Cheatsheet: Descriptive Statistics

1. Measures of Central Tendency

1.1 Mean

• Mean is sensitive to outliers and extreme values
• Geometric mean used for growth rates and ratios
• Harmonic mean used for rates and ratios (e.g., average speed)

1.2 Median

• Middle value when data is ordered from smallest to largest
• For odd n: median is the ((n+1)/2)^th value
• For even n: median is the average of (n/2)^th and ((n/2)+1)^th values
• Not affected by outliers; robust measure of central tendency
• Preferred for skewed distributions

1.3 Mode

• Most frequently occurring value in the dataset
• Can have no mode, one mode (unimodal), two modes (bimodal), or multiple modes (multimodal)
• Only measure of central tendency applicable to categorical data
• Not affected by extreme values

1.4 Relationships Between Mean, Median, and Mode

2. Measures of Dispersion

2.1 Range

• Range = Maximum value - Minimum value
• Simplest measure of spread; highly sensitive to outliers
• Interquartile Range (IQR) = Q₃ - Q₁ (more robust)

2.2 Variance

• Sample variance uses (n-1) denominator (Bessel's correction) for unbiased estimator
• Units are squared units of original data

2.3 Standard Deviation

• Most common measure of dispersion; same units as original data
• Approximately 68% of data within ±1σ, 95% within ±2σ, 99.7% within ±3σ (for normal distribution)

2.4 Coefficient of Variation

• CV = (s / x̄) × 100% for sample data
• CV = (σ / μ) × 100% for population data
• Dimensionless measure; useful for comparing variability between datasets with different units or means

2.5 Mean Absolute Deviation

• MAD = Σ|x_i - x̄| / n
• Average absolute distance from the mean
• Less sensitive to outliers than standard deviation

3. Measures of Position

3.1 Percentiles

• P_k is the value below which k% of the data falls
• Position: L_p = (p/100) × (n + 1), where p is the percentile
• If L_p is not an integer, interpolate between adjacent values

3.2 Quartiles

• IQR = Q₃ - Q₁
• Outlier detection: values below Q₁ - 1.5×IQR or above Q₃ + 1.5×IQR

3.3 Z-Score (Standard Score)

• z = (x - μ) / σ for population
• z = (x - x̄) / s for sample
• Indicates how many standard deviations a value is from the mean
• Positive z-score: value above mean; negative z-score: value below mean
• |z| > 3 often considered an outlier

4. Measures of Shape

4.1 Skewness

• Skewness = 0: symmetric distribution
• Skewness > 0: right-skewed (positive skew); tail extends to the right
• Skewness < 0:="" left-skewed="" (negative="" skew);="" tail="" extends="" to="" the="">
• |Skewness| < 0.5:="" approximately="">
• 0.5 ≤ |Skewness| ≤ 1: moderately skewed
• |Skewness| > 1: highly skewed

4.2 Kurtosis

• Excess kurtosis = 0: mesokurtic (normal distribution)
• Excess kurtosis > 0: leptokurtic (heavy tails, peaked)
• Excess kurtosis < 0:="" platykurtic="" (light="" tails,="">

5. Grouped Data

5.1 Frequency Distributions

5.2 Mean for Grouped Data

• x̄ = Σ(f_i × x_m,i) / n, where x_m,i is class midpoint and f_i is frequency

5.3 Variance for Grouped Data

• s² = [Σf_i(x_m,i - x̄)²] / (n - 1)
• Computational formula: s² = [Σf_ix_m,i² - (Σf_ix_m,i)²/n] / (n - 1)

5.4 Modal Class

• Class interval with the highest frequency
• Mode approximated by midpoint of modal class

6. Data Visualization

6.1 Graphical Methods

6.2 Five-Number Summary

• Consists of: Minimum, Q₁, Median (Q₂), Q₃, Maximum
• Used to construct box plots
• Provides robust description of data distribution

7. Correlation and Covariance

7.1 Covariance

• Measures direction of linear relationship between two variables
• Positive covariance: variables increase together
• Negative covariance: one variable increases as other decreases
• Units depend on units of x and y; difficult to interpret magnitude

7.2 Correlation Coefficient (Pearson's r)

• r = s_xy / (s_x × s_y) = Σ(x_i - x̄)(y_i - ȳ) / √[Σ(x_i - x̄)² × Σ(y_i - ȳ)²]
• Dimensionless; range: -1 ≤ r ≤ 1
• r = 1: perfect positive linear relationship
• r = -1: perfect negative linear relationship
• r = 0: no linear relationship
• |r| > 0.8: strong correlation; 0.5 < |r|="">< 0.8:="" moderate;="" |r|="">< 0.5:="">
• r² (coefficient of determination): proportion of variance in y explained by x

7.3 Spearman's Rank Correlation

• r_s = 1 - [6Σd_i²] / [n(n² - 1)], where d_i is difference in ranks
• Used for ordinal data or when relationship is non-linear but monotonic
• Less sensitive to outliers than Pearson's r

8. Empirical Rule and Chebyshev's Theorem

8.1 Empirical Rule (68-95-99.7 Rule)

• Applies to bell-shaped (normal) distributions
• Approximately 68% of data within μ ± σ
• Approximately 95% of data within μ ± 2σ
• Approximately 99.7% of data within μ ± 3σ

8.2 Chebyshev's Theorem

• Applies to any distribution shape
• At least [1 - (1/k²)] of data within μ ± kσ, where k > 1
• k = 2: at least 75% within μ ± 2σ
• k = 3: at least 89% within μ ± 3σ

9. Common Data Transformations

9.1 Linear Transformation

• y = a + bx
• Mean: ȳ = a + bx̄
• Variance: s_y² = b²s_x²
• Standard deviation: s_y = |b|s_x

9.2 Standardization

• z = (x - x̄) / s
• Standardized data has mean = 0 and standard deviation = 1
• Allows comparison of variables with different units or scales

9.3 Logarithmic Transformation

• y = log(x) or y = ln(x)
• Reduces right skewness
• Stabilizes variance for data with increasing spread
• Useful for data spanning several orders of magnitude

10. Sampling and Data Collection

10.1 Types of Data

10.2 Levels of Measurement

10.3 Sampling Methods