# 10% condition

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The 10 percent condition in statistics says that whenever we wish to draw a sample from a given population we should take the sample size to be at least 10% of the total population. So, if you wanted to conduct a statistical study of a population of size ten thousand you can do so by extracting a random sample of a thousand people.

Why is the 10% condition necessary?

Many of the famous theorems that are used in real life, such as the Central Limit Theorem, are derived under the assumption that the sample is chosen with replacement from the population. But, in real life, we almost always choose our random sample without replacement.

Hence we need to provide some justification for using the classical theorems of statistics in such a situation. It can be proved that if choose a sample according to the 10 percent condition then the standard error of the random variable in the without replacement case differs from the standard error of the variable in the with replacement case merely by a factor of 0.05 which is considered negligible.

When should one use the 10% condition?

One should use the 10 percent condition in the following situations:

1. Using sample mean to estimate the population mean.
2. Using sample variance to estimate population variance.
3. Conducting tests of hypothesis for mean of the population.
4. Calculate probabilities for successive Bernoulli trials that are not independent (as stated above the obtained probability will be within a factor of 0.05 of the original probability if you apply the 10% rule),

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