As she was listed as aged 6 in the 1861 census, with older siblings aged 10 and 8 and younger siblings aged 4, 2 and 1, she has to have been born in 1854 or 1855.
In fact, as the 1861 census was carried out on 7 April 1861, she should have been born between 8 April 1854 and 7 April 1855. Therefore she is more likely to have been born in 1854 than in 1855, and indeed the absence of a formal birth certificate confirms that she must indeed have been born before 31 December 1854.
The younger siblings are
Catherine 4 suggests born 1856/1857 - births index says 1856
William 2 suggests born 1858/1859 - births index says 1857
Donald 1 suggests born 1859/1860 - births index says 1859
Going on to 1871 when the family were in Wick
James 19 -> 1851/1852
Margaret 18 -> 1852/1853
Johan 16 -> 1854/1855
Catherine 15 -> 1855/1856 index says 1856
William 14 -> 1856/1857 index says 1857
Donald 12 -> 1858/1859 index says 1859
Mary 10 -> 1860/1861 index says 1861
Adam 8 -> 1862/1863 index says 1863
Jane 3 -> 1867/1868 index says 1868
Ann 0 -> 1870/1871 index says 1870
So you can see that all the ages listed in 1871 are consistent with the dates of birth registrations after 1855. If the parents got all the younger ones' ages right, why would they get the older ones wrong?
If Alexander H and Catherine Calder married on 2 January 1852, then James was born either before their marriage or indecorously soon afterwards. If they were members of the Church of Scotland this would have resulted in a record in the Latheron Kirk Session minutes of the parents being scolded for antenuptial fornication. However the relevant volume is not (yet?) available at Scotland's People.