Practical Statistics for Data Scientists

1. Exploratory Data Analysis: The Foundation of Data Science

The key idea of EDA: the first and most important step in any project based on data is to look at the data.

Origins and importance. Exploratory Data Analysis (EDA), pioneered by John Tukey in the 1960s, forms the cornerstone of data science. It involves summarizing and visualizing data to gain intuition and understanding before formal modeling or hypothesis testing.

Key components of EDA:

• Examining data quality and completeness
• Calculating summary statistics
• Creating visualizations (histograms, scatterplots, etc.)
• Identifying patterns, trends, and outliers
• Formulating initial hypotheses

EDA is an iterative process that helps data scientists understand the context of their data, detect anomalies, and guide further analysis. It bridges the gap between raw data and actionable insights, setting the foundation for more advanced statistical techniques and machine learning models.

2. Understanding Data Types and Structures

The basic data structure in data science is a rectangular matrix in which rows are records and columns are variables (features).

Data types. Understanding different data types is crucial for proper analysis and modeling:

• Numeric: Continuous (e.g., temperature) or Discrete (e.g., count data)
• Categorical: Nominal (unordered) or Ordinal (ordered)
• Binary: Special case of categorical with two categories

Data structures. The most common data structure in data science is the rectangular format:

• Rows represent individual observations or records
• Columns represent variables or features
• This structure is often called a data frame in R or a DataFrame in Python

Understanding data types and structures helps in choosing appropriate visualization techniques, statistical methods, and machine learning algorithms. It also aids in data preprocessing and feature engineering, which are critical steps in the data science workflow.

3. Measures of Central Tendency and Variability

The mean is easy to compute and expedient to use, it may not always be the best measure for a central value.

Central tendency. Measures of central tendency describe the "typical" value in a dataset:

• Mean: Average of all values (sensitive to outliers)
• Median: Middle value when data is sorted (robust to outliers)
• Mode: Most frequent value (useful for categorical data)

Variability. Measures of variability describe the spread of the data:

• Range: Difference between maximum and minimum values
• Variance: Average squared deviation from the mean
• Standard deviation: Square root of variance (same units as data)
• Interquartile range (IQR): Difference between 75th and 25th percentiles (robust)

Choosing appropriate measures depends on the data distribution and the presence of outliers. For skewed distributions or data with outliers, median and IQR are often preferred over mean and standard deviation. These measures provide a foundation for understanding data distributions and are essential for further statistical analysis.

4. Visualizing Data Distributions

Neither the variance, the standard deviation nor the mean absolute deviation are robust to outliers and extreme values.

Importance of visualization. Visual representations of data distributions provide insights that summary statistics alone may miss. They help identify patterns, outliers, and the overall shape of the data.

Key visualization techniques:

• Histograms: Show frequency distribution of a single variable
• Box plots: Display median, quartiles, and potential outliers
• Density plots: Smooth version of histograms, useful for comparing distributions
• Q-Q plots: Compare sample distribution to theoretical distribution (e.g., normal)

These visualizations complement numerical summaries and help data scientists make informed decisions about data transformations, outlier treatment, and appropriate statistical methods. They also aid in communicating findings to non-technical stakeholders, making complex data more accessible and interpretable.

5. Correlation and Relationships Between Variables

Variables X and Y (each with measured data) are said to be positively correlated if high values of X go with high values of Y, and low values of X go with low values of Y.

Understanding correlation. Correlation measures the strength and direction of the relationship between two variables. It ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no linear relationship.

Visualization and measurement:

• Scatter plots: Visualize relationship between two continuous variables
• Correlation matrix: Display correlations between multiple variables
• Pearson correlation coefficient: Measure linear relationship
• Spearman rank correlation: Measure monotonic relationship (robust to outliers)

While correlation doesn't imply causation, it's a valuable tool for exploring relationships in data. It helps identify potential predictors for modeling, detect multicollinearity, and guide feature selection. However, it's important to remember that correlation only captures linear relationships and may miss complex, non-linear patterns in the data.

6. The Importance of Random Sampling and Bias Avoidance

Even in the era of Big Data, random sampling remains an important arrow in the data scientist's quiver.

Random sampling. Random sampling is crucial for obtaining representative data and avoiding bias. It ensures that each member of the population has an equal chance of being selected, allowing for valid statistical inferences.

