The central limit theorem

Introduction to Statistics in Python

Maggie Matsui

Content Developer, DataCamp

Rolling the dice 5 times

die = pd.Series([1, 2, 3, 4, 5, 6])

# Roll 5 times
samp_5 = die.sample(5, replace=True)
print(samp_5)

array([3, 1, 4, 1, 1])

np.mean(samp_5)

2.0

 

six sided die

Introduction to Statistics in Python

Rolling the dice 5 times

# Roll 5 times and take mean
samp_5 = die.sample(5, replace=True)
np.mean(samp_5)

4.4

samp_5 = die.sample(5, replace=True)
np.mean(samp_5)

3.8
Introduction to Statistics in Python

Rolling the dice 5 times 10 times

Repeat 10 times:

  • Roll 5 times
  • Take the mean
sample_means = []

for i in range(10):

  samp_5 = die.sample(5, replace=True)
  sample_means.append(np.mean(samp_5))

print(sample_means)

[3.8, 4.0, 3.8, 3.6, 3.2, 4.8, 2.6,
3.0, 2.6, 2.0]
Introduction to Statistics in Python

Sampling distributions

Sampling distribution of the sample mean

histogram of 10 sample means

Introduction to Statistics in Python

100 sample means

sample_means = []
for i in range(100):
    sample_means.append(np.mean(die.sample(5, replace=True)))

histogram of 100 sample means

Introduction to Statistics in Python

1000 sample means

sample_means = []
for i in range(1000):
    sample_means.append(np.mean(die.sample(5, replace=True)))

histogram of 1000 sample means

Introduction to Statistics in Python

Central limit theorem

The sampling distribution of a statistic becomes closer to the normal distribution as the number of trials increases.

histograms of 10, 100, and 1000 sample means, where higher number of sample means has a more bell-curve shaped distribution

 

* Samples should be random and independent

Introduction to Statistics in Python

Standard deviation and the CLT

sample_sds = []
for i in range(1000):
  sample_sds.append(np.std(die.sample(5, replace=True)))

Distribution of 1000 sample standard deviations of 5 die rolls

Introduction to Statistics in Python

Proportions and the CLT

sales_team = pd.Series(["Amir", "Brian", "Claire", "Damian"])

sales_team.sample(10, replace=True)

array(['Claire', 'Damian', 'Brian', 'Damian', 'Damian', 'Amir', 'Amir', 'Amir', 
      'Amir', 'Damian'], dtype=object)

sales_team.sample(10, replace=True)

array(['Brian', 'Amir', 'Brian', 'Claire', 'Brian', 'Damian', 'Claire', 'Brian', 
      'Claire', 'Claire'], dtype=object)
Introduction to Statistics in Python

Sampling distribution of proportion

Distribution of sample proportions also looks normal

Introduction to Statistics in Python

Mean of sampling distribution

# Estimate expected value of die
np.mean(sample_means)

3.48

# Estimate proportion of "Claire"s
np.mean(sample_props)

0.26

Sampling distribution of sample means with dashed line down the middle  

  • Estimate characteristics of unknown underlying distribution
  • More easily estimate characteristics of large populations
Introduction to Statistics in Python

Let's practice!

Introduction to Statistics in Python

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