Essential Mathematics for Data Analysis and Earth Geometry

Time Series Analysis

Identifying Trends

• Upward trend: Data increases over time.
• Downward trend: Data decreases over time.
• Stationary trend: No clear increase or decrease.

Smoothing Data (Reducing Fluctuations)

• 3-point moving mean: Average of 3 consecutive values.
• 5-point moving mean: Average of 5 consecutive values.

Seasonal Indices (SI)

Formula:

Seasonal Index = Actual Value ÷ Trend Value

• SI > 1 → Above-average season.
• SI < 1 → Below-average season.

Deseasonalising Data

Formula:

Deseasonalised Value = Actual Value ÷ Seasonal Index

Regression Line (Trend Line)

Equation of the line:

y = mx + c

• m = gradient (rate of change).
• c = y-intercept (starting value).

Bivariate Data Analysis

Types of Correlation

• Positive → Both variables increase together.
• Negative → One increases, the other decreases.
• No correlation → No clear relationship.

Correlation Coefficient (Pearson’s r)

• Measures the strength & direction of correlation.
• Range:
  • r > 0.7 → Strong positive.
  • r ≈ 0 → Weak or no correlation.
  • r < −0.7 → Strong negative.

Interpolation vs. Extrapolation

• Interpolation (✔️ reliable) → Predicting within the given data range.
• Extrapolation (⚠️ risky) → Predicting beyond the given data range.

Line of Best Fit (Linear Regression)

Equation:

y = mx + c

Residuals: Differences between actual and predicted values (lower values indicate a better fit).

Growth & Decay in Sequences

Arithmetic Sequences (Linear Growth)

Formula:

T_n = a + (n − 1)d

• a = first term.
• d = common difference.
• n = term number.

Example: 2, 5, 8, 11... (d = 3).

Geometric Sequences (Exponential Growth/Decay)

Formula:

T_n = a × r⁽ⁿ⁻¹⁾

• a = first term.
• r = common ratio.
• n = term number.

Example: 3, 6, 12, 24... (r = 2).

Compound Interest (Exponential Growth)

Formula:

A = P(1 + r)^t

• A = final amount.
• P = initial amount (principal).
• r = interest rate (decimal).
• t = time (years).

Depreciation (Exponential Decay)

Formula:

A = P(1 − r)^t

• r = depreciation rate.

Earth Geometry & Time Zones

Latitude & Longitude

• Latitude (°N/°S) → Horizontal lines (equator = 0°).
• Longitude (°E/°W) → Vertical lines (Prime Meridian = 0°).

Great Circle Distance

Formula:

d = 6371 × cos⁻¹(sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ)

• φ₁, φ₂ = latitudes (in degrees).
• Δλ = difference in longitudes (degrees).
• 6371 km = Earth’s radius.

Steps:

1. Convert latitudes & longitudes to degrees.
2. Find the difference in longitude (Δλ).
3. Use the calculator’s cos⁻¹ function.

Time Zones & GMT

• Earth rotates 360° in 24 hours → 15° = 1-hour time difference.
• East → Add time
• West → Subtract time

Time Zone Conversion Formula

Local Time = GMT ± Time Zone Offset

Example:

• Sydney (AEST) = GMT +10.
• London (GMT 0).
• If it’s 2 PM in London, the time in Sydney = 2 + 10 = 12 AM (the next day).

International Date Line (IDL)

• Crossing west → go forward a day.
• Crossing east → go back a day.