The stock Intel cooler can be tricky to install.
A poor installation will result in higher cpu temperatures.
If properly mounted, you should expect temperatures at idle to be 10-15c. over ambient.
Nudge the cooler, it should not wiggle.
Check the back of the motherboard to verify that all 4 pushpins are through and locked.
To mount the Intel stock cooler properly, place the motherboard on top of the foam or cardboard backing that was packed with the motherboard.
The stock cooler will come with paste pre applied, it looks like three grey strips.
The 4 push pins should come in the proper position for installation, that is with the pins rotated in the opposite direction of the arrow,(clockwise)
and pulled up as far as they can go.
Take the time to play with the pushpin mechanism until you know how they work.
Orient the 4 pins so that they are exactly over the motherboard holes.
If one is out of place, you will damage the pins which are delicate.
Push down on a DIAGONAL pair of pins at the same time. Then the other pair.
When you push down on the top black pins, it expands the white plastic pins to fix the cooler in place.
If you do them one at a time, you will not get the cooler on straight.
Lastly, look at the back of the motherboard to verify that all 4 pins are equally through the motherboard, and that the cooler is on firmly.
This last step must be done, which is why the motherboard should be out of the case to do the job. Or you need a case with a opening that lets you see the pins.
It is possible to mount the cooler with the motherboard mounted in the case, but you can then never be certain that the push pins are inserted properly
unless you can verify that the pins are through the motherboard and locked.
If you should need to remove the cooler, turn the pins counter clockwise to unlock them.
You will need to clean off the old paste and reapply new if you ever take the cooler off.