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Central Limit Theorem
• We know from our study of populations that if a population has a
mean
and a standard deviation
then it can be normally
distributed.
• The central limit deals with what happens when we take a sample
from this population.
Central Limit Theorem
• The mean of an extremely large sample is very hard to obtain.
• Therefore the easiest thing to do is to divide the population into
different samples and get the mean of these samples.
• The mean of these smaller groups will then be approximately the
same as getting the mean of the population.
Population
Population
Large Sample
Sample Means
Mean
𝜇
𝑥
𝜇𝑥
Standard Deviation
𝜎
𝑠
𝜎𝑥 (Standard Error)
Population
S ample 1
S ample 2
S ample 3
S ample 6
S ample 4
S ample 5
Central Limit theorem
• The central limit theorem tells us that if the sample we take is large
(n > 30) and we get the means of these samples they will form a
normal distribution.
• And the mean of these samples will be the same as the mean of the
original population.
• In simple English. The mean of the samples form a normal
distribution when the sample size is large
Central Limit Theorem
• However the standard deviation of the samples will not be the same
as the standard deviation of the population
• So we must account for this error.
Notes on the central limit theorem
• The means of all the samples are sometimes refereed to as the
sampling distribution.
• We can still use the central limit theorem for samples less than 30 as
long it states that the population is normally distributed.
Hom
e
The distribution of sample means
• The centre of the distribution (𝝁𝒙 ) is
identical to the population centre (𝝁).
Hom
e
The distribution of sample means
• The mean of the samples is the same as
the mean of the population
Hom
e
The distribution of sample means
• The distribution is more compact than
the population.
Hom
e
The distribution of sample means
Why is the distribution of sample means
more compact than the population?
Hom
e
The distribution of sample means
How does the spread of the distribution of
sample means compare to the population?
𝟐. 𝟓𝟕
≅𝟖
𝟎. 𝟑𝟐
𝝈
𝝈𝒙 =
𝒏
Summary of the Central Limit Theorem
• The mean of random samples can be approximated by a Normal
Distribution.
• The larger the sample, the better the approximation will be
• Basically, when you split a population into lots of smaller groups of equal
size, if you get the mean of all their means, it will equal the mean of the
whole group.
• However, the standard deviation of the means of these groups is different
to the overall standard deviation.
Summary
Population
Large Sample
Sample Means
Mean
𝜇
𝑥
𝜇𝑥
Standard Deviation
𝜎
𝑠
𝜎𝑥 (Standard Error)
In practice, from the table above, we can say that for n  30
1. The sample means are normally distributed.
2. The mean of the sample means is the same as the population mean.  x  
3. The standard deviation of the sample means is equal to
this is called the standard error.  x 

n

n
Question
A random sample of size 49 is chosen from a population mean that is
known to be normal. The mean of the population is 9 with a standard
deviation of 2.
Find the probability that the sample mean is greater than 10.