Believe it or not it is possible and, dare I say, common to experience such a discrepancy. There are a million variables that can each make a large difference, and they might all just be adding up badly for you. However, as somebody who is concerned about fuel economy it does seem quite unlikely that that would happen to you...if you were posting on RaceAroundLikeAJerk.com it would be more likely. Just keep in mind that it's possible.
Can you tell us about your car, roads, route, traffic conditions, driving style, and fuel filling procedure? Looking at your log, it does look like possibly short trips, which could contribute to low fuel economy.
What was your last car, and what was your fuel economy for that car?
Does your car roll well? If you put it in neutral while driving at a decent speed on a mild to moderate descent, does it coast at the same speed or even pick up a little speed, or does it lose speed?
First of all I'm sorry to hear you are reporting such low MPG numbers. That must be especially frustrating with a new car which so many others are reporting much better numbers for. I've read Mazda in particular is usually very truthful when it comes to setting EPA estimates for MPG, which definitely is not the case for all manufacturers.
That being said, your numbers do seem extremely low. I took an average of all the other MPG numbers for 2014 Mazda 3 sedans on Fuelly and came up with an overall average of 32.7 MPG. This would seem to be in line with Mazda's own estimates of 29 city / 40 highway for this model, which makes your totals look even more perplexing.
A few questions for you and a couple of things I thought of that may be worth considering in an attempt to decipher this mystery.
First, I notice you live in Toronto. I am in Minnesota and we had an unusually cold winter. I'm assuming maybe you did as well? Do you spend a lot of time letting your car warm up before driving when it is bitter cold? I also noticed your fuel-ups are listed 90% city driving. Do you use the car almost exclusively to commute to/from work? Do you drive a lot of very short trips? If so, does the engine have time to completely warm up and come up to standard operating temperature before you complete your "normal" trips? Personally, I live only 4 miles from work and I know my MPG average drops quite a bit in the winter due to these factors. Since all 7 of your fuel-ups fell during a particularly cold winter I'm assuming that has to drag your numbers down considerably. Hopefully, once the weather gets warmer you will see a significant increase in MPG.
Does your car have the 2.0 liter or 2.5 liter Skyactiv engine? I've read the 2.5s get slightly less gas mileage though it should be almost negligible.
Have all your fuel-ups been at the same gas station? I realize this is a long shot but if so and if their pumps are incorrectly calibrated it is possible you are actually putting less fuel in the car each time than what the pump is reading.
Does the engine seem to run OK, i.e. idle correctly, accelerate smoothly, etc.? No drivability issues that would point you to a problem of some kind?
Please keep us posted regarding your numbers as you continue to drive this car. I for one would definitely like to know what the culprit was when you finally solve this mystery.
In addition to the above, check your tire pressure. Low pressures can have a negative affect on fuel economy before they reach the point where the TPMS alerts you, and the tire may not be visibly low looking.