How common is it for male children to masturbate in early childhood (before kindergarten)?
Is it normal for extremely young male boys to fondle their own genitals?
Is that part of normal human biology or not?
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According to one 2010 paper, which actually tried to identify the correlates of the behavior, albeit in a small sample (n=13, 3 males, 10 females) that was referred to a clinic:
The median age at the first incident was 19.5 months (range, 4-36 months); the median masturbation frequency, 4 times/day; and the median duration of each event, 3.9 min. The subjects masturbated in both prone (n=10) and supine positions (n=3); two subjects used the knee-chest position. All subjects showed facial flushing; 6, friction between the thighs; 5, sweating; 9, sleeping after the event; and 12, disturbance on interruption. EEG was abnormal in one of eight subjects tested, and neuroimages were normal in all of nine subjects examined. The case and control groups had comparable levels of all sex hormones, except estradiol, which showed significantly lower levels in the case group (P=.02).
CONCLUSION: Masturbation in children seems to be associated with reduced estradiol levels, but not with other sex hormones. Further studies are needed to confirm our findings.
So, there was nothing too obviously wrong with these children otherwise...
The sample is from a clinic in Jordan but I didn't sense any obvious cultural bias, except for the higher presumed referral rate... and lack of follow-up:
Since Jordan is a sexually conservative country with no formal sex education, childhood masturbation may create more parental concern than in Western societies, and the referral rate may differ. Ten (77%) of our children did not attend any follow-up visits after their parents were informed about the diagnosis of childhood masturbation, possibly due to the concern of stigmatization.
By the way, there is a term for this masturbation-like behavior in children (as mentioned in the paper) namely gratification disorder. Alas the Wikipedia article cites only a handful of even older articles, to say that not much research was done.
Some similar studies conducted in the UK and in Turkey actually had somewhat larger samples, but otherwise there's not a lot to say about this. The UK study had a broader age range... and there's the odd bit about the car seat:
Thirty one patients were diagnosed (11 males and 20 females). [...] The median age at first symptoms was 10.5 months (range 3 months to 5 years 5 months). The median age at diagnosis was 24.5 months (range 5 months to 8 years). [...] Events occurred in any situation in 10 children, and in a car seat in 11.
The discussion section also notes something about stigma:
Masturbation in children is commonly recognised to be a variant of normal behaviour. Once the diagnosis is made and there are no suspicions of child sexual abuse requiring further investigation and management, reassurance seems to be the most effective management. Parents prefer the term gratification (or even benign idiopathic infantile dyskinesia) to infantile masturbation as there is less social stigma attached to these terms.
Classification-wise, one 2008 review says that
CM is defined as self-stimulation of the genitalia in a prepubescent child. It is not included in the DSM-IV, the Diagnostic and Statistical Manual of Mental Disorders (4th edition) of the American Psychiatric Association, which means that it is not classified as a (specific) psychiatric disorder. The WHO places (excessive) CM under the title “Other specified behavioural and emotional disorders with onset usually occurring in childhood and adolescence” (F98.8) in the ICD-10, the International Statistical Classification System of Diseases and Related Health Problems.
And to help answer your title question, this review actually does cite (and even include a table from) a study on prevalence, although this was assessed indirectly (parental reporting):
In a study of Friedrich, the frequencies of a lot of different sexual behaviours in 1,114 2- to 12-year-old children were rated by their mothers (by means of the CSBI and the CBCL). Friedrich excluded children with a history or suspicion of sexual abuse in order to compose a normative sample. His results were consistent with earlier research and showed that a broad spectrum of sexual behaviours appears in children with varying frequencies. Self-stimulation is one of the most frequently seen sexual behaviours (Table 1).
That study actually used a combined sample from Los Angeles and Minnesota, with the stated reason for doing so to average over possible cultural bias of the respondents.
There's yet another 2009 review which is open-access, but on the broader topic sexual-like behaviors in childhood including such interactions with other children, exhibitionism etc. This review notes that "more than 50% of children will engage in some type of sexual behavior before their 13th birthday" but this is across the aforementioned broader range of behaviors. For childhood masturbation proper, they don't cite any other study besides Friedrich's already discussed above. But (I guess mainly from that) they summarize in a table of "Examples of Sexual Behaviors in Children 2 to 6 Years of Age" that "Normal, Common Behaviors" include "Touching/masturbating genitals in public/private".
Actually Friedrich is a co-author to another study that (using a translated questionnaire) had roughly similar findings in a Dutch sample, but a bit lower e.g. only 9.2% for "masturbates with hand" in the 2-5 y.o. I've cropped the most relevant part of their questionnaire report:
They also note some parental correlates of higher reporting, which not surprisingly include a more liberal attitude towards nudity, homosexuality etc. A more recent study mentions that results turned out not to be so easily generalizable because what parents report seems to vary across cultures...
Comparison of the available research based on US, European, and Asian samples, reveals that European parents score highest on the CSBI total (de Graaf & Rademakers, 2007), whereas Asian children hardly show any sexual behavior according to the CSBI total score (Jin, Chung, Jeong, & Lee, 2013).
One good news (perhaps) is that within the same culture, mothers and fathers make consistent reports on their children. The latter paper does note, culture-wise:
we must take into consideration the conservative Korean culture based on Confucian strictures including sexual taboos even among the adults, which likely affected the parents' ability to observe and report their children's sexual behavior.