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Chapter 13

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Class Mark (Mid-point)

Topic: Logical ReasoningChapter: Chapter 13
Class Mark=Upper Class Limit+Lower Class Limit2\text{Class Mark} = \frac{\text{Upper Class Limit} + \text{Lower Class Limit}}{2}
This formula calculates the mid-point of a class interval. It is used to represent the class in further statistical calculations, such as finding the mean from a frequency distribution.

Class Width (for Continuous Data)

Topic: Logical ReasoningChapter: Chapter 13
Class Width=Upper Class Boundary−Lower Class Boundary\text{Class Width} = \text{Upper Class Boundary} - \text{Lower Class Boundary}
This formula determines the size of a class interval. For continuous data, class boundaries are used to ensure no gaps or overlaps between classes. For discrete data, it's often (Upper Limit - Lower Limit + 1).

Relative Frequency

Topic: Logical ReasoningChapter: Chapter 13
Relative Frequency=Frequency of a ClassTotal Frequency\text{Relative Frequency} = \frac{\text{Frequency of a Class}}{\text{Total Frequency}}
Relative frequency expresses the proportion of observations falling into a particular class interval. It is useful for comparing frequency distributions of different sizes and can be converted into a percentage.

Sturges' Rule (for Number of Classes)

Topic: Logical ReasoningChapter: Chapter 13
K=1+3.322log⁡10(N)K = 1 + 3.322 \log_{10}(N)
Sturges' Rule is an empirical formula used to determine an appropriate number of classes (K) when constructing a frequency distribution from a dataset of N observations. It provides a guideline rather than a strict rule.

Range of Data

Topic: Logical ReasoningChapter: Chapter 13
Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}
The range is the simplest measure of dispersion, representing the difference between the highest and lowest values in a dataset. It is often used in conjunction with Sturges' Rule to estimate an appropriate class width (Range / K).

Probability of Selection in Simple Random Sampling (SRS)

Topic: Types of Data and SamplingChapter: Chapter 13
P(selection of any specific unit)=nNP(\text{selection of any specific unit}) = \frac{n}{N}
This formula calculates the probability that any single unit from the population will be selected into the sample when using Simple Random Sampling. 'N' represents the total size of the population, and 'n' represents the desired size of the sample. In SRS, every unit has an equal and independent chance of being selected.

Future Value (FV) of a Single Sum

Topic: Time Value of MoneyChapter: Chapter 13
FV=P×(1+i)nFV = P \times (1 + i)^n
This formula calculates the future value of a single lump sum investment or deposit made today, compounded over a period. 'P' is the principal amount, 'i' is the interest rate per period, and 'n' is the number of compounding periods.

Present Value (PV) of a Single Sum

Topic: Time Value of MoneyChapter: Chapter 13
PV=FV(1+i)norPV=FV×(1+i)−nPV = \frac{FV}{(1 + i)^n} \quad \text{or} \quad PV = FV \times (1 + i)^{-n}
This formula calculates the present value of a single future amount, discounted back to the present. 'FV' is the future value, 'i' is the discount rate per period, and 'n' is the number of periods.

Future Value of an Ordinary Annuity (FVA)

Topic: Time Value of MoneyChapter: Chapter 13
FVA=A×[(1+i)n−1i]FVA = A \times \left[\frac{(1 + i)^n - 1}{i}\right]
This formula calculates the future value of a series of equal payments (annuity) made at the end of each period, compounded over 'n' periods. 'A' is the annuity payment per period, 'i' is the interest rate per period, and 'n' is the number of periods.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 13
FVAD=A×[(1+i)n−1i]×(1+i)FVAD = A \times \left[\frac{(1 + i)^n - 1}{i}\right] \times (1 + i)
This formula calculates the future value of a series of equal payments (annuity) made at the beginning of each period, compounded over 'n' periods. 'A' is the annuity payment per period, 'i' is the interest rate per period, and 'n' is the number of periods.

Present Value of an Ordinary Annuity (PVA)

Topic: Time Value of MoneyChapter: Chapter 13
PVA=A×[1−(1+i)−ni]PVA = A \times \left[\frac{1 - (1 + i)^{-n}}{i}\right]
This formula calculates the present value of a series of equal payments (annuity) received at the end of each period, discounted back to the present. 'A' is the annuity payment per period, 'i' is the discount rate per period, and 'n' is the number of periods.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 13
PVAD=A×[1−(1+i)−ni]×(1+i)PVAD = A \times \left[\frac{1 - (1 + i)^{-n}}{i}\right] \times (1 + i)
This formula calculates the present value of a series of equal payments (annuity) received at the beginning of each period, discounted back to the present. 'A' is the annuity payment per period, 'i' is the discount rate per period, and 'n' is the number of periods.

Present Value of a Perpetuity

Topic: Time Value of MoneyChapter: Chapter 13
PVperpetuity=AiPV_{\text{perpetuity}} = \frac{A}{i}
This formula calculates the present value of an infinite series of equal payments (perpetuity) received at the end of each period. 'A' is the periodic payment, and 'i' is the discount rate per period.

Effective Annual Rate (EAR) / Effective Interest Rate (EIR)

Topic: Time Value of MoneyChapter: Chapter 13
EAR=(1+im)m−1EAR = \left(1 + \frac{i}{m}\right)^m - 1
This formula calculates the actual annual rate of interest earned or paid on an investment or loan, considering the effect of compounding more frequently than once a year. 'i' is the nominal annual interest rate, and 'm' is the number of compounding periods per year.
Chapter 13 Formulas — CA Foundation | Kartavvya