How to calculate average and standard deviation for minutes:seconds?

Your data might actually be a date instead of a duration. This occurs automatically if you enter something like "1:30" in a cell. Numbers will understand it as today's date at 1:30 AM. You can check that this is the case by formatting the cell to display the date (instead of "None").

To enter a duration, use the "Xm Ys" where X is the number of minutes and Y the number of seconds, e.g. "1m 30s" for 1 minute and 30 seconds. If the cells are properly a duration, you won't have issue with the statistics functions (see below). Note that once the cells do contain a duration, you can modify their format to display them as "X:Y" (see column E below, where E2 is set to B2).

You are welcome. Another way to enter duration is to prefix the duration with a single quote mark, e.g. '1:30. The cell will automatically possess the Text format but you can change it to the Duration format.