Descriptive Statistics Cheatsheet

Cheatsheet Content

Measures of Central Tendency Arithmetic Mean (A.M.): For ungrouped data: $\bar{x} = \frac{\sum x_i}{n}$ For grouped data: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ Median: The middle value of an ordered dataset. For $n$ observations, Median position: $\left(\frac{n+1}{2}\right)^{\text{th}}$ item. For grouped data: $M = L + \frac{\frac{N}{2} - C_f}{f} \times h$ $L$: lower boundary of median class, $N$: total frequency, $C_f$: cumulative frequency of class before median class, $f$: frequency of median class, $h$: class width. Mode: The most frequently occurring value. For grouped data: $M_0 = L + \frac{f_m - f_1}{2f_m - f_1 - f_2} \times h$ $L$: lower boundary of modal class, $f_m$: frequency of modal class, $f_1$: frequency of preceding class, $f_2$: frequency of succeeding class, $h$: class width. Geometric Mean (G.M.): $G.M. = \sqrt[n]{x_1 x_2 ... x_n}$ or $G.M. = \text{antilog}\left(\frac{\sum \log x_i}{n}\right)$ Harmonic Mean (H.M.): $H.M. = \frac{n}{\sum \frac{1}{x_i}}$ Relationship: $A.M. \ge G.M. \ge H.M.$ (for positive data) Weighted Averages: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$ Partition Values Quartiles ($Q_k$): Divide data into 4 equal parts. $Q_1$: 25th percentile, $Q_2$: 50th percentile (Median), $Q_3$: 75th percentile. Position for ungrouped data: $Q_k = \left(\frac{k(n+1)}{4}\right)^{\text{th}}$ item. For grouped data: $Q_k = L + \frac{\frac{kN}{4} - C_f}{f} \times h$ Deciles ($D_k$): Divide data into 10 equal parts. Position for ungrouped data: $D_k = \left(\frac{k(n+1)}{10}\right)^{\text{th}}$ item. For grouped data: $D_k = L + \frac{\frac{kN}{10} - C_f}{f} \times h$ Percentiles ($P_k$): Divide data into 100 equal parts. Position for ungrouped data: $P_k = \left(\frac{k(n+1)}{100}\right)^{\text{th}}$ item. For grouped data: $P_k = L + \frac{\frac{kN}{100} - C_f}{f} \times h$ Measures of Dispersion Range: $R = X_{\text{max}} - X_{\text{min}}$ Quartile Deviation (Q.D.): $Q.D. = \frac{Q_3 - Q_1}{2}$ Coefficient of Quartile Deviation: $C.Q.D. = \frac{Q_3 - Q_1}{Q_3 + Q_1}$ Mean Deviation (M.D.): About Mean: $M.D.(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}$ About Median: $M.D.(M) = \frac{\sum |x_i - M|}{n}$ (usually minimum) For grouped data: $M.D. = \frac{\sum f_i |x_i - A|}{N}$ (where $A$ is mean or median) Standard Deviation (S.D.): $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$ or $\sqrt{\frac{\sum x_i^2}{n} - \bar{x}^2}$ For grouped data: $\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{N}}$ Variance ($\sigma^2$): Square of Standard Deviation. Coefficient of Variation (C.V.): $C.V. = \frac{\sigma}{\bar{x}} \times 100\%$ (Relative measure) Moments Raw Moments about Origin ($A=0$): $m_r' = \frac{\sum x_i^r}{n}$ (for ungrouped data) $m_r' = \frac{\sum f_i x_i^r}{N}$ (for grouped data) $m_1' = \bar{x}$ Central Moments about Mean ($\bar{x}$): $\mu_r = \frac{\sum (x_i - \bar{x})^r}{n}$ (for ungrouped data) $\mu_r = \frac{\sum f_i (x_i - \bar{x})^r}{N}$ (for grouped data) $\mu_1 = 0$ $\mu_2 = \sigma^2$ (Variance) Relation between Raw and Central Moments: $\mu_1 = m_1' - m_1' = 0$ $\mu_2 = m_2' - (m_1')^2$ $\mu_3 = m_3' - 3m_2'm_1' + 2(m_1')^3$ $\mu_4 = m_4' - 4m_3'm_1' + 6m_2'(m_1')^2 - 3(m_1')^4$ Skewness and Kurtosis Skewness: Degree of asymmetry of a distribution. Positive Skewness: Tail to the right, Mean > Median > Mode. Negative Skewness: Tail to the left, Mean Symmetric: Mean = Median = Mode. Measures of Skewness: Pearson's Coefficient of Skewness: $SK_p = \frac{\bar{x} - M_0}{\sigma}$ $SK_p = \frac{3(\bar{x} - M)}{\sigma}$ Bowley's Coefficient of Skewness: $SK_B = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$ Moment-based Coefficient of Skewness ($\beta_1$ and $\gamma_1$): $\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ $\gamma_1 = \sqrt{\beta_1} = \frac{\mu_3}{\sigma^3}$ $\gamma_1 = 0$ for symmetric distribution. Kurtosis: Degree of peakedness or flatness of a distribution. Mesokurtic: Normal distribution ($\gamma_2 = 0$). Leptokurtic: More peaked than normal ($\gamma_2 > 0$). Platykurtic: Flatter than normal ($\gamma_2 Measures of Kurtosis: Moment-based Coefficient of Kurtosis ($\beta_2$ and $\gamma_2$): $\beta_2 = \frac{\mu_4}{\mu_2^2}$ $\gamma_2 = \beta_2 - 3 = \frac{\mu_4}{\sigma^4} - 3$