Types of sampling:

• Simple random sampling: Each unit has an equal probability of selection
• Stratified sampling: Population divided into subgroups, then sampled randomly
• Cluster sampling: Groups are randomly selected, then all units within the group are studied

Avoiding bias. Bias can lead to incorrect conclusions and poor decision-making. Common types of bias include:

• Selection bias: Non-random sample selection
• Survivorship bias: Focusing only on "survivors" of a process
• Confirmation bias: Seeking information that confirms pre-existing beliefs

Proper sampling techniques and awareness of potential biases are essential for ensuring the validity and reliability of data science projects, even when working with large datasets.

7. Sampling Distributions and the Bootstrap Method

The bootstrap does not compensate for a small sample size - it does not create new data, nor does it fill in holes in an existing dataset.

Sampling distributions. A sampling distribution is the distribution of a statistic (e.g., mean, median) calculated from repeated samples of a population. Understanding sampling distributions is crucial for estimating uncertainty and making inferences.

The bootstrap method. Bootstrap is a powerful resampling technique that estimates the sampling distribution by repeatedly sampling (with replacement) from the original dataset:

1. Draw a sample with replacement from the original data
2. Calculate the statistic of interest
3. Repeat steps 1-2 many times
4. Use the distribution of calculated statistics to estimate standard errors and confidence intervals

Advantages of bootstrap:

• Does not require assumptions about the underlying distribution
• Applicable to a wide range of statistics
• Particularly useful for complex estimators without known sampling distributions

The bootstrap provides a practical way to assess the variability of sample statistics and model parameters, especially when theoretical distributions are unknown or difficult to derive.

8. Confidence Intervals: Quantifying Uncertainty

Presenting an estimate not as a single number but as a range is one way to counteract this tendency.

Purpose of confidence intervals. Confidence intervals provide a range of plausible values for a population parameter, given a sample statistic. They quantify the uncertainty associated with point estimates.

Interpretation. A 95% confidence interval means that if we were to repeat the sampling process many times, about 95% of the intervals would contain the true population parameter.

Calculation methods:

• Traditional method: Uses theoretical distributions (e.g., t-distribution)
• Bootstrap method: Uses resampling to estimate the sampling distribution

Factors affecting confidence interval width:

• Sample size: Larger samples lead to narrower intervals
• Variability in the data: More variability leads to wider intervals
• Confidence level: Higher confidence levels (e.g., 99% vs. 95%) lead to wider intervals

Confidence intervals provide a more nuanced view of statistical estimates, helping data scientists and decision-makers understand the precision of their findings and make more informed decisions.

9. The Normal Distribution and Its Limitations

It is a common misconception that the normal distribution is called that because most data follow a normal distribution, i.e. it is the normal thing.

The normal distribution. Also known as the Gaussian distribution, it's characterized by its symmetric, bell-shaped curve. Key properties:

• Mean, median, and mode are all equal
• 68% of data falls within one standard deviation of the mean
• 95% within two standard deviations, 99.7% within three

Importance in statistics:

• Central Limit Theorem: Sampling distributions of means tend to be normal
• Many statistical tests assume normality
• Basis for z-scores and standardization

Limitations:

• Many real-world phenomena are not normally distributed
• Assuming normality can lead to underestimation of extreme events
• Can be inappropriate for inherently skewed data (e.g., income distributions)

While the normal distribution is theoretically important and useful in many contexts, data scientists must be cautious about assuming normality without checking their data. Techniques like Q-Q plots and normality tests can help assess whether the normal distribution is appropriate for a given dataset.

10. Long-Tailed Distributions in Real-World Data

Assuming a normal distribution can lead to underestimation of extreme events ("black swans").

Long-tailed distributions. Many real-world phenomena exhibit long-tailed distributions, where extreme events occur more frequently than predicted by normal distributions. Examples include:

• Income distributions
• City populations
• Word frequencies in natural language
• Stock market returns

Characteristics:

• Higher kurtosis (more peaked) than normal distributions
• Heavier tails, indicating higher probability of extreme events
• Often asymmetric (skewed)

Implications for data science:

• Need for robust statistical methods that don't assume normality
• Importance of considering extreme events in risk analysis and modeling
• Potential for using alternative distributions (e.g., log-normal, power law)

Recognizing and properly handling long-tailed distributions is crucial for accurate modeling and decision-making in many domains. It requires a shift from traditional statistical thinking based on normal distributions to more flexible approaches that can capture the complexity of real-world data.

Last updated: August 9, 2024