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# Statistical Functions
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Statistical analysis functions for correlation, regression, and statistical measures of price data. These functions provide mathematical insights into price relationships, trends, and variability patterns.
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## Capabilities
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### Beta Coefficient
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Measures the sensitivity of a security's returns relative to market returns, indicating systematic risk.
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```python { .api }
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def BETA(real0, real1, timeperiod=5):
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    """
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    Beta
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    Calculates beta coefficient measuring correlation and relative volatility between two price series.
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    Parameters:
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    - real0: array-like, dependent variable (typically individual security)
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    - real1: array-like, independent variable (typically market index)
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    - timeperiod: int, number of periods for calculation (default: 5)
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    Returns:
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    numpy.ndarray: Beta coefficient values
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    """
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```
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### Pearson Correlation Coefficient
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Measures the linear correlation between two price series, ranging from -1 to +1.
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```python { .api }
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def CORREL(real0, real1, timeperiod=30):
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    """
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    Pearson's Correlation Coefficient (r)
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    Measures linear relationship strength between two data series.
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    Parameters:
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    - real0: array-like, first data series
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    - real1: array-like, second data series  
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    - timeperiod: int, number of periods for calculation (default: 30)
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    Returns:
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    numpy.ndarray: Correlation coefficient values (-1 to +1)
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    """
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```
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### Linear Regression
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Calculates linear regression line values, providing trend line based on least squares method.
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```python { .api }
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def LINEARREG(real, timeperiod=14):
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    """
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    Linear Regression
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    Calculates linear regression line values using least squares method.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods for regression (default: 14)
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    Returns:
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    numpy.ndarray: Linear regression line values
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    """
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```
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### Linear Regression Angle
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Calculates the angle of the linear regression line, indicating trend steepness.
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```python { .api }
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def LINEARREG_ANGLE(real, timeperiod=14):
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    """
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    Linear Regression Angle
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    Calculates the angle of the linear regression line in degrees.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods for regression (default: 14)
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    Returns:
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    numpy.ndarray: Regression line angle values (degrees)
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    """
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```
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### Linear Regression Intercept
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Calculates the y-intercept of the linear regression line.
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```python { .api }
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def LINEARREG_INTERCEPT(real, timeperiod=14):
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    """
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    Linear Regression Intercept
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    Calculates the y-intercept of the linear regression line.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods for regression (default: 14)
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    Returns:
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    numpy.ndarray: Regression line intercept values
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    """
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```
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### Linear Regression Slope
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Calculates the slope of the linear regression line, indicating trend direction and strength.
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```python { .api }
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def LINEARREG_SLOPE(real, timeperiod=14):
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    """
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    Linear Regression Slope
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    Calculates the slope of the linear regression line.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods for regression (default: 14)
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    Returns:
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    numpy.ndarray: Regression line slope values
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    """
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```
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### Standard Deviation
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Measures the dispersion of price data around the mean, indicating volatility.
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```python { .api }
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def STDDEV(real, timeperiod=5, nbdev=1):
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    """
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    Standard Deviation
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    Calculates standard deviation of price data over specified period.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods (default: 5)
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    - nbdev: float, number of deviations (default: 1)
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    Returns:
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    numpy.ndarray: Standard deviation values
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    """
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```
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### Time Series Forecast
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Projects future price values based on linear regression analysis.
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```python { .api }
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def TSF(real, timeperiod=14):
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    """
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    Time Series Forecast
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    Forecasts next period value using linear regression.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods for forecast calculation (default: 14)
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    Returns:
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    numpy.ndarray: Forecasted values
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    """
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```
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### Variance
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Calculates the variance of price data, measuring the average of squared deviations from the mean.
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```python { .api }
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def VAR(real, timeperiod=5, nbdev=1):
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    """
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    Variance
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    Calculates variance of price data over specified period.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods (default: 5)
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    - nbdev: float, number of deviations (default: 1)
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    Returns:
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    numpy.ndarray: Variance values
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    """
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```
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### Average Deviation
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Calculates the average deviation from the mean, providing a measure of dispersion that is less sensitive to outliers than standard deviation.
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```python { .api }
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def AVGDEV(real, timeperiod=14):
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    """
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    Average Deviation
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    Calculates average deviation from the mean over specified period.
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    Provides a robust measure of dispersion less sensitive to outliers.
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    Parameters:
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    - real: array-like, input price data
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    - timeperiod: int, number of periods (default: 14)
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    Returns:
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    numpy.ndarray: Average deviation values
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    """
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```
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## Usage Examples
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```python
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import talib
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import numpy as np
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# Sample data - individual stock and market index
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stock_prices = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])
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market_prices = np.array([1000, 1020, 1010, 1025, 1040, 1035, 1050, 1065, 1055, 1070])
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# Calculate statistical measures
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beta = talib.BETA(stock_prices, market_prices, timeperiod=5)
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correlation = talib.CORREL(stock_prices, market_prices, timeperiod=5)
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linear_reg = talib.LINEARREG(stock_prices, timeperiod=5)
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reg_slope = talib.LINEARREG_SLOPE(stock_prices, timeperiod=5)
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reg_angle = talib.LINEARREG_ANGLE(stock_prices, timeperiod=5)
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std_dev = talib.STDDEV(stock_prices, timeperiod=5)
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variance = talib.VAR(stock_prices, timeperiod=5)
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forecast = talib.TSF(stock_prices, timeperiod=5)
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print("Statistical Analysis Results:")
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print(f"Beta: {beta[-1]:.3f}")
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print(f"Correlation: {correlation[-1]:.3f}")
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print(f"Current Price: {stock_prices[-1]}")
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print(f"Linear Regression: {linear_reg[-1]:.2f}")
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print(f"Trend Slope: {reg_slope[-1]:.4f}")
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print(f"Trend Angle: {reg_angle[-1]:.2f} degrees")
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print(f"Standard Deviation: {std_dev[-1]:.3f}")
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print(f"Variance: {variance[-1]:.3f}")
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print(f"Next Period Forecast: {forecast[-1]:.2f}")
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# Interpretation:
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# - Beta > 1: Stock is more volatile than market
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# - Correlation near 1: Strong positive relationship with market
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# - Positive slope/angle: Upward trend
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# - High std dev/variance: High volatility
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```
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## Practical Applications
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### Risk Management
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- **Beta**: Portfolio construction and risk assessment
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- **Standard Deviation**: Position sizing and volatility measurement
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- **Correlation**: Diversification analysis and pair trading
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### Trend Analysis
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- **Linear Regression**: Identify trend direction and strength
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- **Regression Angle**: Measure trend steepness
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- **TSF**: Short-term price projection
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### Market Relationships
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- **Correlation**: Identify related instruments for spread trading
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- **Beta**: Hedge ratio calculation and market exposure measurement
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### Statistical Arbitrage
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- **Variance**: Volatility-based trading strategies
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- **Regression Analysis**: Mean reversion strategy development