The Edinburgh Scale is a test that is commonly used in geriatric studies in order to assess the cognitive performances of elderly people. The score of the scale ranges from $0$ to $500$ point. The normal score for an elderly person (without specific mental or cognitive disorders) can be modelled by a Normal distribution with mean $266$ points and standard deviation $16$. In a retirement home of the city of Berlin to a random sample of $61$ elderly people the Edinburgh Scale was administered, and the score collected. The relevant statistics were
$$\sum x = 15568,\quad \sum x^2 = 4054484.$$
At $5\%$ significance, is there statistical evidence that the variance of the sampled population differs from that of the normal score for an elderly person?
Let me show you how I attempted to find the variance for the problem. I have no issue with knowing how to solve the actual problem itself, but I am getting the wrong variance somehow.
$$\sum\frac{x}{n} = \frac{15568}{61} = 255.2131147541 = \text{mean}$$
$$V(x) = \frac{4054484 - 255.2131147541^2}{61} = 65,399.184$$
However, this is wrong as per my professor's notes. The correct value for variance is $1355.4372$.