Box plot and violin plot maker with statistics

Box plot. The box runs from the first quartile ($Q_1$) to the third ($Q_3$) with a line at the median; the interquartile range is $\mathrm{IQR}=Q_3-Q_1$. Quartiles use linear interpolation between ordered values (the default in R, NumPy and Excel’s QUARTILE.INC): with $n$ values sorted, the $p$-th quantile sits at position $1+(n-1)p$. Whiskers reach the most extreme values within $Q_1-1.5\times \mathrm{IQR}$ and $Q_3+1.5\times \mathrm{IQR}$, and points beyond are outliers (Tukey’s rule; McGill, Tukey and Larsen 1978).

Violin plot. A violin is a box plot with the density of the data drawn on both sides (Hintze and Nelson 1998). The density is a Gaussian kernel estimate $f(x)=(1/nh)\sum \phi ((x-x_i)/h)$, with Silverman’s rule-of-thumb bandwidth $h=0.9\cdot min(\mathrm{SD},\mathrm{IQR}/1.34)\cdot n^{-1/5}$ (Silverman 1986, eq. 3.31; the default in R’s density() and ggplot2’s geom_violin). Each violin is scaled to the same maximum width and cut at the smallest and largest value.

Two groups. Welch’s t-test compares means without assuming equal variances, with Welch–Satterthwaite degrees of freedom (Welch 1947):

$$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$$$$\mathrm{df}=\frac{(s_1^2/n_1+s_2^2/n_2{)}^2}{\frac{(s_1^2/n_1{)}^2}{n_1-1}+\frac{(s_2^2/n_2{)}^2}{n_2-1}}$$

It is the better default than Student’s t-test, losing little power when variances are equal (Delacre et al. 2017). The Mann–Whitney U test ranks all values together: $U_1=R_1-n_1(n_1+1)/2$, where $R_1$ is the rank sum of group 1 (Mann and Whitney 1947). Without ties the p-value is exact, from the null distribution of $U$; with ties it uses the normal approximation with tie and continuity corrections, $z=(|U-n_1{n}_2/2|-0.5)/{\sigma }_U$.

Three or more groups. One-way ANOVA:

$$F=\frac{\sum n_i({\overline{x}}_i-\overline{x}{)}^2/(k-1)}{\sum \sum (x-{\overline{x}}_i{)}^2/(N-k)}$$

with $p$ from the F distribution with $(k-1,N-k)$ degrees of freedom; pairs are compared with the Tukey–Kramer test,

$$q=\frac{|{\overline{x}}_i-{\overline{x}}_j|}{\sqrt{{\mathrm{MS}}_{\mathrm{within}}/2\cdot (1/n_i+1/n_j)}}$$

with $p$ from the studentised range distribution. Kruskal–Wallis:

$$H=\frac{12}{N(N+1)}\sum \frac{R_i^2}{n_i}-3(N+1)$$

divided by the tie correction $1-\sum (t^3-t)/(N^3-N)$, with $p$ from ${\chi }^2$ with $k-1$ degrees of freedom (Kruskal and Wallis 1952); pairs are compared with Dunn’s z test on mean ranks, Holm-adjusted. All tests are two-sided.

Accuracy. The t, F, ${\chi }^2$, normal and studentised range distributions are those of the LabCanvas Graphs app, validated against SciPy and R; they reproduce the table values $t_{0.975,10}=2.228$, $F_{0.95;2,12}=3.885$, ${\chi }_{0.95,2}^2=5.991$, $z_{0.975}=1.960$ and $q_{0.95;3,21}=3.565$. Every statistic and p-value in the worked example matches SciPy 1.18 to 8 or more significant figures.

Sources: Welch BL. The generalization of ‘Student’s’ problem when several different population variances are involved. Biometrika 1947;34:28–35, doi:10.1093/biomet/34.1-2.28. Mann HB, Whitney DR. On a test of whether one of two random variables is stochastically larger than the other. Ann Math Stat 1947;18:50–60, doi:10.1214/aoms/1177730491. Kruskal WH, Wallis WA. Use of ranks in one-criterion variance analysis. J Am Stat Assoc 1952;47:583–621, doi:10.1080/01621459.1952.10483441. McGill R, Tukey JW, Larsen WA. Variations of box plots. Am Stat 1978;32:12–16, doi:10.1080/00031305.1978.10479236. Hintze JL, Nelson RD. Violin plots: a box plot-density trace synergism. Am Stat 1998;52:181–184, doi:10.1080/00031305.1998.10480559. Delacre M, Lakens D, Leys C. Why psychologists should by default use Welch’s t-test instead of Student’s t-test. Int Rev Soc Psychol 2017;30:92–101, doi:10.5334/irsp.82. Silverman BW. Density Estimation for Statistics and Data Analysis. London: Chapman & Hall; 1986.

The built-in data are tumour volumes (mm³) of three groups of 8 mice: vehicle, Drug A and Drug B.

Box plot by hand (vehicle). Sorted: 768, 812, 890, 945, 976, 1020, 1103, 1350. $Q_1$ is at position $1+7\times 0.25=2.75$, between 812 and 890: $812+0.75\times 78=870.5$. The median is $(945+976)/2=960.5$ and $Q_3=1020+0.25\times 83=1040.75$, so $\mathrm{IQR}=170.25$. The upper fence is $1040.75+1.5\times 170.25=1296.1$, so 1350 is drawn as an outlier and the whisker stops at 1103. The violin’s Silverman bandwidth is $h=0.9\times min(183.6,170.25/1.34)$ $\times 8^{-1/5}=75.4{\mathrm{mm}}^3$.

Three groups, parametric. One-way ANOVA gives $F(2,21)=15.42$, $p=0.000076$ (${\eta }^2=0.59$). Tukey–Kramer: vehicle vs Drug A $p=0.0031$ (**), vehicle vs Drug B $p=0.000064$ (****), Drug A vs Drug B $p=0.25$ (ns).

Three groups, non-parametric. Kruskal–Wallis $H(2)=14.61$, $p=0.00067$ (mean ranks 19.75, 11.38 and 6.38). Dunn’s test, Holm-adjusted: vehicle vs Drug A $p=0.036$ (*), vehicle vs Drug B $p=0.00046$ (***), Drug A vs Drug B $p=0.16$ (ns). The outliers weigh less in the rank test, but the conclusion is the same.

Two groups. Delete the Drug B column to compare vehicle and Drug A only. Welch’s t-test: the standard error is $\sqrt{{183.6}^2/8+{84.0}^2/8}=71.39$, so $t=(983-667.5)/71.39=4.42$ with $\mathrm{df}=9.80$, $p=0.0014$; the difference is 315.5 mm³ (95% CI 156.0 to 475.0). Mann–Whitney: vehicle has a rank sum of 99, so $U=99-36=63$ of a possible 64, and the exact two-sided $p=0.00031$.

Not sure which test fits your design (paired data, counts, correlations)? Use the statistical test chooser.