I am trying to understand how to average certain data, interpret the result of that, and potentially convert the results to percentages. I have four categories for the data:1: (0-50%): Not Well2: (50-75%): Okay3: (75-90%): Well Done4: (90-100%): ExcellentThis is being used in a rating system. Say, for instance, we have 3 ratings each of category 4, and 7 ratings of category 3: {3, 3, 3, 3, 3, 3, 3, 4, 4, 4}. I don't believe doing a straight arithmetic mean will be useful in this case since category 1 is larger than category 2, category 2 is larger than category 3, and category 3 is larger than category 4. For instance, what would a result of 3.3 mean if we took the arithmetic mean? I'm trying to understand how to interpret the results of the data, but am at a loss for how to do this. Is there a way to convert the results back to one of the 4 categories or a percentage?Edit: I think I've made some progress, but I could be off the mark. Taking a few different examples...{1,1,1,1,1,1,1,1,4,4}.They got a rating of 1 80% of the time, and a rating of 4 20% of the time .8*1+.2*4 = 1.6, possibly a Not Well result?{1,1,1,3,3,4,4,4,4,4}: .3*1+.5*4+.2*3 = 2.9, maybe an Okay result?{1,1,1,1,2,2,2,4,4,4}: .4*1+.3*2+.3*4 = 2.2, maybe an Okay result?The problem with using that method is that Excellent will only come up if every rating is a 4. I am not sure how to fix that or if what I've tried is the correct method.

Question

I am trying to understand how to average certain data, interpret the result of that, and potentially convert the results to percentages. I have four categories for the data:1: (0-50%): Not Well2: (50-75%): Okay3: (75-90%): Well Done4: (90-100%): ExcellentThis is being used in a rating system. Say, for instance, we have 3 ratings each of category 4, and 7 ratings of category 3: {3, 3, 3, 3, 3, 3, 3, 4, 4, 4}. I don&#039;t believe doing a straight arithmetic mean will be useful in this case since category 1 is larger than category 2, category 2 is larger than category 3, and category 3 is larger than category 4. For instance, what would a result of 3.3 mean if we took the arithmetic mean? I&#039;m trying to understand how to interpret the results of the data, but am at a loss for how to do this. Is there a way to convert the results back to one of the 4 categories or a percentage?Edit: I think I&#039;ve made some progress, but I could be off the mark. Taking a few different examples...{1,1,1,1,1,1,1,1,4,4}.They got a rating of 1 80% of the time, and a rating of 4 20% of the time .8*1+.2*4 = 1.6, possibly a Not Well result?{1,1,1,3,3,4,4,4,4,4}: .3*1+.5*4+.2*3 = 2.9, maybe an Okay result?{1,1,1,1,2,2,2,4,4,4}: .4*1+.3*2+.3*4 = 2.2, maybe an Okay result?The problem with using that method is that Excellent will only come up if every rating is a 4. I am not sure how to fix that or if what I&#039;ve tried is the correct method.

Josie Stephenson · Accepted Answer

The standard way to calculate such a mean is to multiply the midpoint of each interval by the frequency, sum these and divide by the total frequency.{1,1,1,1,1,1,1,1,4,4}:             8      ×      0.25      +      2      ×      0.95        10    =  0.39 :Not well{1,1,1,3,3,4,4,4,4,4}:             3      ×      0.25      +      2      ×      0.825      +      5      ×      0.95        10    =  0.735:Okay{1,1,1,1,2,2,2,4,4,4}:            4      ×      0.25      +      3      ×      0.625      +      3      ×      0.95        10    =  0.5725:Okay{3,3,3,4,4,4,4,4,4,4}:             3      ×      0.825      +      7      ×      0.95        10    =  0.9125:ExcellentThis method is usually called &quot;estimating the mean of a grouped frequency table&quot; because the coded values 1, 2, 3 and 4 hide the true percentage values. There is thus a margin of error in each calculation.It is possible to give a margin of error for your mean, too, which you could use to decide whether the overall result was truly &quot;excellent&quot; or not.{1,1,1,1,2,2,2,4,4,4}: lower bound is             4      ×      0      +      3      ×      0.5      +      3      ×      0.9        10    =  0.42{1,1,1,1,2,2,2,4,4,4}: upper bound is             4      ×      0.5      +      3      ×      0.75      +      3      ×      1        10    =  0.725{3,3,3,4,4,4,4,4,4,4}: lower bound is             3      ×      0.75      +      7      ×      0.9        10    =  0.855{3,3,3,4,4,4,4,4,4,4}: upper bound is             3      ×      0.9      +      7      ×      1        10    =  0.97

I am trying to understand how to average certain data, interpret the result of that, and potentially

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