Grouped Variance

Starting my stats course and it gives the formula for variance - but I don't understand how they've formed grouped variance in the example attached?

I've literally put the numbers in my calculator and it's given a different value for variance.

Also not quite sure what the difference between grouped and upgrouped is? Thought it was just data in bins vs raw data...

There are two equivalent formulae for the usual variance
https://en.wikipedia.org/wiki/Variance#Definition
so this one is based on the
E(X^2) - (E(X))^2
with the extra bit (weighting by the frequency) for the grouping.

Grouping is as you say, you treat subsets of the data as if theyre in single bins.

Just not quite sure how the second bit in brackets was squared? Like why is the denominator also squared?

The formula
E(X^2) - (E(X))^2
is a standard one for variance and you must have come across it/derived it (done in the wiki link)? E() is the expected value or simply average when you have a finite data set. The second term is
(E(X))^2
so simply the square of the mean. You could have used the 156.4 you work out (and square it) or think about it as the numerator and denominator of E(X) both squared, then divide. The former is easier.

I've seen it as attached... but how come one version uses the mean and one doesn't?

Starting my stats course and it gives the formula for variance - but I don't understand how they've formed grouped variance in the example attached?

I've literally put the numbers in my calculator and it's given a different value for variance.