14.1 Descriptive statistics and summary measures

Descriptive statistics and summary measures are the backbone of data analysis. They help you understand your dataset's central tendencies, spread, and shape. These tools give you a quick snapshot of what's going on in your data.

In exploratory data analysis, these measures are your first step. They reveal patterns, outliers, and relationships in your data. By using means, medians, standard deviations, and correlations, you can start to uncover the story your data is telling.

Central Tendency Measures

Calculating Average Values

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  Skewness and the Mean, Median, and Mode – Adapted By Darlene Young Introductory Statistics View original

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  summary statistics - Explaining Mean, Median, Mode in Layman's Terms - Cross Validated View original

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  Average - Wikipedia View original

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  Skewness and the Mean, Median, and Mode – Adapted By Darlene Young Introductory Statistics View original

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• Mean represents the arithmetic average of a dataset, calculated by summing all values and dividing by the number of observations
• Median identifies the middle value in a sorted dataset, less affected by extreme outliers than the mean
• Mode pinpoints the most frequently occurring value in a dataset, particularly useful for categorical data

• [summary()](https://www.fiveableKeyTerm:summary())

  function in R provides a quick overview of central tendency measures for numeric variables, including mean and median

Choosing Appropriate Measures

• Mean works best for symmetrical distributions without significant outliers
• Median proves more robust for skewed distributions or datasets with extreme values
• Mode applies effectively to categorical data or discrete numerical data with clear peaks
• Multiple modes can occur in datasets, referred to as bimodal (two modes) or multimodal (more than two modes)

Dispersion Measures

Quantifying Data Spread

• Range measures the difference between the maximum and minimum values in a dataset, providing a simple measure of spread
• Variance calculates the average squared deviation from the mean, offering a comprehensive measure of data dispersion
• Standard deviation, the square root of variance, expresses dispersion in the same units as the original data
• Quartiles divide a dataset into four equal parts, with Q1 (25th percentile), Q2 (median), and Q3 (75th percentile)
• Interquartile range (IQR) measures the spread of the middle 50% of data, calculated as Q3 minus Q1

Interpreting Dispersion Statistics

• Larger ranges, variances, or standard deviations indicate greater data spread
• Standard deviation often preferred over variance due to its interpretability in original data units
• IQR proves useful for identifying outliers, with values beyond 1.5 times the IQR below Q1 or above Q3 considered potential outliers
• Coefficient of variation (CV) allows comparison of dispersion across datasets with different units or scales, calculated as (standard deviation / mean) * 100

Distribution Shape

Analyzing Symmetry and Tails

• Skewness measures the asymmetry of a distribution, with positive skew indicating a longer right tail and negative skew a longer left tail
• Symmetric distributions have a skewness close to zero (normal distribution)
• Right-skewed distributions have mean > median > mode, while left-skewed distributions have mode > median > mean
• Kurtosis quantifies the "tailedness" of a distribution, comparing it to a normal distribution

Interpreting Distribution Characteristics

• Mesokurtic distributions have kurtosis similar to a normal distribution (kurtosis ≈ 3)
• Leptokurtic distributions have higher peaks and heavier tails than normal (kurtosis > 3)
• Platykurtic distributions have lower, flatter peaks and thinner tails than normal (kurtosis < 3)
• Skewness and kurtosis help identify potential outliers and inform choices for appropriate statistical tests

Relationship Measures

Quantifying Variable Associations

• Correlation measures the strength and direction of linear relationships between two variables
• Pearson correlation coefficient ranges from -1 to 1, with -1 indicating perfect negative correlation and 1 perfect positive correlation
• Covariance measures how two variables vary together but is sensitive to the scale of the variables
• Spearman's rank correlation assesses monotonic relationships, useful for non-linear associations

Analyzing and Visualizing Relationships

• Scatter plots visually represent relationships between two continuous variables
• Correlation matrices display pairwise correlations for multiple variables

• [describe()](https://www.fiveableKeyTerm:describe())

  function from the

  psych

  package in R provides detailed descriptive statistics, including correlations and covariances
• Interpreting correlation requires caution, as correlation does not imply causation and may be influenced by outliers or non-linear relationships