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TGPSC MPSO / ASO Paper II: The Statistics Formulas Worth Memorising

A compact formula sheet for the TGPSC MPSO / ASO Statistics paper, organised by syllabus unit, with the shortcut each formula enables and one worked example per unit.

By · Published · Facts checked · 4 min read

Which formulas save the most time?

One minute per question leaves no time to derive anything. The formulas below are the ones that turn a three-step problem into one step. Learn them with the condition attached: most wrong answers in Statistics papers come from using a correct formula where it does not apply.

Averages

\bar{x} = \frac{\sum f x}{\sum f} \qquad \text{Mode} \approx 3\,\text{Median} - 2\,\bar{x} \qquad \text{A.M.} \ge \text{G.M.} \ge \text{H.M.}

The empirical mode relation holds only for moderately skewed distributions. For two positive numbers, G.M.² = A.M. × H.M., a frequent one-line question. Replacing one observation changes the total, not the count: if one value rises by 10 in a set of 10, the mean rises by 1.

Dispersion, skewness and kurtosis

\sigma = \sqrt{\frac{\sum f (x-\bar{x})^2}{\sum f}} = \sqrt{\overline{x^2} - \bar{x}^2} \qquad \text{C.V.} = \frac{\sigma}{\bar{x}} \times 100

S_k = \frac{\bar{x} - \text{Mode}}{\sigma} = \frac{3(\bar{x} - \text{Median})}{\sigma} \qquad \beta_2 = \frac{\mu_4}{\mu_2^{2}}, \quad \beta_2 = 3 \text{ for a normal curve}

Adding a constant to every value leaves σ unchanged; multiplying by k multiplies σ by |k|. The series with the lower coefficient of variation is the more consistent one.

Probability and distributions

P(A \cup B) = P(A) + P(B) - P(A \cap B) \qquad P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Distribution facts asked most often
DistributionMeanVarianceRemember
Binomial (n, p) np npq Variance < mean, because q < 1
Poisson (λ) λ λ Mean = variance
Normal (μ, σ) μ σ² 68.27 %, 95.45 % and 99.73 % lie within 1, 2 and 3 σ of the mean

Worked example. A binomial variable has mean 4 and variance 3. Then q = 3/4, p = 1/4 and n = 4 ÷ (1/4) = 16. If an option gives a variance larger than the mean for a binomial, it is wrong without any calculation.

Correlation and regression

r = \frac{\operatorname{cov}(x,y)}{\sigma_x \sigma_y} \qquad b_{yx} = r\,\frac{\sigma_y}{\sigma_x}, \quad b_{xy} = r\,\frac{\sigma_x}{\sigma_y} \qquad r^2 = b_{yx}\, b_{xy}

\rho = 1 - \frac{6\sum d^2}{n(n^2 - 1)} \qquad \text{P.E.} = 0.6745\,\frac{1 - r^2}{\sqrt{n}}

Both regression coefficients have the same sign as r, and their product cannot exceed 1. The two regression lines meet at (x̄, ȳ). Worked example: if b(yx) = 0.8 and b(xy) = 0.45, then r² = 0.36 and r = +0.6.

Index numbers and time series

P_{01}^{L} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 \qquad P_{01}^{P} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 \qquad P_{01}^{F} = \sqrt{P^{L} \times P^{P}}

Fisher's index satisfies both the time reversal and factor reversal tests, which is why it is called ideal; Laspeyres and Paasche satisfy neither. A time series has four components: secular trend, seasonal, cyclical and irregular variation.

Sampling and tests of significance

\text{S.E.}(\bar{x}) = \frac{\sigma}{\sqrt{n}} \qquad z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \qquad \chi^2 = \sum \frac{(O - E)^2}{E}

To halve the standard error, multiply the sample size by four. Use z for large samples, t for small samples with unknown σ, χ² for goodness of fit and independence, and F for comparing two variances. A type I error rejects a true null hypothesis; its probability is the level of significance.

How to use these TGPSC MPSO ASO statistics formulas

Write each formula once by hand, then use it on ten timed questions before moving on. The TGPSC MPSO ASO test series alternates Paper II with Paper I, and the calendar shows which Statistics unit each test covers.

Frequently asked questions

Is TGPSC ASO Paper II only Statistics?

Yes, at degree level, with a short computer-fundamentals unit. Paper I is General Studies and General Abilities.

How many questions are in Paper II?

150 questions in 150 minutes, for 150 marks.

Is there negative marking in TGPSC MPSO / ASO?

The notification text does not specify one. Check the final instructions on tgpsc.gov.in before the exam.

Sources and official links

DPR Prep is independent. Always confirm dates, eligibility and rules on the official notification.