The number the date range hides
Checked at noon in every year from 1930 to 2035, the Sun on this date stood between about ten and a half and twelve degrees of Cancer. That is the whole spread, not a rounded average with outliers behind it.
A date range gives you one bit of information and disagrees with itself between sources by a day at either end. A degree that holds inside a band of roughly a degree and a half over four generations is both narrower and checkable, and it is the figure that feeds every other technical judgement about a chart.
Why the figure refuses to wander
A common year is about a quarter of a day shorter than the Earth's circuit of the Sun, so each year the Sun reaches any given longitude about six hours later on the clock. On its own that would march this birthday steadily through Cancer and out the far side.
The extra day in February corrects it, pulling everything back by roughly eighteen hours in a single step every four years. Creep and correction very nearly cancel, and the rule that withholds a leap day in 1900 and 2100 stops the small remainder from piling up.
Eleven degrees means the middle face
Each sign divides into three faces of ten degrees. Eleven degrees of Cancer falls just inside the second, the band running from ten to twenty. In the Chaldean order that band belongs to Mercury, and in the triplicity scheme it is assigned to Scorpio and so to Mars.
Tradition reads the middle face of a water sign as the more analytical stretch of it, with the Moon still ruling the sign and Mercury colouring the face. That is interpretation and should be taken as interpretation. The measurable part is only the position, and the position is what decides which face is in play.
Concretely, what the Mercury face is taken to add is a habit of naming things. The planet is associated in the old material with speech, counting, reckoning and going back over a thing to check it, so a Moon ruled sign carrying a Mercury face is written up as one that puts its impressions into words rather than leaving them as impressions. Under the triplicity reading the same degrees come out as Scorpio and Mars, which the tradition pushes towards holding on rather than talking. The two accounts do not reconcile, and need not.
What eleven degrees cannot tell you
The Sun is the slowest moving thing a birth date pins down. The Moon covers the zodiac in under a month and can be in either of two signs within a single 3 July, depending on the hour. The ascendant runs through all twelve signs in a day, so two births a few hours apart share this solar degree and agree on almost nothing else.
That is the honest limit of a birthday page. It fixes one placement precisely and leaves the rest open, which is why the widget above asks for the year, the hour and the birthplace before it reports your own degree and your distance from the nearest sign boundary.
It is worth adding that the Moon is not just any other body here. It rules Cancer, so the sign's ruler is also the quickest thing in the chart, moving further in a day than the Sun does in a fortnight. In most signs the ruler is slow enough that everyone born within a few days shares it in one sign. Cancer is the exception: the ruler of a 3 July chart could be sitting in any of the twelve signs depending on the year, and in either of two on the day itself. That single fact does more to separate two people born on this date than anything the sun sign carries.
Questions people ask about this
What sign is 3 July?
Cancer, without qualification. The Sun entered Cancer around 21 June and does not leave until roughly 22 July, so this date sits well clear of both edges in every year sampled.
What degree of Cancer is 3 July?
Between about ten and a half and twelve degrees at noon, depending on the year. Your exact figure also depends on the hour and the place of birth, which the widget above accounts for.
Which decan does that put me in?
The second decan of Cancer, ten to twenty degrees. Chaldean order gives it to Mercury, while the triplicity scheme assigns it to Scorpio and therefore to Mars.
Why do sources give slightly different dates for Cancer?
Because the ingress happens at a clock moment that shifts by hours each year and changes calendar day depending on the time zone. A fixed date range has to pick one version and live with it.
Does the second decan change what Cancer means?
In the older material it shades it rather than replacing it. The sign ruler stays the Moon and the face ruler adds a second layer, which is the layer that separates one Cancer birthday from another.
Is my birth time needed for this date?
Not to settle the sign, which is fixed. It is needed for your exact degree, for the Moon, and for the ascendant, none of which the calendar date carries on its own.