Standard Error: Central Limit Theorem

Sample statistics (sample mean/ stdev) are…

Random variables

Take on certain values depending on the distribution of data in the sample

Sample variation

Concept that some samples may represent the population better than others

Inevitable

Sample means tend to be close to the population mean as we increase our sample size → taking more from the population = covering more area

Sampling distribution of the sample meanp

Probability distribution of all possible sample means from all possible samples

This distribution is normal → allows use of the empirical rule to describe what percentage of all sample means are within certain values

Central Limit Theorem (CLT)

For any population regardless of its distribution, as we increase the size of our samples, the samples’ means will be normally distributed (binomial, uniform, normal, etc.)

Using CLT to estimate

• As long as we have a sufficiently large sample, (roughly 30+) allows us to use the sample mean in place of an unknown population mean and standard deviation equal to the standard error of the mean

• Allows us to define an interval where the sample means are expected to fall as n is sufficiently large enough

Sampling error

Sample mean, etc. not represented by the population mean, etc. → can describe by standard deviation

Standard error of the mean

Population standard deviation/ square root of the number of observations (n)

Measure of uncertainty

Difference between standard error and standard deviation

• Error: assess how far a sample statistic likely falls from the population parameter

  • quantify variability between samples drawn from the same population

• Deviation: assess how far a particular data point is likely to fall from the mean

  • quantify variability of values in a dataset

When to use proportions?

When our random variable is nominal and has 2 mutually exclusive groups, rather than the mean, we look at proportion

Proportion

Fraction, ratio, or percentage that indicates what point of the sample/ population has a particular trait of interest

P = x/n

You can apply the CLT and say the distribution of a sample proportion is normal, given the sample size is sufficiently large. You can determine if the sample size is large enough by:

nπ ≥ 5 and nπ(1-π) ≥ 5

Not pi as in 3.14, proportion of the population