The definition:

The sample mean will approximately be normally distributed for large sample sizes, regardless of the distribution from which we are sampling.

Suppose we are sampling from a population with a finite mean and a finite standard-deviation(sigma). Then

Mean and standard deviation of the sampling distribution of the sample mean can be given as:

Where represents the sampling distribution of the sample mean of size n each, and are the mean and standard deviation of the population respectively.

The distribution of the sample tends towards the normal distribution as the sample size increases.

**Code: Python implementation of the Central Limit Theorem**

`import` `numpy ` `import` `matplotlib.pyplot as plt ` ` ` `# number of sample ` `num ` `=` `[` `1` `, ` `10` `, ` `50` `, ` `100` `] ` `# list of sample means ` `means ` `=` `[] ` ` ` `# Generating 1, 10, 30, 100 random numbers from -40 to 40 ` `# taking their mean and appending it to list means. ` `for` `j ` `in` `num: ` ` ` `# Generating seed so that we can get same result ` ` ` `# every time the loop is run... ` ` ` `numpy.random.seed(` `1` `) ` ` ` `x ` `=` `[numpy.mean( ` ` ` `numpy.random.randint( ` ` ` `-` `40` `, ` `40` `, j)) ` `for` `_i ` `in` `range` `(` `1000` `)] ` ` ` `means.append(x) ` `k ` `=` `0` ` ` `# plotting all the means in one figure ` `fig, ax ` `=` `plt.subplots(` `2` `, ` `2` `, figsize ` `=` `(` `8` `, ` `8` `)) ` `for` `i ` `in` `range` `(` `0` `, ` `2` `): ` ` ` `for` `j ` `in` `range` `(` `0` `, ` `2` `): ` ` ` `# Histogram for each x stored in means ` ` ` `ax[i, j].hist(means[k], ` `10` `, density ` `=` `True` `) ` ` ` `ax[i, j].set_title(label ` `=` `num[k]) ` ` ` `k ` `=` `k ` `+` `1` |

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**Output:**

It is evident from the graphs that as we keep on increasing the sample size from 1 to 100 the histogram tends to take the shape of a normal distribution.

**Rule of thumb: **

Of course, the term “large” is relative. Roughly, the more “abnormal” the basic distribution, the larger n must be for normal approximations to work well. The rule of thumb is that a sample size n of at least 30 will suffice.

**Why is this important?**

The answer to this question is very simple, as we can often use well developed statistical inference procedures that are based on a normal distribution such as 68-95-99.7 rule and many others, even if we are sampling from a population that is not normal, provided we have a large sample size.