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I have data where each person can take 5 potential values of $(0, 0.25, 0.5, 0.75, 1)$. Each person is part of a larger group of 3 people, so that the reported value for each group is the average of 3 people's values, which is between 0 and 1.

For example,

Group Name  Scores              Average of Scores
Group 1     (0.50, 0, 0.50)     0.333
Group 2     (1.0, 0.5, 1.0)     0.833    
Group 3     (0.75, 0.5, 0.25)   0.500
Group 4     (0.5, 1, 0.5)       0.666

I would like to construct a confidence interval for the "Average of Scores" column. Is it fair to use a confidence interval for binomial proportion $\hat{p}=(0.333+0.833+0.5+0.666)/4$

$$ \hat{p} \pm z_{\alpha}\sqrt{\dfrac{p(1-p)}{N}} $$ ?

I have data where each person can take 5 potential values of $(0, 0.25, 0.5, 0.75, 1)$. Each person is part of a larger group of 3 people, so that the reported value for each group is the average of 3 people's values, which is between 0 and 1.

For example,

Group Name  Scores              Average of Scores
Group 1     (0.50, 0, 0.50)     0.333
Group 2     (1.0, 0.5, 1.0)     0.833    
Group 3     (0.75, 0.5, 0.25)   0.500
Group 4     (0.5, 1, 0.5)       0.666

I would like to construct a confidence interval for the "Average of Scores" column. Is it fair to use a confidence interval for binomial proportion $\hat{p}=(0.333+0.833+0.5+0.666)/4$

$$ \hat{p} \pm z_{\alpha}\sqrt{\dfrac{p(1-p)}{N}} $$ ?

The histogram of "Average of Scores" looks like the following for 200 scores:

I have data where each person can take 5 potential values of $(0, 0.25, 0.5, 0.75, 1)$. Each person is part of a larger group of 3 people, so that the reported value for each group is the average of 3 people's values, which is between 0 and 1.

For example,

Group Name  Scores              Average of Scores
Group 1     (0.50, 0, 0.50)     0.333
Group 2     (1.0, 0.5, 1.0)     0.833    
Group 3     (0.75, 0.5, 0.25)   0.500
Group 4     (0.5, 1, 0.5)       0.666

I would like to construct a confidence interval for the "Average of Scores" column. Is it fair to use a confidence interval for binomial proportions such as

$$ p \pm z_{\alpha}\sqrt{\dfrac{p(1-p)}{N}} $$ ?

I have data where each person can take 5 potential values of $(0, 0.25, 0.5, 0.75, 1)$. Each person is part of a larger group of 3 people, so that the reported value for each group is the average of 3 people's values, which is between 0 and 1.

For example,

Group Name  Scores              Average of Scores
Group 1     (0.50, 0, 0.50)     0.333
Group 2     (1.0, 0.5, 1.0)     0.833    
Group 3     (0.75, 0.5, 0.25)   0.500
Group 4     (0.5, 1, 0.5)       0.666

I would like to construct a confidence interval for the "Average of Scores" column. Is it fair to use a confidence interval for binomial proportion $\hat{p}=(0.333+0.833+0.5+0.666)/4$

$$ \hat{p} \pm z_{\alpha}\sqrt{\dfrac{p(1-p)}{N}} $$ ?