Lots of n=3 samples of a common measurement type: how to do a single assessment of normality? I have 123 samples total.  116 samples are of sample size 3; 7 samples are of sample size 2.  They definitely have different population means, but may or may not have a common population variance.
If I were to hypothesise that each originates from an identical normal distribution except for different means, is there a way (possibly assuming a common population variance) to do a single assessment (with the result in the form of a likelihood or a test result) for whether this is the case or not?
(This, testing the normality of lots of samples from a common measurement procedure with a small sample size each, being in contrast with testing the normality of a single sample with a large sample size.)
Edit:  Relevant regarding my statistics comprehension:  the below webpage looks relevant to my question, but when I try to read and understand the answer my mind goes blank.
How to test for normality of growth disturbances in chemo treatment?
 A: I don't think you will get much information from the datasets with only two observations. Here is an example with six datasets with three replications
(instead of your 116).
set.seed(1234)
x1 = rnorm(3, 100, 15);  x2 = rnorm(3, 104, 15)
x3 = rnorm(3,  90, 15);  x4 = rnorm(3, 102, 15)
x5 = rnorm(3, 100, 15);  x6 = rnorm(3, 105, 15)
x = c(x1,x2,x3,x4,x5,x6)
g = as.factor(rep(1:6, each=3))

In the ANOVA table below MS(Resid) = 186.8 estimates the common variance
$\sigma^2 = 15^2 = 256.$ With so little data, this is not a very good estimate,
but it should be a better estimate for your more extensive data.
aov.out = aov(x ~ g)
summary(aov.out)
            Df Sum Sq Mean Sq F value Pr(>F)
g            5  853.6   170.7   0.914  0.504
Residuals   12 2241.1   186.8 

We can obtain the residuals and test them for normality as follows:
A Shapiro-Wilk test of normality does not reject the null hypothesis
that data are from a normal distribution. A normal probability plot
of the residuals is reasonably close to linear.
r = aov.out$resi
shapiro.test(r)

        Shapiro-Wilk normality test

data:  r
W = 0.95288, p-value = 0.4719

qqnorm(r); qqline(r)


Most intermediate-level statistics texts discuss testing residuals from
an ANOVA model for normality. The model for a one-way ANOVA is
$$Y_{ij} = \mu + a_i + e_{ij},$$
where $i = 1,2 \dots, G,$ for $G$ groups (6 above) and $j=1,2,3$ (above).
The $e_{ij} \stackrel{}{\sim} \mathsf{Norm}(0, \sigma),$ where $\sigma^2$ is the common
group variance. Residuals are $r_{ij} = Y_{ij} - \bar Y_i,$ where $\bar Y_i$ are
the $G$  group sample means. The residuals $r_{ij}$ emulate the normal random
errors $e_{ij},$ except that residuals in each group must add to $0,$ so that
the $r_{ij}$ are not exactly independent.
