Samacheer Kalvi 12th Geography Statistics in Geography Exercise Solutions

Statistics plays a vital role in geography by providing tools to collect, organise, analyse, and interpret spatial data. Geographers use statistics to: 1. Summarise large volumes of data (rainfall, temperature, population). 2. Compare different regions quantitatively. 3. Identify patterns, trends, and relationships between variables. 4. Test hypotheses about geographical phenomena. 5. Make predictions using regression and correlation analysis.

Mean: The arithmetic average, calculated as the sum of all values divided by the number of values. Example: Rainfall (mm) for 5 stations: 100, 120, 80, 90, 110 → Mean = (100+120+80+90+110)/5 = 100 mm. Median: The middle value when data is arranged in ascending or descending order. Example: Arranging 80, 90, 100, 110, 120 → Median = 100 (3rd value). Mode: The most frequently occurring value. Example: If rainfall data is 80, 90, 100, 100, 110, 120 → Mode = 100 (appears twice).

Standard deviation (σ) measures the average distance of each data point from the mean. It indicates how spread out the data is. Formula: σ = √[Σ(xi - x̄)² / N] Where xi = each value, x̄ = mean, N = number of values. Steps: 1. Calculate the mean (x̄). 2. Find the deviation of each value from the mean (xi - x̄). 3. Square each deviation (xi - x̄)². 4. Sum all squared deviations: Σ(xi - x̄)². 5. Divide by N to get variance. 6. Take the square root to get standard deviation.

Correlation measures the strength and direction of the relationship between two variables. Types of correlation: 1. Positive correlation: Both variables increase together (e.g., temperature and evaporation). 2. Negative correlation: One variable increases while the other decreases (e.g., altitude and temperature). 3. Zero correlation: No relationship between the variables (e.g., shoe size and intelligence). Degree: - Perfect correlation: r = ±1 - High correlation: |r| > 0.7 - Moderate correlation: 0.4 < |r| < 0.7 - Low correlation: |r| < 0.4

The coefficient of variation (CV) is the ratio of the standard deviation to the mean, expressed as a percentage: CV = (σ / x̄) x 100 Uses: 1. It allows comparison of variability between datasets with different units (e.g., comparing rainfall variability in mm with temperature variability in °C). 2. It helps determine which dataset is more consistent. Lower CV = more consistent. 3. It is dimensionless, making it ideal for comparing heterogeneous data. Example: If rainfall in Region A has CV = 20% and Region B has CV = 40%, Region A's rainfall is more consistent.

| Variance | Standard Deviation | |---|---| | Average of squared deviations from the mean | Square root of variance | | Expressed in squared units (e.g., mm²) | Expressed in original units (e.g., mm) | | σ² = Σ(xi - x̄)² / N | σ = √[Σ(xi - x̄)² / N] | | Not directly interpretable | Directly interpretable measure of spread | | Intermediate step in calculation | Final measure of dispersion |

A frequency distribution is a tabular arrangement of data showing how many times each value (or group of values) occurs. It organises raw data into classes with corresponding frequencies. | Rainfall (mm) | No. of Stations (Frequency) | |---|---| | 0 – 200 | 3 | | 200 – 400 | 7 | | 400 – 600 | 5 | | 600 – 800 | 3 | | 800 – 1000 | 2 | This table shows a grouped frequency distribution of rainfall data for 20 stations.

Regression analysis is a statistical method used to predict the value of a dependent variable (Y) based on the known value of an independent variable (X). The linear regression equation is: Y = a + bX Where: - Y = dependent variable (predicted) - X = independent variable (known) - a = Y-intercept (value of Y when X = 0) - b = slope (change in Y per unit change in X) Example: If the regression equation for rainfall (X) and crop yield (Y) is Y = 2 + 0.5X, then for X = 100 mm rainfall, the predicted yield Y = 2 + 0.5(100) = 52 units.

Measures of central tendency represent the centre or typical value of a dataset. The three main measures are: 1. Mean (Arithmetic Average): Formula: x̄ = Σxi / N Geographical Example: The mean annual rainfall of Tamil Nadu: Chennai (1400 mm), Coimbatore (700 mm), Madurai (850 mm), Trichy (900 mm), Salem (1050 mm). Mean = (1400 + 700 + 850 + 900 + 1050) / 5 = 4900/5 = 980 mm Advantage: Uses all values; suitable for further statistical calculations. Disadvantage: Affected by extreme values (outliers). 2. Median (Middle Value): Arrange data in order, find the middle value. For even N, median = average of two middle values. Geographical Example: Temperatures in °C: 22, 25, 28, 30, 35 Median = 28°C (middle value) If an outlier is added: 22, 25, 28, 30, 35, 48 Median = (28 + 30)/2 = 29°C (unaffected by the outlier 48) Advantage: Not affected by extreme values; preferred for skewed data. Disadvantage: Does not use all data values; not suitable for further calculations. 3. Mode (Most Frequent Value): Geographical Example: In a study of land use types across 20 Tamil Nadu districts, if 8 districts have "agriculture" as the dominant land use, 6 have "forest," 4 have "urban," and 2 have "barren" — the mode is "agriculture." Advantage: Useful for categorical data; easy to identify. Disadvantage: May not exist or may not be unique; not suitable for further calculations. When to use which measure: - Use mean for symmetric distributions without outliers. - Use median for skewed distributions or data with outliers. - Use mode for categorical data or to find the most common value.

