students show more low-end unusual values. However, because both groups share a wide portion of the grade distribution, this chart should be interpreted as a descriptive comparison rather than proof of a large group gap.

This chart is also helpful because it separates central tendency from variability. A one-point median difference can be meaningful in context, but the overlap reminds us that many male and female students have similar grades. The distribution matters as much as the median.

Why this chart matters: It shows whether the sex difference is visible across the distribution, not only in the mean. It is a useful companion to effect-size charts such as Cohen’s d or Hedges’ g.

Reporting decision: State that female students have a slightly higher G3 median than male students, but the distributions overlap, so the practical difference appears modest.

Specific interpretation: The study-time boxplot compares final grades across four study-time categories. Students studying less than 2 hours have a median around 11. Students studying 2 to 5 hours have a median around 12. Students studying 5 to 10 hours and more than 10 hours both have medians around 13. This shows a positive descriptive pattern: higher study-time categories tend to have higher typical final grades.

The lower study-time group has more low-end unusual values, including scores close to 0. The 5 to 10 hours group has a higher central location and appears more concentrated around stronger grades. The more than 10 hours group has a similar median to the 5 to 10 hours group but shows a wider spread, which may be due to smaller group size or more mixed student outcomes.

This chart should not be interpreted as causal. It does not prove that increasing study time automatically causes higher grades. It does show that, in this dataset, higher study-time groups are associated with higher median G3 outcomes. That makes it a valuable descriptive result and a useful starting point for further modeling.

Why this chart matters: It connects a behavioral variable, study time, with the final grade distribution. It also shows that grouped five-number summaries can reveal educational patterns that a single overall summary would hide.

Reporting decision: State that G3 median increases from 11 in the lowest study-time group to 13 in the higher study-time groups, suggesting a positive descriptive relationship between study time and final grade.

Specific interpretation: The output summary panels confirm that the analysis used 649 rows and produced five-number summary tables, grade-focused summaries, grouped summaries, an absences outlier-fence table, a clean SPSS-ready data file and publication-ready charts. The key G3 five-number summary is minimum = 0, Q1 = 10, median = 12, Q3 = 14, maximum = 19 and IQR = 4.

Why these panels matter: They make the workflow transparent. A reader can quickly see what dataset was used, what outputs were generated and what the central G3 result is before reading the full interpretation.

Reporting decision: Use the summary panel values as the main numerical result in the report, then use the boxplots and distribution charts to explain the pattern visually.