A binary variable.
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Not necessarily. You can have True, False and Indeterminate.
No. Here's one set of data where the mean is not one of the values: a set of 250,000 numbers. 125,000 of them are "1", 125,000 are "3". The mean of this data set is "2", which is not among the data.
You will need to put the un-grouped data in ascending or descending order. If you have an odd number of data values the formula for the median value is (n+1)/2. Example my data in ascending order is 0, 2, 4, 5, 7, 8, 9. I have 7 data values. The median is the value (7+1)/2 = 4th value from left or right which is 5. For an even number of data values, you will need to calculate the median and it may not be a data value. It will be the mean of the two center values. Use the formula n/2 to get the left most value. Example my data in ascending order is 0, 2, 4, 5, 7, 8. I have 6 data values. The left most value I will use to calculate the median is 6/2 = 3rd. The 3rd value from the left is 4. The next value is 5. Median is (4+5)/2 = 4.5.
It is the mean average of number of collected data values.
mean