Often the values 1, 2, 3, 4 and 5 are assigned to categories such as "Strongly Disagree," "Disagree Somewhat," "Neutral," "Agree Somewhat," "Strongly Agree." This question has to do with such a "scaling." (The scale is called a Likert scale.)
Of all random variables taking the just values 1, 2, 3, 4, 5, the one with Pr{X = 1} = 1/2 and Pr{X = 5} = 1/2 has maximum standard deviation. What is the value of the variance of this random variable?