CA Foundation Statistics Formulas Cheat Sheet: Quick Reference Guide

Statistics is a crucial component of Quantitative Aptitude in CA Foundation. Having all important formulas at your fingertips accelerates problem-solving. Here's your comprehensive cheat sheet.

Measures of Central Tendency

Mean (Average)

Simple Mean = Σx / n

Where: Σx = Sum of all values, n = Number of observations

Grouped Data Mean = Σ(f × m) / Σf

Where: f = Frequency, m = Midpoint of class

Weighted Mean = Σ(w × x) / Σw

Where: w = Weight, x = Value

**Example**:

Values: 10, 20, 30, 40, 50

Mean = (10+20+30+40+50) / 5 = 30

Median

For Ungrouped Data:

For Grouped Data:

Median = L + [(N/2 - CF) / f] × h

Where:

Mode

Mode = Most frequently occurring value

For Grouped Data:

Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × h

Where:

Measures of Dispersion

Range

Range = Highest Value - Lowest Value

Coefficient of Range = (H - L) / (H + L)

Variance and Standard Deviation

Variance (σ²) = Σ(x - μ)² / n

Standard Deviation (σ) = √Variance

For Grouped Data:

Variance = [Σ(f × m²) / Σf] - [Σ(f × m) / Σf]²

**Example**:

Values: 2, 4, 6, 8, 10

Mean = 6

Deviations²: (2-6)² = 16, (4-6)² = 4, (6-6)² = 0, (8-6)² = 4, (10-6)² = 16

Variance = (16+4+0+4+16)/5 = 8

Std Dev = √8 = 2.83

Coefficient of Variation

CV = (σ / Mean) × 100%

Useful to compare variability of two datasets with different means.

Probability and Distributions

Basic Probability

P(A) = Number of Favorable Outcomes / Total Possible Outcomes

Conditional Probability

P(A|B) = P(A ∩ B) / P(B)

Multiplication Rule

P(A ∩ B) = P(A) × P(B|A)

For independent events: P(A ∩ B) = P(A) × P(B)

Addition Rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Normal Distribution

Standard Normal Variable: Z = (X - μ) / σ

Where:

Binomial Distribution

P(X = r) = ⁿCᵣ × p^r × q^(n-r)

Where:

Mean = n × p

Variance = n × p × q

Std Dev = √(n × p × q)

Poisson Distribution

P(X = r) = (e^(-λ) × λ^r) / r!

Where:

Mean = λ

Variance = λ

Correlation and Regression

Correlation Coefficient (Pearson's r)

r = [n × Σ(xy) - Σx × Σy] / √{[n × Σ(x²) - (Σx)²] × [n × Σ(y²) - (Σy)²]}

Range: -1 to +1

Covariance

Cov(X,Y) = Σ(x - mean_x)(y - mean_y) / n

Alternative: Cov(X,Y) = [Σ(xy) / n] - mean_x × mean_y

Regression Line (Least Squares)

y = a + bx

Where:

b = [n × Σ(xy) - Σx × Σy] / [n × Σ(x²) - (Σx)²]

a = mean_y - b × mean_x

Coefficient of Determination (R²)

R² = r²

Explains percentage of variance in y explained by x.

Index Numbers

Simple Index (Base Year = 100)

Price Index = (Current Price / Base Year Price) × 100

Quantity Index = (Current Quantity / Base Year Quantity) × 100

Laspeyre's Index

Price: Σ(P₁ × Q₀) / Σ(P₀ × Q₀) × 100

Uses base year quantities

Paasche's Index

Price: Σ(P₁ × Q₁) / Σ(P₀ × Q₁) × 100

Uses current year quantities

Fisher's Ideal Index

= √(Laspeyre's × Paasche's)

Time Series Analysis

**Components of Time Series**:

**Additive Model**: Y = T + S + C + I

**Multiplicative Model**: Y = T × S × C × I

**Trend Analysis Methods**:

**Method of Semi-Averages**:

Divide data into two periods, find average of each. Trend line passes through these averages.

**Method of Least Squares**:

T = a + bt

Where t = time period (0, 1, 2, 3...)

b = Σ(tY) / Σ(t²)

a = mean_Y - b × mean_t

Hypothesis Testing Quick Reference

Standard Error

SE = σ / √n

t-Test Statistic

t = (x̄ - μ) / (s / √n)

Degrees of freedom = n - 1

Chi-Square Test

χ² = Σ[(O - E)² / E]

Where O = Observed frequency, E = Expected frequency

ANOVA (Analysis of Variance)

Tests if means of multiple groups are equal.

F = Mean Square Between / Mean Square Within

Common Exam Problems

Problem 1: Mean and Variance

Data: 5, 10, 15, 20, 25

Find mean and standard deviation.

Solution:

Mean = (5+10+15+20+25)/5 = 15

Deviations²: 100, 25, 0, 25, 100

Variance = 250/5 = 50

Std Dev = √50 = 7.07

Problem 2: Correlation

Given: n=5, Σx=25, Σy=30, Σ(xy)=160, Σ(x²)=150, Σ(y²)=200

Find correlation coefficient.

Solution:

r = [5×160 - 25×30] / √[(5×150-625)×(5×200-900)]

r = [800-750] / √[125×100]

r = 50 / √12,500 = 50/111.8 = 0.447

Problem 3: Regression

Using above data, find regression equation of y on x.

b = [5×160 - 25×30] / [5×150 - 625]

b = 50/125 = 0.4

a = 6 - 0.4×5 = 4

Equation: y = 4 + 0.4x

Problem 4: Probability

Probability of getting at least 2 heads in 4 coin tosses.

P(At least 2) = P(2) + P(3) + P(4)

Using binomial: P(r) = ⁴Cᵣ × (0.5)^r × (0.5)^(4-r)

P(2) = 6 × 0.0625 = 0.375

P(3) = 4 × 0.0625 = 0.25

P(4) = 1 × 0.0625 = 0.0625

P(At least 2) = 0.375 + 0.25 + 0.0625 = 0.6875

Memorization Tips

Create formula cards for each category: Central Tendency, Dispersion, Probability, Correlation-Regression.

Group formulas by similar structure. Notice correlation and regression formulas are related.

Practice problems repeatedly. Knowing when to apply which formula is more important than memorization.

Draw graphs. Visualizing normal distribution, regression line, and time series helps conceptual understanding.

Common Mistakes to Avoid

Don't confuse sample and population formulas. Sample variance divides by (n-1), population by n.

Don't use wrong correlation formula. Know Pearson's r formula by heart.

Don't forget to standardize (using z-score) when comparing different distributions.

Never mix up regression directions. y = a + bx is different from x = a + by.

Quick Reference Memory Aids

Mean = Average = Sum/Count

Variance = Average Squared Deviation

SD = √Variance

Correlation = How together variables move (-1 to +1)

Regression = Predicting one variable from another

With CA Saarthi's statistics practice problems and solved examples, master every formula through application. Access our formula cards, printable cheat sheets, and practice problems designed for quick mastery!