Measures of dispersion indicate how spread out the data is around the central value. They tell us whether the values are closely clustered or widely scattered. 1. Range: Range = Maximum value - Minimum value Example: Tamil Nadu district rainfall: Max = 3400 mm (Nilgiris), Min = 600 mm (Dharmapuri). Range = 3400 - 600 = 2800 mm. Significance: Simplest measure; gives a quick idea of spread. But it only uses two extreme values and ignores all others. 2. Quartile Deviation (QD): QD = (Q3 - Q1) / 2 Where Q1 = 25th percentile, Q3 = 75th percentile. Significance: More reliable than range as it excludes extreme values. Useful for understanding the spread of the middle 50% of data. 3. Mean Deviation: MD = Σ|xi - x̄| / N (or Σ|xi - median| / N) Significance: Uses all values and considers absolute deviations. Less affected by outliers than standard deviation. 4. Standard Deviation (σ): σ = √[Σ(xi - x̄)² / N] Geographical Example: Consider rainfall data for two regions (in mm): Region A: 900, 910, 890, 905, 895 → Mean = 900, σ ≈ 8.37 Region B: 500, 1200, 600, 1100, 700 → Mean = 820, σ ≈ 302.8 Region A has low variability (rainfall is consistent); Region B has high variability (unreliable rainfall). Significance: The most important and widely used measure of dispersion. It uses all values, is suitable for further statistical work, and is the basis for many advanced techniques. 5. Coefficient of Variation (CV): CV = (σ / x̄) x 100 Significance: Allows comparison of variability across datasets with different units or scales. Example: Comparing rainfall variability (mm) with temperature variability (°C).

Karl Pearson's correlation coefficient (r) measures the linear relationship between two variables. It ranges from -1 to +1. Formula: r = Σ(dx x dy) / √(Σdx² x Σdy²) Where dx = (X - X̄) and dy = (Y - Ȳ) Interpretation: - r = +1: Perfect positive correlation - r = -1: Perfect negative correlation - r = 0: No correlation - 0 < r < 1: Positive correlation (closer to 1 = stronger) - -1 < r < 0: Negative correlation (closer to -1 = stronger) Worked Example: Calculate the correlation between annual rainfall (X, in cm) and rice yield (Y, in quintals/hectare): | Station | X (Rainfall) | Y (Yield) | dx=X-X̄ | dy=Y-Ȳ | dx×dy | dx² | dy² | |---|---|---|---|---|---|---|---| | A | 100 | 30 | -10 | -5 | 50 | 100 | 25 | | B | 110 | 35 | 0 | 0 | 0 | 0 | 0 | | C | 120 | 40 | 10 | 5 | 50 | 100 | 25 | | D | 90 | 25 | -20 | -10 | 200 | 400 | 100 | | E | 130 | 45 | 20 | 10 | 200 | 400 | 100 | | Total | 550 | 175 | | | 500 | 1000 | 250 | X̄ = 550/5 = 110; Ȳ = 175/5 = 35 r = Σ(dx x dy) / √(Σdx² x Σdy²) r = 500 / √(1000 x 250) r = 500 / √250000 r = 500 / 500 r = +1.0 (perfect positive correlation — as rainfall increases, rice yield increases proportionally) Geographical significance: This high correlation indicates that rainfall is a strong determinant of rice yield in these regions, which is expected for a rain-dependent crop.

Regression analysis predicts the value of a dependent variable (Y) based on the known value of an independent variable (X). In geography, it is used to establish functional relationships between phenomena. Linear Regression Equation: Y = a + bX Where: - Y = dependent variable (predicted) - X = independent variable (known) - a = Y-intercept - b = regression coefficient (slope) Formulas: b = Σ(dx x dy) / Σdx² a = Ȳ - bX̄ Where dx = X - X̄, dy = Y - Ȳ Worked Example: Predict crop yield (Y, in quintals/hectare) based on rainfall (X, in cm): Using the data from the correlation example: X̄ = 110, Ȳ = 35, Σ(dx x dy) = 500, Σdx² = 1000 b = 500 / 1000 = 0.5 a = 35 - (0.5 x 110) = 35 - 55 = -20 Regression equation: Y = -20 + 0.5X Prediction: If rainfall X = 140 cm: Y = -20 + 0.5(140) = -20 + 70 = 50 quintals/hectare Geographical Applications: 1. Predicting crop yield from rainfall data. 2. Estimating population growth from past trends. 3. Predicting flood levels from rainfall intensity. 4. Estimating urban sprawl from population growth. 5. Predicting temperature changes from altitude data. Limitations: - Regression assumes a linear relationship, which may not always hold. - Extrapolation beyond the data range can give unreliable results. - Correlation does not imply causation — a high r value does not mean X causes Y.

The normal distribution is a continuous probability distribution that is symmetric about the mean, forming a bell-shaped curve. It is one of the most important distributions in statistics. Characteristics: 1. The curve is symmetric about the mean. 2. Mean = Median = Mode (all three measures of central tendency coincide). 3. The curve is bell-shaped and asymptotic (approaches but never touches the x-axis). 4. The total area under the curve = 1 (or 100%). 5. Approximately 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. Importance in Geography: 1. Rainfall distribution: Annual rainfall at many stations follows a near-normal distribution, allowing prediction of wet and dry years. 2. Temperature data: Daily temperatures at a station approximate a normal distribution, enabling climate classification. 3. Population characteristics: Heights, weights, and ages in a population often follow normal distribution. 4. Error analysis: Measurement errors in surveying and mapping tend to be normally distributed, allowing quality control. 5. Hypothesis testing: Many statistical tests (z-test, t-test, ANOVA) assume normal distribution of data. 6. Standard scores (z-scores): Allow comparison of values from different normal distributions. Skewed Distributions: - Positive skew (right-skewed): Mean > Median > Mode — e.g., income distribution, where a few high values pull the mean right. - Negative skew (left-skewed): Mean < Median < Mode — e.g., exam scores when most students score high. In geographical studies, skewed distributions are common. Rainfall in arid regions is often positively skewed (a few very wet years), while population density data across districts can be highly skewed.