SSC CGL (Combined Graduate Level) Statistics (Tier II Paper II) Syllabus

How is the median of a grouped frequency distribution calculated?

Median = L + [ (N/2 − cf) / f ] × h, where L = lower boundary of median class, N = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

How is the mode of a grouped frequency distribution calculated?

Mode = L + [ (f1 − f0) / (2f1 − f0 − f2) ] × h, where L = lower boundary of modal class, f1 = modal-class frequency, f0 = previous class frequency, f2 = next class frequency, h = class width.

State the empirical relationship between mean, median, and mode.

Mode = 3·Median − 2·Mean (approximately, for moderately skewed distributions).

What is the relationship between Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)?

AM ≥ GM ≥ HM (equality only when all values are equal), and GM² = AM × HM (GM is the geometric mean of AM and HM).

When are the geometric mean and harmonic mean preferred?

GM is used for ratios, rates of growth, and index numbers; HM is used for averaging rates and speeds (e.g., average speed over equal distances).

Which measure of central tendency is most affected by extreme values, and which is unaffected?

The arithmetic mean is most affected by extreme values; the median and mode are not affected by extreme values.

Name the four main absolute measures of dispersion.

Range, quartile deviation (semi-interquartile range), mean deviation, and standard deviation.

What is the formula for range and the coefficient of range?

Range = L − S (largest − smallest value). Coefficient of range = (L − S)/(L + S).

What is quartile deviation and its coefficient?

Quartile deviation (Q.D.) = (Q3 − Q1)/2. Coefficient of Q.D. = (Q3 − Q1)/(Q3 + Q1).

Define standard deviation and its formula for ungrouped data.

Standard deviation (σ) is the positive square root of the mean of squared deviations from the mean: σ = √[ Σ(x − x̄)² / N ].

What is variance and how does it relate to standard deviation?

Variance is the mean of the squared deviations from the mean (σ²); the standard deviation is the square root of the variance.

What is the coefficient of variation (CV) and what is it used for?

CV = (σ / x̄) × 100. It is a relative measure of dispersion used to compare variability/consistency of two or more series; lower CV means more consistent (less variable).

What is the relationship among Mean Deviation, Standard Deviation, and Quartile Deviation for a normal distribution?

Q.D. = (2/3)σ and M.D. = (4/5)σ, giving the ratio Q.D. : M.D. : S.D. = 10 : 12 : 15.

In statistics, what is a moment about the mean (central moment)?

The r-th central moment μr = Σf(x − x̄)^r / N. The first central moment is zero, the second equals the variance, the third measures skewness, and the fourth measures kurtosis.

Define skewness and what its sign indicates.

Skewness measures the asymmetry of a distribution. Positive skew: longer right tail, Mean > Median > Mode. Negative skew: longer left tail, Mean < Median < Mode. Zero skew means symmetrical.

State Karl Pearson's coefficient of skewness.

Skp = (Mean − Mode) / σ, or when mode is ill-defined, Skp = 3(Mean − Median) / σ. Its value generally lies between −3 and +3.

State Bowley's coefficient of skewness (quartile measure).

SkB = (Q3 + Q1 − 2·Median) / (Q3 − Q1). It ranges from −1 to +1.

What is the moment coefficient of skewness (β1 and γ1)?

β1 = μ3² / μ2³ and γ1 = μ3 / μ2^(3/2) = √β1. γ1 carries the sign of skewness (μ3).

Define kurtosis and the three types of curves.

Kurtosis measures the peakedness of a distribution. Leptokurtic: more peaked than normal (β2 > 3); Mesokurtic: normal peak (β2 = 3); Platykurtic: flatter than normal (β2 < 3).

State the moment coefficient of kurtosis (β2 and γ2).

β2 = μ4 / μ2². The excess kurtosis γ2 = β2 − 3; γ2 = 0 for a normal (mesokurtic) distribution.

What does the correlation coefficient measure and what is its range?

It measures the degree and direction of linear relationship between two variables. Its value ranges from −1 to +1: +1 perfect positive, −1 perfect negative, 0 no linear correlation.

