Central Limit Theorem
Why do standardized sample means approach a standard normal distribution?
How this lesson unfolds.
Begin with What object does the Central Limit Theorem describe? The sequence then connects Sample means are centered at the population mean and builds toward The Central Limit Theorem connects sampling to inference.
What object does the Central Limit Theorem describe?
The CLT describes the distribution of a statistic across repeated samples. One sample gives one mean; repeated samples give a sampling distribution of means.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionSample means are centered at the population mean
Why do we subtract μ from the sample mean?
Because E[X̄]=μ, subtracting μ moves the sampling distribution's center to zero without changing its shape.
Because E[X̄]=μ, subtracting μ moves the sampling distribution's center to zero without changing its shape.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionThe sample mean has its own spread
Why is the spread of sample means σ/√n?
Averaging n independent observations reduces variance from σ² to σ²/n, so the standard deviation of X̄ is σ/√n. This is the standard error.
Averaging n independent observations reduces variance from σ² to σ²/n, so the standard deviation of X̄ is σ/√n.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionStandardization creates a common scale
Why subtract the center and divide by the spread?
Subtracting μ centers the error at zero. Dividing by σ/√n measures it in standard-error units, creating a unitless variable with mean 0 and variance 1.
Dividing by σ/√n measures it in standard-error units, creating a unitless variable with mean 0 and variance 1.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionN(0,1) names shape, center and spread
What does N(0,1) actually mean?
N indicates a normal distribution, 0 is its mean, and 1 is its variance. Standardization supplies the center and scale; the CLT supplies the limiting bell shape.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionDividing by standard error makes variance one
Why does the standardized error have variance 1?
The numerator has standard deviation σ/√n. Dividing it by that same spread gives standard deviation 1 and therefore variance 1.
The numerator has standard deviation σ/√n.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionRead the theorem from left to right
What exactly approaches the bell curve as n grows?
The standardized sample-mean error—not raw observations and not one average—converges in distribution to N(0,1), under the theorem's assumptions.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionThe Central Limit Theorem connects sampling to inference
What is the reusable CLT story?
Repeated independent sampling creates means centered at μ with standard error σ/√n. After standardization, their distribution approaches the standard normal as n increases.
Repeated independent sampling creates means centered at μ with standard error σ/√n.
- Learning goalExplain the CLT through repeated sampling, centering, standard error, scaling and convergence.
- Helpful foundationsexpectation · variance · independent samples
- Common trapThe CLT concerns standardized sampling distributions, not a claim that raw observations become normally distributed.
- Next connectionNormal approximations and confidence intervals