Central Limit Theorm Lecture 5, Biostatistics

Lecture

1. ميحرلا نمحرلا للها مسبميحرلا نمحرلا للها مسب

2. Sampling Distribution & Central Sampling Distribution & Central
Limit TheoremLimit Theorem
Dr. Mubashir Ahmed
Research Coordinator & Head of Research
Department
Kharadar General Hospital, Karachi

3. Central Limit Theorem
The CLT describes the relationship b/w the sampling distribution of
sample means and the population.
The central limit theorem states that if you take sufficiently large
samples from a population, the samples’ means will be 
normally distributed, even if the population isn’t normally distributed.

Example: Central limit theorem Example: Central limit theorem
A population follows a Poisson distribution (left image). If we take
10,000 samples from the population, each with a sample size of 50,
the sample means follow a normal distribution, as predicted by
the central limit theorem (right image).

4. Central Limit Theorem

5. Conditions of the central limit theorem
The central limit theorem states that the sampling distribution
of the mean will always follow a normal distribution under the
following conditions:
The sample size is sufficiently large. This condition is usually
met if the sample size is n ≥ 30.
The samples are independent and identically distributed
(i.i.d.) random variables. This condition is usually met if
the sampling is random.
The population’s distribution has finite variance. Central limit
theorem doesn’t apply to distributions with infinite variance,
such as the Cauchy distribution. Most distributions have finite
variance.

6. Sample size and the central limit
theorem
Sample size and normality
The larger the sample size, the more closely the sampling distribution will follow
a normal distribution.
When the sample size is small, the sampling distribution of the mean is sometimes
non-normal. That’s because the central limit theorem only holds true when the sample
size is “sufficiently large.”
By convention, we consider a sample size of 30 to be “sufficiently large”
When n < 30, the central limit theorem doesn’t apply. The sampling distribution will
follow a similar distribution to the population. Therefore, the sampling distribution will
only be normal if the population is normal.
When n ≥ 30, the central limit theorem applies. The sampling distribution will
approximately follow a normal distribution.

7. Sample size and standard
deviations
The sample size affects the standard deviation of the
sampling distribution. Standard deviation is a measure of
the variability or spread of the distribution (i.e., how wide or
narrow it is).
When n is low, the standard deviation is high. There’s a lot of
spread in the samples’ means because they aren’t precise
estimates of the population’s mean.
When n is high, the standard deviation is low. There’s not
much spread in the samples’ means because they’re precise
estimates of the population’s mean.

8. Sampling Distribution of the Mean
•When the population is When the population is normally distributednormally distributed

Shape: Shape: Regardless of sample size, the Regardless of sample size, the
distribution of sample means will be normally distribution of sample means will be normally
distributed.distributed.

Center: Center: The mean of the distribution of sample The mean of the distribution of sample
means is similar as the mean of the population. means is similar as the mean of the population.
Sample size does not affect the center of the Sample size does not affect the center of the
distribution.distribution.

Spread: Spread: The standard deviation of the distribution The standard deviation of the distribution
of sample means, is similar as population of sample means, is similar as population
standard deviation. standard deviation.

9. Sampling Distribution of the Mean
•When the population is When the population is not normally distributednot normally distributed

Shape: Shape: When the sample size taken from such When the sample size taken from such
a population is sufficiently large, the distribution a population is sufficiently large, the distribution
of its sample means will be approximately of its sample means will be approximately
normally distributed regardless of the shape of normally distributed regardless of the shape of
the underlying population those samples are the underlying population those samples are
taken from. According to the taken from. According to the Central Limit Central Limit
TheoremTheorem, the larger the sample size, the more , the larger the sample size, the more
normal the distribution of sample means normal the distribution of sample means
becomes.becomes.

10. Sampling Distribution of the Mean
•When the population is When the population is not normally distributednot normally distributed

Center: Center: The mean of the distribution of sample The mean of the distribution of sample
means is the mean of the population, µ. means is the mean of the population, µ.
Sample size affect the center of the distribution.Sample size affect the center of the distribution.

Spread: Spread: The standard deviation of the The standard deviation of the
distribution of sample means, or the standard distribution of sample means, or the standard
error, iserror, is
.
n
x



11. Standardizing a Sample Mean
on a Normal Curve
•The The standardized standardized zz-score -score is how far above or below is how far above or below
the sample mean is compared to the population the sample mean is compared to the population
mean in units of standard error.mean in units of standard error.

““How far above or below” = How far above or below” = sample mean minus sample mean minus
population mean. population mean.

““In units of standard error” = In units of standard error” = divide by divide by
•Standardized sample meanStandardized sample mean
n
x
z




 –

error standard
mean sample

n


12. Example: Standardizing a Mean
•Problem Problem : When a production machine is properly calibrated, it requires : When a production machine is properly calibrated, it requires
an average of 25 seconds per unit produced, with a standard deviation an average of 25 seconds per unit produced, with a standard deviation
of 3 seconds. For a simple random sample of of 3 seconds. For a simple random sample of nn = 36 units, the sample = 36 units, the sample
mean is found to be 26.2 seconds per unit. When the machine is mean is found to be 26.2 seconds per unit. When the machine is
properly calibrated, what is the probability that the mean for a simple properly calibrated, what is the probability that the mean for a simple
random sample of this size will be at least 26.2 seconds?random sample of this size will be at least 26.2 seconds?



Standardized sample mean:Standardized sample mean:
3,25,2.26  x
40.2
36
3
252.26


z
0082.0)40.2()2.26(  zPxP