My table stores the same variable for each year in separate fields - ex. Pop2010, Pop2011, Pop2012. I used the Transpose Fields tool to transpose the years to long format. It transposes as expected except that the Value Field is stored as text.

Is there a way to set this as integer or double?

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It doesn't appear that you can change the type setting for the Value Field when outputted by the Transpose Fields tool.

However, you could always create a "new" Value Field by adding a field in the attribute table, setting the type to an integer or double then using the field calculator to add all values from the "old" Value field, in that new field.

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