It need not be if:
the number of samples is small;
the elements within each sample, and the samples themselves are not selected independently.
The distribution of sample means will not be normal if the number of samples does not reach 30.
The central limit theorem basically states that for any distribution, the distribution of the sample means approaches a normal distribution as the sample size gets larger and larger. This allows us to use the normal distribution as an approximation to binomial, as long as the number of trials times the probability of success is greater than or equal to 5 and if you use the normal distribution as an approximation, you apply the continuity correction factor.
If the samples are drawn frm a normal population, when the population standard deviation is unknown and estimated by the sample standard deviation, the sampling distribution of the sample means follow a t-distribution.
the means does not change
No, it is not.
The distribution of sample means will not be normal if the number of samples does not reach 30.
Not necessarily. It needs to be a random sample from independent identically distributed variables. Although that requirement can be relaxed, the result will be that the sample means will diverge from the Normal distribution.
The distribution of the sample means will, as the sample size increases, follow the normal distribution. This is true for any given distribution (e.g. does not need to be a normal distribution). This concept is from the central limit theorem. It is one of the most important concepts in statistics, along with the law of large numbers. An applet to help you understand this concept is located at: http:/www.stat.sc.edu/~west/javahtml/CLT.html
The central limit theorem basically states that for any distribution, the distribution of the sample means approaches a normal distribution as the sample size gets larger and larger. This allows us to use the normal distribution as an approximation to binomial, as long as the number of trials times the probability of success is greater than or equal to 5 and if you use the normal distribution as an approximation, you apply the continuity correction factor.
According to the Central Limit Theorem if the sample size is large enough then the means will tend towards a normal distribution regardless of the distribution of the actual sample.
If the samples are drawn frm a normal population, when the population standard deviation is unknown and estimated by the sample standard deviation, the sampling distribution of the sample means follow a t-distribution.
Frequently it's impossible or impractical to test the entire universe of data to determine probabilities. So we test a small sub-set of the universal database and we call that the sample. Then using that sub-set of data we calculate its distribution, which is called the sample distribution. Normally we find the sample distribution has a bell shape, which we actually call the "normal distribution." When the data reflect the normal distribution of a sample, we call it the Student's t distribution to distinguish it from the normal distribution of a universe of data. The Student's t distribution is useful because with it and the small number of data we test, we can infer the probability distribution of the entire universal data set with some degree of confidence.
The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30.
the means does not change
The sample mean is distributed with the same mean as the popualtion mean. If the popolation variance is s2 then the sample mean has a variance is s2/n. As n increases, the distribution of the sample mean gets closer to a Gaussian - ie Normal - distribution. This is the basis of the Central Limit Theorem which is important for hypothesis testing.
No, it is not.
It means distribution is flater then [than] a normal distribution and if kurtosis is positive[,] then it means that distribution is sharper then [than] a normal distribution. Normal (bell shape) distribution has zero kurtosis.