State Karl Pearson's coefficient of correlation formula.

r = Σ(x − x̄)(y − ȳ) / [ √Σ(x − x̄)² · √Σ(y − ȳ)² ] = Cov(x,y) / (σx·σy).

State Spearman's rank correlation coefficient formula.

R = 1 − [ 6·Σd² / (n(n² − 1)) ], where d is the difference between the ranks of paired observations and n is the number of pairs.

Is the correlation coefficient affected by change of origin and scale?

No. The correlation coefficient is independent of both change of origin and change of scale (it is a pure number).

What is the difference between correlation and regression?

Correlation measures the degree/direction of association between variables (symmetric, a pure number). Regression measures the average functional dependence of one variable on another and is used for estimation/prediction (asymmetric, has units).

State the two regression equations and their coefficients.

Regression of Y on X: (Y − ȳ) = byx(X − x̄), where byx = r·σy/σx. Regression of X on Y: (X − x̄) = bxy(Y − ȳ), where bxy = r·σx/σy.

What is the relationship between the two regression coefficients and r?

r = ±√(byx · bxy); the correlation coefficient is the geometric mean of the two regression coefficients, taking the sign of the coefficients. Also |r| ≤ 1 implies byx·bxy ≤ 1.

At what point do the two lines of regression intersect, and what does the angle between them indicate?

They intersect at the mean point (x̄, ȳ). The smaller the angle between them, the higher the correlation; when r = ±1 the two lines coincide, and when r = 0 they are perpendicular.

Define an index number and state its base-year value convention.

An index number is a statistical measure showing relative change in a variable (or group) over time, place, or characteristic. The base period value is taken as 100.

State Laspeyres', Paasche's, and Fisher's price index formulas.

Laspeyres = (Σp1q0 / Σp0q0)×100 (base-year weights). Paasche = (Σp1q1 / Σp0q1)×100 (current-year weights). Fisher = √(Laspeyres × Paasche), the 'ideal' index.

Why is Fisher's index called the 'ideal' index number?

Because it is the geometric mean of Laspeyres and Paasche indices and satisfies both the time reversal test and the factor reversal test.

What are the time reversal test and factor reversal test for index numbers?

Time reversal: P01 × P10 = 1 (index from 0 to 1 times index from 1 to 0 equals unity). Factor reversal: P01 × Q01 = ΣP1Q1 / ΣP0Q0 (product of price and quantity indices equals the value ratio).

What are the four components of a time series?

Secular trend (long-term movement), seasonal variation (regular within-year periodic), cyclical variation (multi-year business cycles), and irregular/random variation.

Name the methods used to measure the secular trend in a time series.

Freehand (graphic) method, method of semi-averages, method of moving averages, and method of least squares (fitting a trend line).

What is the method of moving averages used for, and what is its limitation?

It smooths a time series to reveal the trend by averaging successive sets of values. Limitation: trend values cannot be computed for the periods at the start and end of the series, and it is unsuitable for forecasting beyond the data.

What are the two models for decomposition of a time series?

Additive model: Y = T + S + C + I (components independent, in same units). Multiplicative model: Y = T × S × C × I (components interdependent, expressed as ratios/percentages).

Define probability of an event and give its range.

Probability P(E) = (number of favourable outcomes) / (total number of equally likely outcomes). It ranges from 0 (impossible) to 1 (certain), so 0 ≤ P(E) ≤ 1.

State the addition theorem of probability for two events.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

State the multiplication theorem of probability and the condition for independence.

P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B). If A and B are independent, P(A ∩ B) = P(A)·P(B).

State Bayes' theorem.

P(Ai | B) = [ P(Ai)·P(B|Ai) ] / Σ[ P(Aj)·P(B|Aj) ], used to find the posterior probability of a cause Ai given that event B has occurred.

What is conditional probability and its formula?

Conditional probability is the probability of event A given that B has occurred: P(A|B) = P(A ∩ B) / P(B), provided P(B) ≠ 0.