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Sampling Distributions & the Limits of the CLT

Separate the distribution of the data from the distribution of a statistic, and find where the central limit theorem stops applying.

Ch 1 1.4
Simulator
Foundations & Data Quality
15 mindifficulty 2/5

Controls

Population distribution
Statistic per sample
5

How many observations go into each replicate.

1200

Replicates sharpen the picture of the sampling distribution; they do not change it.

Random seed

Every result on this page is a deterministic function of the seed and the controls.

Population

One large draw from the generative model you selected.

00.200.400.600.80012345[-0.31, -0.16) n=0[-0.16, -0.01) n=0[-0.01, 0.13) n=476[0.13, 0.28) n=498[0.28, 0.43) n=424[0.43, 0.57) n=332[0.57, 0.72) n=322[0.72, 0.87) n=265[0.87, 1.02) n=248[1.02, 1.16) n=211[1.16, 1.31) n=174[1.31, 1.46) n=134[1.46, 1.60) n=109[1.60, 1.75) n=112[1.75, 1.90) n=84[1.90, 2.04) n=79[2.04, 2.19) n=76[2.19, 2.34) n=64[2.34, 2.49) n=58[2.49, 2.63) n=42[2.63, 2.78) n=34[2.78, 2.93) n=34[2.93, 3.07) n=34[3.07, 3.22) n=20[3.22, 3.37) n=20[3.37, 3.51) n=22[3.51, 3.66) n=15[3.66, 3.81) n=17[3.81, 3.96) n=14[3.96, 4.10) n=9[4.10, 4.25) n=14[4.25, 4.40) n=9[4.40, 4.54) n=9[4.54, 4.69) n=3[4.69, 4.84) n=7[4.84, 4.99) n=2[4.99, 5.13) n=7[5.13, 5.28) n=3[5.28, 5.43) n=3[5.43, 5.57) n=2μxdensity

Sampling distribution of the mean

1200 replicates of size n = 5.

00.200.400.600.8010.5011.5022.50[0.11, 0.18) n=3[0.18, 0.26) n=4[0.26, 0.33) n=25[0.33, 0.41) n=40[0.41, 0.48) n=46[0.48, 0.55) n=76[0.55, 0.63) n=73[0.63, 0.70) n=92[0.70, 0.78) n=69[0.78, 0.85) n=70[0.85, 0.93) n=91[0.93, 1) n=85[1, 1.07) n=74[1.07, 1.15) n=72[1.15, 1.22) n=53[1.22, 1.30) n=43[1.30, 1.37) n=54[1.37, 1.45) n=45[1.45, 1.52) n=40[1.52, 1.59) n=26[1.59, 1.67) n=16[1.67, 1.74) n=21[1.74, 1.82) n=16[1.82, 1.89) n=15[1.89, 1.96) n=10[1.96, 2.04) n=10[2.04, 2.11) n=6[2.11, 2.19) n=3[2.19, 2.26) n=3[2.26, 2.34) n=2[2.34, 2.41) n=1[2.41, 2.48) n=2[2.48, 2.56) n=3[2.56, 2.63) n=5[2.63, 2.71) n=2[2.71, 2.78) n=2μx̄density
Simulated statisticMatched normal densityPopulation mean μ
Mean of statistic
0.9998
μ = 1
Std. error (observed)
0.4597
σ/√n (CLT)
0.4472
prediction for the mean
Observed ÷ predicted
1.028
≈ 1 when the CLT scaling holds