Sampling Distributions Chapter 9 Central Limit Theorem

Take a random sample of size n from any population with mean  and standard deviation . When n is large, the sampling distribution of the sample mean is close to the normal distribution. How large a sample size is needed depends on the shape of the population distribution.

Uniform distribution Sample size 1

Uniform distribution Sample size 2

Uniform distribution Sample size 3

Uniform distribution Sample size 4

Uniform distribution Sample size 8

Uniform distribution Sample size 16

Uniform distribution Sample size 32

Triangle distribution Sample size 1

Triangle distribution Sample size 2

Triangle distribution Sample size 3

Triangle distribution Sample size 4

Triangle distribution Sample size 8

Triangle distribution Sample size 16

Triangle distribution Sample size 32

Inverse distribution Sample size 1

Inverse distribution Sample size 2

Inverse distribution Sample size 3

Inverse distribution Sample size 4

Inverse distribution Sample size 8

Inverse distribution Sample size 16

Inverse distribution Sample size 32

Parabolic distribution Sample size 1

Parabolic distribution Sample size 2

Parabolic distribution Sample size 3

Parabolic distribution Sample size 4

Parabolic distribution Sample size 8

Parabolic distribution Sample size 16

Parabolic distribution Sample size 32

Loose ends An unbiased statistic falls sometimes above and sometimes below the actual mean, it shows no tendency to over or underestimate. As long as the population is much larger than the sample (rule of thumb, 10 times larger), the spread of the sampling distribution is approximately the same for any size population.

Loose ends As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean  ? As the sample size increases, the mean of the observed sample gets closer and closer to . (law of large numbers)