According to a Khan Academy's lecture, the Central Limit Theorem is defined as follows:
Central limit theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution.
To test this definition I considered a population of 100,000 random numbers with the following parameters (see the image below)
Population Parameters:
Mean: 503.76, Median: 503.0, Mode: 338, and Standard Deviation: 285.72
Then, plotting the sampling distribution of the sample mean with varying sample sizes and sample counts resulted in the following observations (each graph is accordingly labeled).
Question: It seems to me that simply increasing the sample size is not sufficient for the distribution to become normal (based on visual observation). The number of samples also should be more.
- Then how do I reconcile these observations with the formal definition of the theorem?
- And are these conclusions correct (given the plots)?
- Increasing the sample size simply reduces the standard error
- As long as the sample size is above a minimum value, increasing the sample count seems sufficient for normality