How do you know of sampling distribution is large enough from a histogram?
Determining if a sampling distribution is "large enough" from a histogram generally involves looking for certain characteristics that would indicate the distribution is approaching a normal distribution, assuming the central limit theorem applies. This is based on the idea that as the sample size grows, the sampling distribution of the mean will tend to be normally distributed regardless of the shape of the population distribution, as long as the population has a finite variance.
Here are a few visual cues from a histogram that suggest the sampling distribution may be large enough:
1. **Shape**: The histogram should appear bell-shaped, with most data concentrated in the middle and symmetrically tapering off towards either end. This symmetry suggests normality, which is expected in a sufficiently large sampling distribution according to the central limit theorem.
2. **Spread**: The spread of the histogram should reflect that of a normal distribution, where approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and nearly all (99.7%) within three standard deviations.
3. **Outliers**: There should be fewer outliers and extreme values than you might see in a small sample size. Large sample sizes tend to produce more reliable estimates with fewer extreme values, due to the law of large numbers.
4. **Consistency**: With random sampling, the histograms of different sample sets should be relatively consistent with each other if the sample size is sufficiently large. If the histograms look wildly different from one sample to another, this could suggest that the sample size is not large enough to produce a stable estimate of the population distribution.
5. **Sample size**: While visual cues are useful, the actual numerical sample size is important. A good rule of thumb for the central limit theorem to hold is that the sample size should be at least 30, assuming individual observations are independent and identically distributed. However, for populations with extreme skewness or heavy tails, larger sample sizes may be necessary.
One should note that a histogram alone does not provide comprehensive proof of a "large enough" sample size or normality of the sampling distribution. It's a visual tool that needs to be supplemented with other methods, such as statistical tests for normality (e.g., Shapiro-Wilk test, Anderson-Darling test, or Kolmogorov-Smirnov test) or examination of sample size relative to the particular dataset and context. Additionally, some level of subjectivity is involved in interpreting histograms, so it's important to use them as just one part of a broader analysis.