How to fit a model to self-reported number of friend interactions over a 20 day period? I am a novice in statistics so please correct me if I am doing something fundamentally wrong. After wrestling for a long time with R in trying to fit my data to a good distribution, I figured out that it fits the Cauchy distribution with the following parameters: 
   location      scale  
  37.029894   18.678936 
 ( 3.405665) ( 2.779136)

The data was from a survey where 100 people were asked how many friends they talked to over a period of 20 days and I am trying to see if it fits a known distribution. I generated the QQ-plot with the reference line and it looks like the image given below. From what I have been reading on the web, if the points fall close to the reference line then it is a good evidence that the data comes from this distribution. 
So, is this a good evidence to say that the distribution is Cauchy or do I need to run any more tests? If so, can someone tell me the physical interpretation of this result? I mean, I read that if the data falls into a Cauchy distribution, then it will not have a mean and standard deviation but can someone help me understand this in plain English? If it does not have a mean then from what I understand, I cannot sample from this distribution. What is one supposed to infer about the population based on this result? Or should I be looking at other models?
UPDATE: What am I trying to achieve?
I am trying to evaluate how much time it takes for some arbitrary piece of information to propagate for a population of size X. As this depends on the communication patterns of people, what I was trying to do was to build a model that could use the information from the 100 people I surveyed to give me patterns for the X number where X could be 500 or 1000. 
QQ-Plot

Density Distribution of my data

Cauchy Distribution

QQ-Plot when trying to fit a Normal distribution to my data

UPDATE:
From all the suggestions, I think I now understand why this cannot be a Cauchy distribution. Thanks to everyone. @HairyBeast suggested that I look at a negative binomial distribution so I plotted the following as well:
QQ-Plot when Negative Binomial Distribution was used

Negative Binomial Distribution

 A: Agree with HairyBeast (+1) that Cauchy is not appropriate here (it's symmetric for one thing) and that negative binomial might well be better.
Disagree about QQ-plot though. You can do a QQ-plot for any distribution, not just normal. What you say about interpretation of a QQ-plot is correct, but note that 2 of your points lie very far indeed from the straight line.
On the Cauchy's lack of moments: this doesn't affect sampling. Once you know the parameters of the distribution sampling from it is easy (as the quantile function has a closed form) and the lack of moments is irrelevant. But the fact that the Cauchy distribution doesn't even have a mean does indicate that it's inappropriate here, as clearly it is meaningful to ask what's the expected number of friends with whom a person has a conversation in a 20-day period.
A: First off, your response variable is discrete. The Cauchy distribution is continuous. Second, your response variable is non-negative. The Cauchy distribution with the parameters you specified puts about 1/5 of its mass on negative values. Whatever you have been reading about the QQ norm plot is false. Points falling close to the line is evidence of normality, not evidence in favor of being Cauchy distributed (EDIT: Disregard these last 2 sentences; a QQ Cauchy plot - not a QQ norm plot - was used, which is fine.) The Poisson distribution, used for modeling count data, is inappropriate since the variance is much larger than the mean. The Binomial distribution is also inappropriate since theoretically, your response variable has no upper bound. I'd look into the negative binomial distribution. 
As a final note, your data does not necessarily have to come from a well known, "named" distribution. It may have come from a mixture of distributions, or may have a "true" distribution whose mass function is not a nice transformation of x to P(X=x). Don't try too hard to "force" a distribution to the data.
