Python has become a leading language for statistical analysis due to its versatility, extensive libraries, and active community support. Its strengths lie in:
Several Python modules are crucial for statistical analysis. Here’s a brief overview:
NumPy: NumPy (Numerical Python) forms the foundation for many scientific computing tasks in Python. It provides powerful N-dimensional array objects and tools for working with these arrays efficiently. NumPy is essential for numerical operations, linear algebra, and handling large datasets. Many other statistical libraries build upon NumPy.
SciPy: SciPy (Scientific Python) builds on NumPy, providing a vast collection of algorithms and mathematical tools for scientific and engineering applications. Its scipy.stats module is particularly important for statistical analysis, offering functions for probability distributions, hypothesis testing, statistical measures, and more.
Statsmodels: Statsmodels is a powerful module specifically designed for statistical modeling. It provides classes and functions for estimating various statistical models, including linear regression, generalized linear models (GLMs), time series analysis, and more. It also offers comprehensive diagnostic tools for assessing model fit and assumptions.
Pandas: Pandas is a crucial library for data manipulation and analysis. It provides high-performance, easy-to-use data structures like DataFrames, which are particularly well-suited for organizing and working with tabular data. Pandas excels at data cleaning, transformation, and preparation for statistical analysis. It frequently works in conjunction with NumPy and SciPy.
The easiest way to set up your Python environment for statistics is using Anaconda. Anaconda is a free and open-source distribution of Python that includes many of the essential libraries needed for data science and statistical analysis (including NumPy, SciPy, Statsmodels, and Pandas). Download and install Anaconda from the official website. Anaconda also provides a package manager (conda) to easily install and manage additional packages. Alternatively, you can use pip, Python’s built-in package manager, to install individual packages. For example, to install NumPy using pip, you would open your terminal or command prompt and run pip install numpy. Repeat this process for other libraries as needed.
The primary data structure for statistical analysis in Python is the array, often a NumPy array or a Pandas DataFrame.
NumPy arrays: NumPy arrays are efficient for storing and manipulating numerical data. They are homogeneous, meaning all elements must be of the same data type. This allows for optimized numerical operations.
Pandas DataFrames: DataFrames are more versatile and user-friendly for tabular data. They can hold different data types within a single DataFrame, making them ideal for real-world datasets that often contain a mix of numerical and categorical variables. DataFrames allow for easy data manipulation, filtering, and aggregation. They often serve as the input for statistical analyses performed using SciPy and Statsmodels. Pandas Series, which are one-dimensional labeled arrays, are also commonly used.
NumPy provides efficient functions for calculating measures of central tendency:
numpy.mean().import numpy as np
data = np.array([1, 2, 3, 4, 5])
mean = np.mean(data) # mean will be 3.0
print(f"Mean: {mean}")numpy.median(). For even-numbered datasets, the median is the average of the two middle values.data = np.array([1, 2, 3, 4, 5, 6])
median = np.median(data) # median will be 3.5
print(f"Median: {median}")scipy.stats.mode().from scipy import stats
data = np.array([1, 2, 2, 3, 3, 3, 4, 4, 4, 4])
mode = stats.mode(data)[0][0] # mode will be 4
print(f"Mode: {mode}")NumPy functions efficiently compute measures of data dispersion:
numpy.var().data = np.array([1, 2, 3, 4, 5])
variance = np.var(data)
print(f"Variance: {variance}")numpy.std().data = np.array([1, 2, 3, 4, 5])
std_dev = np.std(data)
print(f"Standard Deviation: {std_dev}")numpy.max() and numpy.min().data = np.array([1, 2, 3, 4, 5])
data_range = np.max(data) - np.min(data)
print(f"Range: {data_range}")NumPy’s numpy.percentile() function calculates quantiles (percentiles) of a dataset.
data = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
percentile_25 = np.percentile(data, 25) # 25th percentile
percentile_75 = np.percentile(data, 75) # 75th percentile
print(f"25th Percentile: {percentile_25}")
print(f"75th Percentile: {percentile_75}")While NumPy itself doesn’t directly create visualizations, it provides the data foundation for plotting libraries like Matplotlib.
