Please Help!!! Real World Issue

#1
Hello All,

It has been 20+ years since I have had to breakout the AP stats knowledge, and I have hit a huge brick wall! Thus far my search for stats help has come up empty, so I am praying someone here can help me solve my problem.

We receive 9 different types of calls in a given month. Some call types occur more frequently than others. We are evaluating two different call centers on their effectiveness at resolving customers issues. Each call type that comes in is scored on a scale of 1 to 100, with 1 being utterly ineffective at customer resolution and 100 being a perfect customer resolution. We have a sample size of 30 calls for each of the 9 different call types for each center (30*9*2 = 540 samples), for which we can calculate the mean and standard deviation for each call type for each Center, thus allowing us to calculate the probability that Center A is better than Center B. (I am using the NORM.DIST function in excel to find the probability that Center A is better than Center B)

What I need to know is: For a random call, what is the overall probability that Center A is better than Center B?

Example data
  • Call type 1 - 12% of monthly Call Volume - 88% probability Center A is better than Center B
  • Call type 2 - 01% of monthly Call Volume - 32% probability Center A is better than Center B
  • Call type 3 - 02% of monthly Call Volume - 60% probability Center A is better than Center B
  • Call type 4 - 22% of monthly Call Volume - 75% probability Center A is better than Center B
  • Call type 5 - 22% of monthly Call Volume - 82% probability Center A is better than Center B
  • Call type 6 - 14% of monthly Call Volume - 62% probability Center A is better than Center B
  • Call type 7 - 06% of monthly Call Volume - 61% probability Center A is better than Center B
  • Call type 8 - 16% of monthly Call Volume - 71% probability Center A is better than Center B
  • Call type 9 - 05% of monthly Call Volume - 42% probability Center A is better than Center B
 

Dason

Ambassador to the humans
#2
Before getting to that can you explain your process quoted here more

thus allowing us to calculate the probability that Center A is better than Center B. (I am using the NORM.DIST function in excel to find the probability that Center A is better than Center B)
 
#3
I'd be happy to explain. Assuming a normal distribution, if the Mean of Center B is 88.78 and the Standard Dev is 5.17, and the Mean of Center A is 94.04, we want to know what is the probability of Center B having a score greater than 94.04, or the area to the right of 94.04 on a normal distribution. Using the NORM.DIST function in excel for the variables above we get 88.17%. But the NORM.DIST only gives the area to the left, so I use 1-NORM.DIST to get the area to the right, which is 11.83%. The probability that Center B can have a score better than the mean of Center A is 11.83%; the inverse of this being that there is an 88.17% probability that Center A is better than Center B.

Perhaps that is where I am going wrong? Is this the wrong approach? Is the inverse assumption totally off basis? Any help is greatly appreciated!
 
#4
Is there a reason that you need to attach a probability of which center is "better" than the other? The reason I ask is maybe it would suffice to show the distribution of scores for each call type/center via a histogram or boxplot? Perhaps showing this is enough evidence to suggest that one center is performing better than the other. I'm also curious to know how the score is assigned. Is it a subjective grade from a human being? If this is the case then there is probably inherent variability in scores between (and within) different people.
 
#5
Sadly, yes, I do need to have the probability of which center is "better" given a random call; as this is both a mandate from the CEO and a likely question I will receive from the FCC (super long story, but lets just say my industry is federally regulated). I have the information displayed graphically with a box whisker type; due to the extremely small standard deviations of Center A (in most cases its under 1) it does not lend itself well to the standard box whisker plotting. The score is assigned by a human looking for certain "errors" if you will, on the recorded call; the more errors the lower the score. We have 13 different Scoring Agents, each with a semi evenly distributed number of evaluations for both Center A and Center B; we do have inherent variability within the data, but given the large sample size and the semi-evenly distribution of scoring agents, we are making the assumption to ignore the potential human variability ... after all its close enough for government work!

So I am back to my original query: Is my approach to find which center is "better" using the NORM.DIST function (area to the left under the curve) the correct approach? And how do I go about applying the correct weighting given the disproportionate calling volume of the 9 call types?