Central Limit Theorem explained in Python (with examples) – Part III

See Part I and Part II to get started.

Now we want to see how the sampling distribution looks for this population. We will consider two cases, i.e. with a small sample size (n= 2), and a large sample size (n=500).

First, we will draw 50 random samples from our population of size 2 each. The code to do the same in Python is given below:

# drawing 50 random samples of size 2 from the exponentially distributed population
sample_size = 2
df2 = pd.DataFrame(index= [‘x1’, ‘x2’] )

for i in range(1, 51):
exponential_sample = np.random.exponential((1/rate), sample_size)
col = f’sample {i}’
df2[col] = exponential_sample

# Taking a peek at the samples
df2

	sample 1	sample 2	sample 3	sample 4	sample 5	sample 6	sample 7	sample 8	sample 9	sample 10	…	sample 41	sample 42	sample 43	sample 44	sample 45	sample 46	sample 47	sample 48	sample 49	sample 50
x1	3.308423	7.105807	0.787859	2.811602	0.255161	5.085278	7.253975	2.549191	1.318133	0.659430	…	13.017465	10.280906	1.863208	4.000935	1.119582	1.640825	7.242127	0.807044	11.797688	4.585229
x2	2.969489	1.082994	3.382971	3.474494	8.949835	0.993594	7.335135	5.529222	3.760836	1.690919	…	8.690013	1.468530	0.376954	0.167118	4.100110	0.255927	1.754906	3.647159	1.883523	1.101046

2 rows × 50 columns

For each of the 50 samples, we can calculate the sample mean and plot its distribution as follows:

# Calculating sample means and plotting their distribution
df2_sample_means = df2.mean()
sns.distplot(df2_sample_means);

We can observe that even for a small sample size such as 2, the distribution of sample means looks very different from that of the exponential population, and looks more like a poor approximation of a normal distribution, with some positive skew.

Stay tuned for the next installment, in which Ashutosh will repeat the above process, but with a much larger sample size (n=500).

Visit QuantInsti for additional details and to download full code: https://blog.quantinsti.com/central-limit-theorem/.

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