import matplotlib.pyplot as plt
data = np.random.randn(1000) # Example data: 1000 random numbers from a standard normal distribution
# Histogram
plt.hist(data, bins=30)
plt.title("Histogram")
plt.show()
# Box plot
plt.boxplot(data)
plt.title("Box Plot")
plt.show()NumPy offers a wide array of statistical functions beyond those already mentioned, including:
np.sum(): Calculates the sum of array elements.np.prod(): Calculates the product of array elements.np.cumsum(): Calculates the cumulative sum of array elements.np.ptp(): Finds the peak-to-peak value (range).np.corrcoef()).Remember to consult the official NumPy documentation for a complete list and detailed explanations of all available functions.
SciPy’s scipy.stats module provides a comprehensive suite of functions for performing hypothesis tests. Hypothesis testing involves using sample data to make inferences about a population. The general process involves:
SciPy simplifies these steps by providing functions that directly calculate p-values. The interpretation of the p-value is crucial: a low p-value (typically below alpha) suggests strong evidence against the null hypothesis.
t-tests: Used to compare the means of two groups when the population standard deviation is unknown. SciPy offers various t-tests:
scipy.stats.ttest_ind(): Independent samples t-test (comparing means of two independent groups).scipy.stats.ttest_rel(): Paired samples t-test (comparing means of two related groups).scipy.stats.ttest_1samp(): One-sample t-test (comparing the mean of a single group to a known value).z-tests: Similar to t-tests but used when the population standard deviation is known. SciPy doesn’t have a dedicated z-test function, but you can perform a z-test using the standard normal distribution (scipy.stats.norm) and calculating the z-statistic directly.
ANOVA tests compare the means of three or more groups. SciPy provides:
scipy.stats.f_oneway(): Performs a one-way ANOVA, comparing the means of multiple independent groups.Chi-square tests analyze categorical data. SciPy offers:
scipy.stats.chi2_contingency(): Performs a chi-square test of independence on a contingency table, assessing the association between two categorical variables.scipy.stats.chisquare(): Performs a chi-square goodness-of-fit test, comparing observed frequencies to expected frequencies.SciPy facilitates correlation and regression analysis:
scipy.stats.pearsonr(): Calculates Pearson’s correlation coefficient (for linear relationships between two continuous variables) and its p-value.scipy.stats.spearmanr(): Calculates Spearman’s rank correlation coefficient (for monotonic relationships, less sensitive to outliers).scipy.stats.linregress(): Performs linear regression, providing the slope, intercept, R-squared value, p-value, and standard error.Non-parametric tests are used when assumptions of normality or equal variances are violated. SciPy offers several:
scipy.stats.mannwhitneyu(): Mann-Whitney U test (analogous to the independent samples t-test for non-parametric data).scipy.stats.wilcoxon(): Wilcoxon signed-rank test (analogous to the paired samples t-test for non-parametric data).scipy.stats.kruskal(): Kruskal-Wallis test (analogous to one-way ANOVA for non-parametric data).Important Note: Always carefully consider the assumptions of each statistical test before applying it to your data. Incorrect application can lead to inaccurate or misleading results. Consult statistical literature to ensure you are using the appropriate test for your specific research question and data characteristics. The SciPy documentation provides detailed explanations and examples for each function.
Statsmodels provides comprehensive tools for fitting and analyzing linear regression models. The core function is statsmodels.formula.api.ols(), which uses a formula interface for specifying the model. This makes it easier to define complex models and ensures code readability.
import statsmodels.formula.api as smf
import pandas as pd
# Sample data (replace with your actual data)
data = {'x': [1, 2, 3, 4, 5], 'y': [2, 4, 5, 4, 5]}
df = pd.DataFrame(data)
# Define the model
model = smf.ols('y ~ x', data=df)
# Fit the model
results = model.fit()
# Print the summary
print(results.summary())The summary provides key statistics like R-squared, coefficients, p-values, and standard errors, allowing for a comprehensive evaluation of the model’s fit and the significance of predictors.
