Test for statistical significance in performance variability

I'm analysing data collected from a call centre activity in which agents are calling prospective leads. For each call made, if the lead coverts the call is a success and if the lead does not convert the call is a fail.

The data are for 5 agents. Each agent made a different number of calls in a unspecified period of time, and each has a different success rate. Boiled down, I have 5 percentages for the success rates.

I need to find a way to determine if the variability of the success rates is statistically significant, but can't think of an appropriate test. The main problem being that I have no benchmark value to test against. Of course, the data should have been collected in a better-designed experiment. However this is a typical 'here's some data' in an Excel spreadsheet scenario.

None of the tests I'm more familiar such as t-tests, Binomial tests, ANOVA, CHI Sq GOF, regressions, hypothesis testing against a null value, etc, seem to make sense. Is there a method I'm not thinking of for testing statistical significance in the variability in performance with this type of data?

Why not simply a Chi-Squared test of independence over this kind of table ?

+-----+-----+-----+-----+-----+
| Ag_1| Ag_2| Ag_3| Ag_4| Ag_5|
+----------------+-----+-----+-----+-----+-----|
| SuccessCount   |  20 |  5  |  10 |  40 |  21 |
| FailCount      |  30 |  30 |  10 |  30 |  11 |
+----------------+-----+-----+-----+-----+-----+

Ok, thanks - though I'm not sure I follow the theory of how a Chi-Squared Independence test applies to the data. I've used the 'answer' field to respond as this might be the correct solution. Let me know what you think.

Let's say I set hypotheses as follows:

• Null: variation in signups-per-call by agent is a result of chance.
• Alternative: variation in signups-per-call by agent is statistically significant and not a result of chance.

And in Python (SciPy) construct the test:

ag_1 = [326, 1908]
ag_2 = [263, 1478]
ag_3 = [81, 750]
ag_4 = [53, 339]
ag_5 = [45, 199]

data = [ag_1, ag_2, ag_3, ag_4, ag_5]
stats.chi2_contingency(data)

Results:

X-squared = 21.407
Degrees of freedom = 4
p-value = 0.0002629

So I reject the null: the variability in signups-per-call by agent is statistically significant. At least in theory anyway. In reality, as this is an observational study, I think there's a whole host of confounders that muddy the inference.

• This is it. For an obscure reason, in R I have X-squared = 18.7953 and p-value = 0.0008621 with your data but this is not that far. I don't know what to say about confounders. The most important factor - I think - is that your agents are in the same conditions to be compared. To study how big is your problem, and if you can request another dataset, you may study how variable are the performances across time. For example ask long run data and plot % of success every 100 calls to get an idea about how variable success rate is for a common agent. – brumar Jun 26 '15 at 15:10