Elevate Your Statistical Analysis Using Python#

Quick Environment Setup Example#

Below is a simple guide to setting up an environment with Anaconda:

1. Download and install Anaconda from the official site.
2. Open Anaconda Prompt (Windows) or Terminal (macOS/Linux).
3. Create a new environment:

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conda create --name stats_env python=3.9
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conda activate stats_env
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conda install numpy pandas scipy scikit-learn seaborn matplotlib

4. Launch Jupyter Notebook:

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jupyter notebook

With that, you’re set to begin coding. Alternatively, you can use Google Colab, which is a cloud-based approach requiring only a Google account.

Central Tendency Measures#

1. Mean: The average value.
2. Median: The middle value when data is sorted.
3. Mode: The most frequently occurring value.

Dispersion Measures#

1. Variance: Measures how far each value in the data set is from the mean.
2. Standard Deviation (SD): The square root of the variance, lending it the same units as the original data.
3. Range: The difference between the maximum and minimum values in the dataset.

Example in Python#

Let’s illustrate computing these statistics in Python. Suppose we have a list of exam scores:

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import numpy as np
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scores = [88, 92, 79, 93, 85, 90, 78, 95, 91, 87]
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mean_score = np.mean(scores)
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median_score = np.median(scores)
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mode_score = max(set(scores), key=scores.count)  # Simple approach for mode
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variance = np.var(scores, ddof=1)  # ddof=1 => sample variance
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std_dev = np.std(scores, ddof=1)
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print("Mean:", mean_score)
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print("Median:", median_score)
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print("Mode:", mode_score)
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print("Variance:", variance)
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print("Standard Deviation:", std_dev)

Output might look like this (numbers can vary slightly by rounding):

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Mean: 87.8
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Median: 88.5
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Mode: 78
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Variance: 31.733...
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Standard Deviation: 5.635...

Reading Data#

You can import data from CSV, Excel, SQL databases, and more. Here’s an example of reading a CSV file using pandas:

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import pandas as pd
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# Assuming 'data.csv' is in your current directory
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df = pd.read_csv('data.csv')

Common Cleaning Tasks#

1. Handling Missing Values: Replace or drop missing values (NaN).
2. Handling Outliers: Determine whether outliers are valid data points or measurement errors.
3. Type Conversion: Correct data type mismatches (e.g., numeric data stored as strings).

Example of Data Cleaning#

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# Drop rows where any column has NaN
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df_clean = df.dropna()
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# Fill missing values in a specific column with mean
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df['column_name'] = df['column_name'].fillna(df['column_name'].mean())
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# Convert data type
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df['some_string_column'] = df['some_string_column'].astype(str)
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df['some_numeric_column'] = pd.to_numeric(df['some_numeric_column'], errors='coerce')
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# Handle outliers using IQR
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Q1 = df['some_numeric_column'].quantile(0.25)
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Q3 = df['some_numeric_column'].quantile(0.75)
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IQR = Q3 - Q1
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lower_bound = Q1 - 1.5 * IQR
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upper_bound = Q3 + 1.5 * IQR
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df_filtered = df[(df['some_numeric_column'] >= lower_bound) & (df['some_numeric_column'] <= upper_bound)]

By doing these basic tasks, you lay the foundation for accurate statistical modeling. Skipping or performing them incorrectly can lead to misleading results.

Descriptive Statistics#

Once you have a clean dataset, you can use pandas to get a quick overview:

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df_clean.describe()

This gives you counts, means, medians, and quartile information. You can also use Skewness and Kurtosis to understand the distribution shape.

Data Visualization#

Python offers multiple plotting libraries, but Matplotlib and Seaborn are the most commonly used for EDA.

• Line Plots: For time-series or continuous data trends.
• Scatter Plots: For relationships between two numerical variables.
• Histograms: For distribution analysis.
• Box Plots: For understanding outliers and distribution shape.

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import matplotlib.pyplot as plt
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import seaborn as sns
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# Histogram
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plt.hist(df_clean['some_numeric_column'], bins=20, edgecolor='black')
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plt.title("Histogram of Values")
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plt.xlabel("Value")
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plt.ylabel("Frequency")
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plt.show()
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# Box Plot
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sns.boxplot(x=df_clean['some_numeric_column'])
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plt.title("Box Plot of Values")
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plt.show()

Types of Distributions#

• Normal (Gaussian) Distribution
• Uniform Distribution
• Binomial Distribution
• Poisson Distribution
• Exponential Distribution

SciPy’s stats module provides methods for probability density functions (PDF), cumulative distribution functions (CDF), and random sampling for each distribution.

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from scipy.stats import norm, binom
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# Normal Distribution Example
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mean = 0
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std = 1
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x = norm.rvs(loc=mean, scale=std, size=1000)  # Generate random samples
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# Binomial Distribution Example
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n = 10
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p = 0.5
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y = binom.rvs(n, p, size=1000)
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# PDF and CDF
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pdf_values = norm.pdf(x, mean, std)
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cdf_values = norm.cdf(x, mean, std)

Practical Usage#

1. Modeling random phenomena: Determine how likely events are to occur.
2. Setting confidence intervals: Many parametric tests assume normality.
3. Monte Carlo simulations: Random sampling to simulate complex systems.

Steps in Hypothesis Testing#

1. Formulate Null (H0) and Alternative (Ha) Hypotheses
2. Choose a Significance Level (α)
3. Compute Test Statistic and p-value
4. Reject or Fail to Reject H0 based on the p-value.