Statsmodels supports a wide range of GLMs, extending beyond linear regression to handle different data types and distributions. This includes models like logistic regression (for binary outcomes), Poisson regression (for count data), and others. The statsmodels.genmod module provides the necessary functions.
import statsmodels.api as sm
import numpy as np
# Sample data for logistic regression (replace with your data)
X = np.array([[1, 2], [3, 4], [5, 6], [7, 8]])
y = np.array([0, 1, 0, 1])
# Add a constant to the predictor variables
X = sm.add_constant(X)
# Fit a logistic regression model
logit_model = sm.GLM(y, X, family=sm.families.Binomial())
results = logit_model.fit()
print(results.summary())The family argument specifies the distribution of the response variable.
Statsmodels offers tools for time series analysis, including:
ARIMA models: Autoregressive Integrated Moving Average models are widely used for forecasting time series data. Statsmodels provides functions for fitting and analyzing ARIMA models.
State space models: These models are particularly useful for handling complex time series with multiple components (trend, seasonality, etc.).
The specific functions for time series analysis are located within the statsmodels.tsa module. The exact usage depends on the specific model and analysis required. Refer to the Statsmodels documentation for details on these more advanced techniques.
After fitting a statistical model, it’s crucial to assess its adequacy. Statsmodels offers several diagnostic tools:
Residual analysis: Examining residuals (the differences between observed and predicted values) can reveal issues like non-linearity, heteroscedasticity (unequal variance), and outliers. Functions like results.resid provide access to residuals.
Goodness-of-fit tests: Tests like the chi-squared test (for GLMs) help evaluate how well the model fits the data.
Influence measures: Identifying influential observations that disproportionately affect the model’s results.
Interpreting model results involves understanding the estimated coefficients, their standard errors, p-values, and other statistics generated by Statsmodels.
Coefficients: Represent the effect of each predictor variable on the response variable. For linear regression, the coefficient indicates the change in the response variable for a one-unit increase in the predictor, holding other variables constant.
Standard errors: Measure the uncertainty in the coefficient estimates.
p-values: Indicate the statistical significance of each coefficient. A low p-value (typically below 0.05) suggests that the predictor variable is significantly related to the response variable.
R-squared: In regression models, it represents the proportion of variance in the response variable explained by the model. A higher R-squared indicates a better fit.
The summary() method of the fitted model provides a comprehensive table of these statistics, aiding in their interpretation. Remember to carefully consider the context of the data and research question when interpreting these results.
Pandas provides powerful tools for cleaning and preparing data for statistical analysis. This includes:
isnull() and notnull() to identify missing data. Methods like fillna(), dropna(), and interpolate() allow for imputation or removal of missing values.import pandas as pd
import numpy as np
data = {'A': [1, 2, np.nan, 4], 'B': [5, np.nan, 7, 8]}
df = pd.DataFrame(data)
# Fill NaN with the mean of column A
df['A'] = df['A'].fillna(df['A'].mean())
# Drop rows with any NaN values
df = df.dropna()
print(df)Removing duplicates: The duplicated() method identifies duplicates, and drop_duplicates() removes them.
Data type conversion: Pandas allows easy conversion between data types using functions like astype().
String manipulation: Pandas offers string manipulation methods for cleaning and transforming text data.
Pandas excels at aggregating and summarizing data using the groupby() method. This allows for calculating summary statistics (mean, sum, count, etc.) for different subgroups within the data.
import pandas as pd
data = {'Category': ['A', 'A', 'B', 'B', 'C'], 'Value': [10, 15, 20, 25, 30]}
df = pd.DataFrame(data)
# Group by category and calculate the mean of 'Value'
grouped = df.groupby('Category')['Value'].mean()
print(grouped)While Pandas doesn’t offer the same level of customization as dedicated plotting libraries like Matplotlib or Seaborn, it provides convenient plotting functions for quick visualizations:
plot(): Creates various plots (line, bar, scatter, etc.) directly from a DataFrame or Series.import pandas as pd
import matplotlib.pyplot as plt
data = {'x': [1, 2, 3, 4, 5], 'y': [2, 4, 1, 3, 5]}
df = pd.DataFrame(data)
df.plot(x='x', y='y', kind='scatter') #scatter plot
plt.show()
df.plot(kind='bar') #Bar chart of all columns
plt.show()These basic plots are useful for quick exploratory data analysis. For more sophisticated visualizations, integrating Pandas with Matplotlib or Seaborn is recommended.
