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

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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 1sample 2sample 3sample 4sample 5sample 6sample 7sample 8sample 9sample 10sample 41sample 42sample 43sample 44sample 45sample 46sample 47sample 48sample 49sample 50
x13.3084237.1058070.7878592.8116020.2551615.0852787.2539752.5491911.3181330.65943013.01746510.2809061.8632084.0009351.1195821.6408257.2421270.80704411.7976884.585229
x22.9694891.0829943.3829713.4744948.9498350.9935947.3351355.5292223.7608361.6909198.6900131.4685300.3769540.1671184.1001100.2559271.7549063.6471591.8835231.101046

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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