Common Tests#

• Z-test: For large samples or known population variance.
• T-test: For small samples or unknown population variance.
  • One-Sample T-test: Compare the sample mean to a known value.
  • Two-Sample T-test: Compare means of two groups (Independent or Paired).
• Chi-Square Test: For categorical data.
• Mann-Whitney U or Wilcoxon Signed-Rank: Non-parametric alternatives.

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import numpy as np
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from scipy.stats import ttest_ind
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groupA = [5.1, 5.3, 5.5, 4.9, 5.2]
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groupB = [6.2, 5.9, 6.1, 6.0, 6.3]
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t_stat, p_val = ttest_ind(groupA, groupB)
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print("t-statistic:", t_stat)
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print("p-value:", p_val)

Correlation#

• Pearson’s Correlation Coefficient (r): Measures linear correlation between two variables. Values range from -1 to 1.
• Spearman’s Rank Correlation: A non-parametric measure of rank correlation.

Example of computing Pearson’s r in Python:

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import pandas as pd
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import numpy as np
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data = {
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    'hours_studied': [1, 2, 3, 4, 5, 6],
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    'test_score':    [50, 55, 60, 65, 70, 80]
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}
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df_corr = pd.DataFrame(data)
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corr_matrix = df_corr.corr(method='pearson')
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print(corr_matrix)

Linear Regression#

Regression modeling goes beyond correlation by enabling you to predict a response variable based on one or more predictors. Here’s a simple linear regression example using scikit-learn:

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from sklearn.linear_model import LinearRegression
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X = df_corr[['hours_studied']]
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y = df_corr['test_score']
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model = LinearRegression()
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model.fit(X, y)
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print("Intercept:", model.intercept_)
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print("Coefficient:", model.coef_)
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# Make a prediction
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predicted_score = model.predict([[7]])  # Predict score for 7 hours studied
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print("Predicted Score for 7 hours studied:", predicted_score[0])

This yields an approximate formula of the form:
Test Score = Intercept + (Coefficient × Hours Studied)

One-Way ANOVA#

You have one independent variable (factor) with more than two categories. For instance, comparing mean test scores across three teaching methods:

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import pandas as pd
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from scipy.stats import f_oneway
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methodA = [78, 82, 85, 90, 88]
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methodB = [72, 75, 80, 79, 77]
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methodC = [90, 92, 89, 93, 91]
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f_stat, p_val = f_oneway(methodA, methodB, methodC)
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print("F-Statistic:", f_stat)
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print("p-value:", p_val)

If the p-value is below your significance level, you can conclude at least one method’s mean differs statistically from the others. To determine exactly which groups differ, you can perform post-hoc tests like Tukey’s HSD.

Time Series Analysis#

For data that changes over time (e.g., stock prices, weather data), specialized methods such as ARIMA, SARIMA, and Exponential Smoothing become valuable:

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import pandas as pd
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from statsmodels.tsa.arima.model import ARIMA
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# Assume df_time has columns ['date', 'value']
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df_time = df_time.set_index('date')
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model = ARIMA(df_time['value'], order=(1,1,1))
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results = model.fit()
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print(results.summary())

Key topics include:

• Stationarity checks (ADF test)
• Seasonal decomposition
• Forecast accuracy metrics (MAE, RMSE)

Bayesian Statistics#

Bayesian approaches offer flexible modeling choices and can be more interpretable in certain scenarios. Libraries like PyMC in Python provide MCMC (Markov Chain Monte Carlo) algorithms for Bayesian inference:

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import pymc as pm
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import numpy as np
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with pm.Model() as bayesian_model:
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    mu = pm.Normal('mu', mu=0, sigma=10)
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    sigma = pm.HalfNormal('sigma', sigma=10)
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    obs = pm.Normal('obs', mu=mu, sigma=sigma, observed=np.random.randn(50))
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    trace = pm.sample(1000, tune=500, chains=2)
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pm.summary(trace)

Machine Learning Integration#

Statistical models often serve as a foundation for more advanced machine learning tasks:

• Feature engineering based on statistical insights.
• Adding domain expertise to interpret ML model results.
• Using bagging, boosting, or neural networks for predictive tasks.

Scikit-learn provides a unified interface for both classical ML algorithms and some statistical techniques like penalized regressions (Lasso, Ridge).

Reproducibility and Version Control#

• Reproducible Analysis: Store and manage your code, environment (e.g., environment.yml for conda), and data in a version-controlled repository (Git).
• Notebooks: Use Jupyter or other interactive notebooks with literate programming. Provide sufficient documentation, context, and interpretation alongside code blocks.

Data Pipelines and Automation#

• For large projects, adopt pipeline tools like Airflow or Luigi.
• Schedule periodic data ingestion and cleaning tasks.
• Automate model training and validation for continuous data feeds.

Deployment and Scaling#

• Deploy your statistical models as web services using frameworks like FastAPI or Flask.
• Containerize your environment with Docker to ensure consistent execution anywhere.
• Scale out via cloud environments, such as AWS or GCP, for large datasets or real-time predictions.

Ethical Considerations#

• Bias in Data: Evaluate your data sources to ensure that you are not perpetuating discrimination or inequality.
• Privacy: Follow regulations (like GDPR) and best practices when handling personally identifiable information (PII).
• Transparency: Clearly communicate assumptions and potential model limitations to stakeholders.