Pandas provides functions for reading and writing data in various formats:
read_csv(): Reads data from CSV files.read_excel(): Reads data from Excel files.to_csv(): Writes data to a CSV file.to_excel(): Writes data to an Excel file.Pandas also supports other formats like JSON, SQL databases, and more.
Pandas seamlessly integrates with other statistical modules like NumPy, SciPy, and Statsmodels:
NumPy: Pandas DataFrames are built on top of NumPy arrays, enabling efficient numerical operations.
SciPy: Pandas DataFrames can be easily passed to SciPy’s statistical functions for calculations like hypothesis testing and correlation analysis.
Statsmodels: Pandas DataFrames are often used as input to Statsmodels’ statistical modeling functions, providing a streamlined workflow for data analysis and modeling.
This integration simplifies the process of performing comprehensive statistical analyses using various Python libraries. The ease of data manipulation in Pandas combined with the power of other libraries makes it a central tool for many statistical workflows.
Bayesian statistics offers a different approach to statistical inference compared to frequentist methods. Instead of estimating parameters based solely on observed data, Bayesian methods incorporate prior knowledge or beliefs about the parameters. This prior knowledge is combined with the observed data using Bayes’ theorem to obtain a posterior distribution, representing the updated beliefs about the parameters after observing the data.
Python libraries like PyMC3 and Stan provide tools for performing Bayesian inference. These libraries allow for specifying complex models, incorporating various prior distributions, and sampling from the posterior distribution using Markov Chain Monte Carlo (MCMC) methods.
# Example using PyMC3 (requires installation: pip install pymc3)
import pymc3 as pm
import numpy as np
# Sample data (replace with your data)
y = np.array([1, 0, 1, 1, 0, 0, 1, 1])
with pm.Model() as model:
# Prior distribution for the probability of success
p = pm.Beta('p', alpha=1, beta=1) # Uniform prior
# Likelihood (Bernoulli distribution)
y_obs = pm.Bernoulli('y_obs', p=p, observed=y)
# Posterior sampling
trace = pm.sample(1000)
# Analyze the posterior distribution (e.g., calculate credible intervals)
pm.summary(trace)Bayesian methods are particularly useful when dealing with limited data or when prior information is available.
Machine learning algorithms can be powerful tools for statistical analysis. Many machine learning methods can be viewed as sophisticated statistical models.
Regression: Techniques like linear regression, support vector regression (SVR), and random forests can be used for prediction and inference.
Classification: Logistic regression, support vector machines (SVM), decision trees, and random forests can classify data points into different categories.
Clustering: Algorithms like k-means, hierarchical clustering, and DBSCAN can uncover underlying structures in data.
Libraries like scikit-learn provide efficient implementations of these algorithms. However, it’s crucial to remember that the interpretability of some machine learning models can be challenging, especially compared to simpler statistical models like linear regression.
Survival analysis deals with time-to-event data, where the outcome of interest is the time until a specific event occurs (e.g., death, machine failure, customer churn). Survival analysis techniques account for censoring, where the event time is not observed for all individuals in the study.
The lifelines library in Python provides tools for performing survival analysis, including:
Causal inference focuses on determining cause-and-effect relationships between variables. This goes beyond simple correlation, aiming to establish whether changes in one variable actually cause changes in another.
Techniques like:
While specific Python libraries dedicated to causal inference are emerging, many of these techniques can be implemented using standard statistical packages like Statsmodels or specialized packages based on your specific method of choice. Careful experimental design and rigorous statistical analysis are essential for reliable causal inference.
