it will be tricky to study your own improvement in working memory and I wouldn't suggest it! There is substantial research showing that although training can improve performance on specific cognitive tasks (near transfer), there's not too much evidence that a broad construct like working memory can be enhanced by training in the long term (if ever) (Towards a Science of Effective Cognitive Training (1); Do “Brain-Training” Programs Work? (2)).
Therefore, its very possible that you end up not quite measuring your improvement in working memory as it's commonly understood, but rather task performance (Putting brain training to the test (3)). Given this, I would suggest trying to learn something new or practice a skill that you think is interesting! Those could have real-life benefits for you and would likely be more rewarding than improvement in an specific cognitive task! Some people really enjoy video games, for example, while others might be more inclined toward reading, playing music, meditation, or exercise.
However, if you are undeterred by that advice and still want to test your cognition as a result of training, I might suggest looking at narrower outcomes, such as "attention" or "speed of processing" (Systematic review and meta-analyses of useful field of view cognitive training (4)), rather than working memory specifically (working memory is a very elusive subject and target). You can see a variety of memory tasks in the above reference, including some that are grouped as "executive function" (e.g., the n-back task).
And to finally answer your exact question explicitly, CNSVS (website, validation paper (5)) is one neurocognitive testing battery which has been used in clinical situations and can report a standardized score. You can look at a sample report on their website to see how they create a variety of singular test and composite scores. However, just to emphasize, this is not a "diagnostic" test, despite being used in clinical situations.
Just for a more complete story on the scientific and business angles on cognitive training, here (no guarantee this will be available) is a white paper from a brain training company, which I include because it reports results that paint a different picture than the peer-reviewed literature I referenced at the top of this answer (again, just want to emphasize that this answer is not a comprehensive evaluation of the literature, but hopefully somewhat canonical and current).
Hope this helps!
Editing in an update for VSO's questions in the comment, might as well collect more of the recent info here. I'm also taking the liberty of rephrasing/reordering the questions, hope it's not too liberal.
Q. Can one working memory task improvements transfer to other working memory tasks?
A. Maybe, there seems to be some debate in the literature. This (6, preprint) meta-analysis concludes it's possible, but also that the effect did not depend on type of training or dose, so I wonder what the mechanism was. These (7) meta-analyses posit that the effect is mostly in typically developing children, which this (8) meta-analysis also concluded. However, they also found that the effect size is proportional to the similarity of the memory tasks to the original (in typically developing kids maximum mean age 16) here (9).
Q. Are there long-term effects of n-back training on working memory in general?
A. In typically developing children, for whom there may be an effect of cognitive training tasks on other working memory tasks, the size of the effect might be determined by how similar the task is to other tasks (last reference above, 9). There are examples of improving specific skills at all ages though. Thinking about a musician as example (they even do things with their fingers and brains kind of like cognitive tasks). Someone who first starts will get better at specific chords, which will eventually lead to getting better at other chords, but not very much without practicing those other chords. As one practices the n-back, there could be some "n-back" skill that slightly generalizes but the performance seems to track with how similar the tasks are. Here's (10) a meta-analysis for n-back specifically.
- Smid, C. R., Karbach, J., & Steinbeis, N. (2020). Toward a science of effective cognitive training. Current Directions in Psychological Science, 29(6), 531-537.
- Simons, D. J., Boot, W. R., Charness, N., Gathercole, S. E., Chabris, C. F., Hambrick, D. Z., & Stine-Morrow, E. A. (2016). Do “brain-training” programs work?. Psychological Science in the Public Interest, 17(3), 103-186.
- Owen, A. M., Hampshire, A., Grahn, J. A., Stenton, R., Dajani, S., Burns, A. S., ... & Ballard, C. G. (2010). Putting brain training to the test. Nature, 465(7299), 775-778.
- Edwards, J. D., Fausto, B. A., Tetlow, A. M., Corona, R. T., & Valdés, E. G. (2018). Systematic review and meta-analyses of useful field of view cognitive training. Neuroscience & Biobehavioral Reviews, 84, 72-91.
- Gualtieri, C. T., & Johnson, L. G. (2006). Reliability and validity of a computerized neurocognitive test battery, CNS Vital Signs. Archives of Clinical Neuropsychology, 21(7), 623-643.
- Rodas, J. A., & Greene, C. (2020). Small Improvements in Working Memory After Cognitive Training do not Transfer to Fluid Intelligence: Evidence From a Meta-analysis.
- Sala, G., Aksayli, N. D., Tatlidil, K. S., Tatsumi, T., Gondo, Y., Gobet, F., ... & Verkoeijen, P. (2019). Near and far transfer in cognitive training: A second-order meta-analysis. Collabra: Psychology, 5(1).
- Sala, G., & Gobet, F. (2017). Working memory training in typically developing children: A meta-analysis of the available evidence. Developmental Psychology, 53(4), 671.
- Sala, G., & Gobet, F. (2020). Working memory training in typically developing children: A multilevel meta-analysis. Psychonomic bulletin & review, 1-12.
- Soveri, A., Antfolk, J., Karlsson, L., Salo, B., & Laine, M. (2017). Working memory training revisited: A multi-level meta-analysis of n-back training studies. Psychonomic bulletin & review, 24(4), 1077-1096.