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Exporting Data in the Power BI Service
Exporting data from reports in the Power BI Service is done on a visual-by-visual basis. To extract data from a visual, mouse over the visual until a small menu appears on the visual, usually in one of the corners of the visual:

This presents you with a few functions, the most important of which is the ellipses (...) menu highlighted by the red box. Clicking on this button brings up the 'More Options' menu:

Here, you can implement some sorting, show the visual and data as a table, and export your data. Clicking 'Export Data' will bring you to the following menu:

The most common method for exporting data is the Data with Current Layout option, which is useful if you are looking for an Excel file setup in a similar way to your visual. Another common method is the Summarized Data option, which allows you to choose between a CSV or Excel file. This option summarizes the data behind the visual, presenting sums, averages, and other summary statistics. It is also useful if you do not require the structure of the visual itself to carry over to your file. Once you've selected your method and file type, clicking Export will download a file directly to your machine.
Note: This is not to be confused with the large Export button at the top of the screen on reports:

This option is for exporting the report file itself as numerous file types, not the data itself. For example, you could export pages of a report as PDFs for sharing of insights with stakeholders via email, or as a PowerPoint for a quick presentation slide. The Analyze in Excel option requires additional setup and permissions, and is not as robust an experience as directly downloading your data to your machine for analysis.