Matplotlib is a fundamental plotting library in Python, providing a wide range of tools for creating static, interactive, and animated visualizations. While not specifically a statistical visualization library, its flexibility allows for the creation of many statistical plots.
import matplotlib.pyplot as plt
import numpy as np
# Example: Histogram
data = np.random.randn(1000)
plt.hist(data, bins=30)
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.title("Histogram of Random Data")
plt.show()
# Example: Scatter plot
x = np.linspace(0, 10, 50)
y = 2*x + 1 + np.random.randn(50)
plt.scatter(x, y)
plt.xlabel("X")
plt.ylabel("Y")
plt.title("Scatter Plot")
plt.show()Matplotlib offers fine-grained control over plot aesthetics, allowing for customization of labels, titles, colors, and more. However, creating complex statistical visualizations can be more time-consuming compared to higher-level libraries.
Seaborn builds on top of Matplotlib, providing a higher-level interface for creating statistically informative and visually appealing plots. Seaborn simplifies the creation of common statistical visualizations like:
Distributions: Histograms, kernel density estimates, box plots, violin plots.
Relationships: Scatter plots with regression lines, pair plots (showing relationships between multiple variables).
Categorical data: Bar plots, count plots, point plots.
import seaborn as sns
import matplotlib.pyplot as plt
import pandas as pd
# Sample data (replace with your data)
data = {'Category': ['A', 'A', 'B', 'B', 'C', 'C'], 'Value': [10, 15, 20, 25, 30, 35]}
df = pd.DataFrame(data)
# Example: Box plot
sns.boxplot(x='Category', y='Value', data=df)
plt.show()
# Example: Scatter plot with regression line
sns.regplot(x='Value', y='Value', data=df) #Example using the same column for x and y to show the regression line. Replace with your x and y columns
plt.show()Seaborn automatically handles many aspects of plot aesthetics, making it easier to create publication-quality visualizations quickly.
Effective visualizations for statistical reports should:
Communicate findings clearly: The visualization should directly support the key messages of the report.
Be accurate and unbiased: Avoid misleading presentations of data.
Be easy to interpret: Use appropriate scales and labels.
Be visually appealing: Choose colors and styles that enhance readability.
Be appropriate for the audience: Consider the level of statistical knowledge of the intended readers.
The choice of visualization depends heavily on the type of data being presented:
Univariate numerical data (one variable): Histograms, kernel density estimates, box plots, violin plots.
Bivariate numerical data (two variables): Scatter plots, heatmaps.
Multivariate numerical data (more than two variables): Pair plots, 3D plots (with caution), parallel coordinate plots.
Categorical data: Bar charts, count plots, pie charts (use sparingly).
Time series data: Line plots.
Careful consideration of the data type and the message being conveyed is crucial for choosing the most appropriate visualization. Avoid using visualizations that are unclear or misleading.
Clean and efficient code is crucial for reproducibility, collaboration, and maintainability. Here are some best practices:
Use meaningful variable names: Choose names that clearly describe the purpose of variables.
Add comments: Explain complex logic or steps in the code.
Modularize your code: Break down large tasks into smaller, reusable functions.
Use version control: Employ Git or a similar system to track changes and collaborate effectively.
Follow PEP 8 style guide: Adhere to Python’s style guidelines for consistent and readable code.
Document your code: Write clear documentation explaining how to use your functions and modules.
Use vectorized operations: Leverage NumPy’s vectorized operations for faster computations instead of explicit loops whenever possible.
Missing data is common in real-world datasets. Effective strategies include:
Identification: Use Pandas’ isnull() and notnull() functions to identify missing values.
Imputation: Replace missing values with estimated values. Methods include:
Removal: Remove rows or columns with missing data. This is simpler but can lead to loss of information if many values are missing. The best approach depends on the amount of missing data, the mechanism of missingness, and the nature of the analysis.
Outliers are data points that deviate significantly from the rest of the data. Strategies for handling outliers include:
Identification: Use box plots, scatter plots, or z-scores to identify potential outliers.
