I have a dataset which is markedly left-skewed, and I wonder if it will be inappropriate to use the normal distribution curve to analyses the data given it's highly skewed?
Picture of my data:

I have a dataset which is markedly left-skewed, and I wonder if it will be inappropriate to use the normal distribution curve to analyses the data given it's highly skewed?
Picture of my data:

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Your data seems to be non-negative. The normal distribution is almost never well suited for such case. You might want to try a gamma distribution, possibly just an exponential distribution.
Moreover, you should consider if your data is continuous or discrete. In the latter case, you might prefer Poisson distribution or geometric distribution.
If your data is also bounded from above, then beta distribution and binomial distribution are an option.