Formula Sheet

Statistics Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 6 subtopics

\bar{x} = \frac{\sum x_i}{n} \quad \text{or} \quad \bar{x} = \frac{\sum f_i x_i}{\sum f_i}

💡 For frequency distribution, multiply each observation by its frequency. Always check if grouped or ungrouped.

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

💡 Used when different observations carry different importance (weights). Reduces to arithmetic mean when all weights are equal.

\text{Median} = l + \left(\frac{\frac{N}{2} - F}{f}\right) \times h

💡 l = lower limit of median class, N = total frequency, F = cumulative frequency before median class, f = frequency of median class, h = class width.

⚠ Using the wrong class as the median class. The median class is where cumulative frequency first exceeds N/2.

\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h

💡 f₁ = frequency of modal class, f₀ = frequency of class before modal class, f₂ = frequency of class after modal class.

\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{N} = \frac{\sum f_i x_i^2}{N} - \bar{x}^2

💡 The second form (shortcut) avoids computing deviations. Var = E(X²) - [E(X)]². Always non-negative.

⚠ Forgetting to subtract the square of the mean. Writing Var = E(X²) - E(X) instead of E(X²) - [E(X)]².

\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum f_i x_i^2}{N} - \bar{x}^2}

💡 SD is in the same units as the data. Variance is in squared units. SD = sqrt(Variance).

\bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

💡 Weighted by group sizes. Extends to k groups: numerator = sum of (nᵢ times mean of group i).

\sigma_{12}^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}

💡 where d₁ = x̄₁ - x̄₁₂ and d₂ = x̄₂ - x̄₁₂. The dᵢ terms account for the difference between group means and combined mean.

\text{If } y_i = \frac{x_i - a}{h}, \text{ then } \bar{x} = a + h\bar{y}, \quad \sigma_x = |h| \cdot \sigma_y

💡 Mean changes with both origin and scale. SD changes only with scale (not origin). Variance changes by h².

⚠ Thinking variance also shifts when a constant is added. Adding a constant changes mean but NOT variance or SD.

#10

\text{MD}(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{N}

💡 Mean deviation about the mean. Can also compute about the median. Mean deviation about the median is always minimum.