Removal: Remove outliers if they are clearly due to errors or data entry mistakes. However, be cautious, as removing outliers can also bias the results.
Transformation: Apply transformations like logarithmic or Box-Cox transformations to reduce the influence of outliers.
Robust methods: Use statistical methods that are less sensitive to outliers, such as robust regression or median-based statistics. The best approach depends on the cause and impact of the outliers on your analysis.
Common errors in statistical code include:
Incorrect data types: Ensure your data is in the correct format for the statistical methods you’re using.
Incorrect function arguments: Double-check the arguments you’re passing to functions.
Statistical assumptions violations: Verify that your data meet the assumptions of the statistical tests you are applying.
Incorrect interpretation of results: Ensure you understand the meaning of the statistical results and don’t misinterpret them.
Debugging techniques include:
Print statements: Insert print() statements to check intermediate values.
Debugging tools: Use IDE debuggers to step through code and examine variables.
Unit testing: Write tests to verify that individual functions work correctly.
For large datasets, performance optimization is crucial:
Vectorization: Use NumPy’s vectorized operations to avoid slow Python loops.
Data structures: Choose appropriate data structures (e.g., NumPy arrays for numerical data).
Memory management: Avoid loading the entire dataset into memory at once if possible; use generators or chunking techniques.
Algorithmic efficiency: Choose algorithms that scale well with the size of the data.
Parallel processing: Employ libraries like multiprocessing or Dask to parallelize computations. Consider using specialized libraries designed for big data analysis.
Python’s statistical capabilities are used across diverse fields:
Finance: Analyzing stock market data, risk assessment, portfolio optimization. NumPy, Pandas, and Statsmodels are heavily used for financial modeling and forecasting.
Healthcare: Analyzing patient data, clinical trials, disease prediction. SciPy’s statistical tests and Statsmodels’ GLMs are frequently employed in medical research.
Marketing: Customer segmentation, A/B testing, market research. Pandas is used for data manipulation and visualization, while Scikit-learn’s machine learning algorithms are helpful in predictive modeling.
Engineering: Quality control, process optimization, failure analysis. NumPy and SciPy’s mathematical functions are valuable tools for simulations and data analysis in various engineering disciplines.
Environmental science: Analyzing climate data, pollution modeling, ecological studies. Statsmodels and SciPy’s statistical tests are valuable for analyzing environmental data.
These are just a few examples. The versatility of Python’s statistical libraries enables their application in almost any field involving data analysis.
Here’s a simplified illustration of a common workflow:
Scenario: Analyzing the relationship between advertising spending and sales.
1. Data Acquisition and Preparation:
import pandas as pd
# Load data from a CSV file
data = pd.read_csv("advertising_data.csv")
# Clean and preprocess the data (handle missing values, outliers, etc.)
data.dropna(inplace=True) # remove rows with missing data
# ... further data cleaning and preprocessing steps as needed...2. Exploratory Data Analysis (EDA):
import matplotlib.pyplot as plt
import seaborn as sns
# Visualize the data
sns.scatterplot(x="Advertising_Spend", y="Sales", data=data)
plt.show()
# Calculate descriptive statistics
print(data.describe())3. Statistical Modeling:
import statsmodels.formula.api as smf
# Fit a linear regression model
model = smf.ols("Sales ~ Advertising_Spend", data=data)
results = model.fit()
print(results.summary())4. Interpretation and Conclusion: Examine the model’s coefficients, p-values, and R-squared to determine the strength and significance of the relationship between advertising spending and sales.
Interpreting statistical results requires careful consideration:
Statistical significance: A low p-value (typically below 0.05) indicates that the observed result is unlikely to have occurred by chance alone. However, statistical significance doesn’t necessarily imply practical significance.
Effect size: Quantifies the magnitude of the effect. A statistically significant effect might have a small effect size, which could be practically irrelevant.
Confidence intervals: Provide a range of plausible values for the estimated parameters.
