Hello aspirant,

First, you should work at what you like doing best. Recall the old Irish saying, "A man who loves his job never works a day in his life."

Knowledge of statistics - and the ability to program - can be very valuable to an EE, depending on your specialization.

1. Design of circuits - transistors and other components (devices) do not have fixed parametric values. Due to the nature of the manufacturing process, devices are observed to have a distribution of values. Your circuits need to work across the full range of these distributions.
2. Manufacturing related - all manufacturing has variation, which needs to be measured and controlled (to the extent possible). Here the concepts of quality control (QC) are useful - statistical process control (SPC), average outgoing quality, etc. Design Of Experiments (DOE) is used for improving the manufacturing process.
3. Semiconductor fabrication - a subset of #2, but here any job in a fabrication plant requires the knowledge and application of statistics. Data is collected and analyzed on a continuous basis. In addition to QC and DOE, regression is extensively used. Data mining techniques (C4.5 trees, boosted trees, random forest) are also applied here to find the cause of fabrication problems. Alas, most EEs don't have this knowledge coming in - so they have to learn it on the job. However, if a job candidate did have these skills, it would be a huge plus factor in hiring that person. The ability to program (any language) is also a plus factor, as often the data need to be cleaned and rearranged prior to analysis - and a lot of analysis needs to be done (think of 1000 charts).
4. Telecommunications - real channels have noise. Where there is noise, there are statistics.

EE is a broad field, so I can't tell you the application of statistics to each. However, I can confidently tell you that knowledge of statistics, and some practice applying that knowledge, will be useful for every specialty.

Now if you want to go into data science, your EE background will be helpful in at least ways:

1. EE is all math. So you're good at math and problem solving. You have the proper preparation for the new math of statistics and data mining that you will be learning.
2. I would expect you like computers and have done some programming of one. This is an essential skill for data mining.
3. You have a degree in an application area - in this case, EE. This will help you land a job in that area. Contrary to some of the hype and promises of data mining, domain knowledge is essential to problem solving. You need to understand the data you are analyzing. There is no universal problem solving algorithm.

Finally, and back to the original point, get the job you love doing. If that's EE, go with EE (but learn some statistics). If it's data mining and analysis, go there. A job that you are doing only for a big paycheck will only bring you grief and unhappiness. You spend a lot if time at work - make sure that time is spent doing something you enjoy.

Thank you!