Odd behavior from a normal distribution Currently I have a distribution with the following mean and stdev.
Mean: 85.38
Stdev: 7.737
p(x= < 80% via a cumulative distribution): 4.05%
I have then modified the population by replacing one of the numbers with a deliberately low value which will impact the mean and reduce it to:
Mean: 82.22
Stdev: 16.78
I expected the probability of X occurring <= 80% of the time to increase, however, it actually decreased. I could not understand why. I have attached the two curves below. From 4.05% at <= 80%, it has now dropped to 2.36% which doesn't make sense to me.
My thinking is that it could be skewed because with 3 stdevs away from the mean (already set quite high at 80+), it may have some sort of adverse impact on the result. Could anyone assist me regarding this?


 A: The two graphs you've posted are not CDF's, they're PDF's of a Normal distribution. PDF's are bell curved, and CDF's are non-decreasing functions which approach 1 as $x$ goes to infinity.
You received 4.05% from the lower graph, however since that isn't a CDF this isn't a probability.
If $f(x)$ is the density (PDF) then $F(x) = \int_{-\infty}^{x} f(x) dx$ is the CDF that you need. $F(x)$ will give you the probability that your Normal random variable is less than or equal to $x$, which in this case $x$ is 80%.
Below is what such a CDF will look like (but with a different mean and variance obviously). When you add your excessively low value to the population, this should shift the graph to the left, and so the probability for any $x$ should increase as you suspected.

A: Thanks for the insight Patty - I looked back on how I did my PDF and I realised, as you correctly pointed out, I had not down a CDF but a PDF. I had accidentally set the CDF value to False (0) rather than True (1) which only gave me the probability at point X rather than the whole cumulative distribution %.
Once I applied the CDF as per the logic provided I had the below results:

