  We evaluate asymptotically the variance of the number of squarefree integers
up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the
variance of the number of squarefree integers up to $x$ in arithmetic
progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of
respectively the Lindel\"of Hypothesis and the Generalized Lindel\"of
Hypothesis we show that these ranges can be improved to respectively $H <
x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show
that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full
range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These
results improve on a result of Hall (1982) for short intervals, and earlier
results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of
arithmetic progressions.