Assumptions: Ensure the assumptions of the statistical methods used are met. Violations of these assumptions can lead to unreliable results.
Contextualization: Interpret statistical results within the broader context of the research question and the limitations of the data.
Always clearly communicate the limitations of the analysis and avoid overinterpreting the results. Focus on drawing conclusions that are supported by the data and the chosen statistical methods. Provide visualizations to support your findings and make them accessible to a wider audience.
This glossary provides definitions of common statistical terms used throughout this manual.
A
Alpha (α): The significance level in hypothesis testing, representing the probability of rejecting the null hypothesis when it is true (Type I error).
ANOVA (Analysis of Variance): A statistical test used to compare the means of three or more groups.
Association: A statistical relationship between two or more variables. Association does not imply causation.
B
Bayes’ Theorem: A theorem that describes how to update the probability of an event based on new evidence.
Bayesian Statistics: A statistical approach that incorporates prior knowledge or beliefs into the analysis.
Bias: A systematic error in a measurement or estimation.
Binomial Distribution: A probability distribution describing the probability of getting a certain number of successes in a fixed number of independent trials.
C
Categorical Data: Data representing categories or groups (e.g., colors, types).
Central Limit Theorem: A theorem stating that the distribution of the sample mean approaches a normal distribution as the sample size increases.
Chi-square Test: A statistical test used to analyze categorical data and test for independence between variables.
Confidence Interval: A range of values that is likely to contain the true population parameter with a certain level of confidence.
Correlation: A measure of the linear association between two variables.
Correlation Coefficient: A numerical measure of the strength and direction of the correlation between two variables (e.g., Pearson’s r, Spearman’s rho).
D
Data Cleaning: The process of identifying and correcting or removing errors, inconsistencies, and missing values in a dataset.
Descriptive Statistics: Summary statistics that describe the main features of a dataset (e.g., mean, median, standard deviation).
Distribution: A mathematical function that describes the probability of different outcomes of a random variable.
F
F-test: A statistical test used in ANOVA to compare variances between groups.
Frequentist Statistics: A statistical approach that focuses on the frequency of events and probabilities based on observed data.
H
Heteroscedasticity: Unequal variance of the errors in a regression model.
Histogram: A graphical representation of the distribution of a numerical variable.
H
I
Inferential Statistics: Statistical methods used to draw inferences about a population based on sample data.
Interquartile Range (IQR): The difference between the 75th and 25th percentiles of a dataset.
K
M
Mean: The average of a set of values.
Median: The middle value in a sorted dataset.
Mode: The most frequent value in a dataset.
N
Normal Distribution: A symmetric, bell-shaped probability distribution.
Null Hypothesis (H0): The hypothesis being tested in a hypothesis test. It typically represents the absence of an effect or relationship.
O
P
P-value: The probability of observing the obtained results (or more extreme results) if the null hypothesis is true.
Percentile: The value below which a given percentage of observations in a group of observations falls.
Population: The entire group of individuals or objects of interest.
Q
R
Regression: A statistical method used to model the relationship between a dependent variable and one or more independent variables.
Regression Analysis: Statistical process for estimating the relationships among variables.
Residuals: The differences between the observed values and the predicted values in a regression model.
S
Sample: A subset of a population.
Standard Deviation: A measure of the dispersion or spread of a dataset around its mean.
Standard Error: A measure of the variability of a sample statistic (e.g., sample mean).
Statistical Significance: The likelihood that an observed result is not due to random chance.
T
t-test: A statistical test used to compare the means of two groups when the population standard deviation is unknown.
Type I Error: Rejecting the null hypothesis when it is actually true.
Type II Error: Failing to reject the null hypothesis when it is actually false.
V
Variance: A measure of the dispersion or spread of a dataset around its mean; it’s the square of the standard deviation.
Violin Plot: A graphical representation of the distribution of a numerical variable, combining aspects of box plots and kernel density estimates.
This glossary is not exhaustive, but it covers many of the key terms used in this manual. For more comprehensive definitions, refer to a statistical textbook or online resources.