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A statistical theory called the central limit theorem states that when a large sample size has a small variance, samples will be normally distributed and their means will be about equal to those of the total population.
The central limit theorem permits one to assume that the sampling distribution of the mean would often be normally distributed hence, it is helpful for examining big data sets. This makes statistical analysis and inference simpler.
A particular random variable of importance in many real-time applications is the sum of several independent random variables. We may utilize the CLT in these circumstances to support the adoption of the normal distribution.
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If the sample size is high enough, the central limit theorem states that the sampling distribution of the mean will always be normally distributed. No matter if the population has a distribution or not, the mean’s sample distribution will be normal.
In simpler terms, the theorem states that the distribution of the sample means will be normally distributed regardless of the underlying distribution of population given that the sample size is large enough typically greater than thirty.
From the theorem, we can conclude the following facts.
It implies,
Let be a random sample from population which has a mean
and variance
.
Then:
is an unbiased estimator of.
As the central limit theorem is used for populations greater than 30 the difference between the calculated sample standard deviation and the population standard deviation becomes very negligible.
The ratio of the sample standard deviation and the population standard deviation is:
And for large populations becomes negligibly small.
If there are not too many extreme values in the distribution, the sum (and hence the mean) of any set of random variables tends asymptotically to a normal distribution and hence we can use the statistical formulas we use for normal distribution.
The formula for sample means:
The proof for the formula:
Considering which are independent and have a mean
and a finite variance
,then taking a random variable
We get
The other formulas used are
and
.
Where,
=sample mean
=population mean
=sample standard deviation
=population standard deviation
=sample size
When we have to usually represent the sample means it would be a continuous curve because it is a continuous probability distribution where the values lie in a symmetrical fashion mostly situated around the mean.
But with the central limit theorem, we can figure the frequency of the occurrence of the values because the means of each value is calculated and hence represent a given data as a histogram.
Example 1: The data of heights of the male population follows a normal distribution. Its mean and standard deviation are 70 inches and 15 inches, respectively. If we consider the records of 45 males, then calculate the standard deviation of the selected sample?
Solution 1:
Mean of the population
Standard deviation of the population = 15
sample size n = 45
Standard deviation is given by:
Hence the standard deviation would be 2.236
Example 2: A set of samples has been collected from a larger sample and the sample mean values are 34.6,67,90,12.8,45.2. Find the population mean.
Solution 2:
The population mean values can be calculated using the formula
Which implies,
Hence the population mean is 49.92
Example 3: The mean age of people living in a city is 40 years and the standard deviation is 9 years. What is the variance and mean for sample sizes 81 and 9.
Solution:
The variance can be calculated by taking the squaring the standard deviation.
is the standard deviation formula.
For the sample size 81,
Hence the variance is 1 and the mean is 40.
For the sample size 9 the central limit theorem cannot be applied as the size needs to be greater than 30.
Example 4: A distribution has a mean of 60 and a standard deviation of 24. If 121 samples are randomly drawn from this population then using the central limit theorem find the value that is five sample deviations above the expected value.
Solution 4:
We know that the mean of the sample equals the mean of the population according to the theorem.
Hence, mean = 60.
Standard deviation formula
Substituting the values
Hence the value that is five sample deviations above is
Example 5: The average weight of a can is 50 pounds which have a standard deviation of 20 pounds. Taking a sample of 50 cans that are selected at random and then their weights are calculated, what is the probability that the mean weight of the sample is less than 30 pounds?
Solution 5:
Population mean: pounds
Population standard deviation: pounds
Sample size:
Now we use , z-score, and we get,
The sample standard deviation:
And,
= 5.66
Finding the z- score for the value of pounds
Using z- score table OR normal cdf function on a statistical calculator,
Hence, the probability that the weight of the can is less than 40 pounds is 0.0207%
The central limit theorem is very widely used because it states that if we have sufficiently large samples from the population the distribution of samples can be normally distributed, and all the statistical calculations can be made for a data set with very large populations. Because of how the theorem helps to come to mathematical conclusions about huge sets of data, it is used in a lot of real-life situations and conditions. We have also seen how it helps in visualizing data which helps with the analysis and interpretation of the said data.
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A sampling distribution refers to a probability distribution of a statistic that comes from choosing random samples of a given population.
Z-score is a statistical measurement that describes a value’s relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean.
In the statistical theory of the design of experiments, randomization involves randomly allocating the sample space across the population.
Standard Deviation is a measure that shows how much variation or how dispersion from the mean exists.
Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.