Add Cells Whose Date Fall Between A Specific Range

I tried =SUMIFS and tried to input date ranges with >= & <= to no avail.
Column A is formatted to display dates; column B displays numbers for the respective date. There are dates added daily.
I need a formula to do the following: Updates automatically when new dates are entered. It would be for the month.
For instance, there are thirty-one days in the month of January. Ergo, the formula would return a value in a cell for the thirty-one days of January.
I tried the following, but it doesn't work:
=SUMIF(A:A,">=DATE(2016,1,1)"&"<=DATE(2016,1,31)",B:B)
where Column A displays dates and Column B displays values, but I always receive a value of 0.

I would apply this separately for each month. I would just change the date value in the formula to match the month needed.
Please help!

STBTC,

Attached has Dates in Col A and Data in COl B

Rather than changing formulae manually every time you want to check something. D2 - D13 are formatted as monthly dates ("Jan-16", etc)

Following formula in E2 (copied down to E13) gives you monthly totals automatically:

=SUMIFS(B:B,A:A,">="&D2,A:A,"<"&EDATE(D2,1))

(Add up the totals in Col B where the date in Col A is between the first and last days of the Month in Col D)

Because it used the entire columns, it will pick up automatically any new dates you enter - even if they are out of sequence.

Ochimus

I have a file that has dates in Sheet 1, Column E. Dates will be added throughout the year as it progresses. I need a formula that will add the corresponding column G for a specific date range--January, February, et cetera--and return a value in Sheet 2, Column B next to the corresponding month. Could someone help? I tried both alansidman and Ochimus' formulae, but to no avail. I know that those are the formulae which I should use, but I'm skipping a step, I think. I surmise that I am doing something incorrect. For example, I return a value of zero (0) for the month of January when the value returned should be forty-two (42).

STBTC

You could not get my solution to work on your file because the months in Col A of your Sheet 2 are not dates, but text.

If you look at the attached, as soon as I changed the cells to dates everything works perfectly.

BUT omitting the year from the description will give you problems going forward.

Cell A2 says "January", (actual "date" 01/01/16). If you add A14 as also being "January" (actual "date" 01/01/17), the wording in Col A "as is" will not tell you which year is being reported?

Which is why I structured the original as "Jan-16", Feb-16", etc.

Ochimus