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Chapter 14

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Future Value of a Single Amount (Compounding)

Topic: Time Value of MoneyChapter: Chapter 14
FV=P(1+i)nFV = P(1 + i)^n
Calculates the future value (FV) of a single present amount (P) invested for 'n' periods at an interest rate 'i' per period. This formula is used to determine how much an investment will grow over time.

Present Value of a Single Amount (Discounting)

Topic: Time Value of MoneyChapter: Chapter 14
PV=FV(1+i)nPV = \frac{FV}{(1 + i)^n}
Calculates the present value (PV) of a future amount (FV) to be received after 'n' periods, discounted at an interest rate 'i' per period. This helps in understanding the current worth of a future cash flow.

Future Value of an Ordinary Annuity

Topic: Time Value of MoneyChapter: Chapter 14
FVAn=A[(1+i)n−1i]FVA_n = A \left[ \frac{(1 + i)^n - 1}{i} \right]
Calculates the future value (FVA_n) of a series of equal payments (A) made at the end of each period for 'n' periods, compounded at an interest rate 'i' per period. This is used for savings plans where contributions are made at the end of the period.

Future Value of an Annuity Due

Topic: Time Value of MoneyChapter: Chapter 14
FVAndue=A[(1+i)n−1i](1+i)FVA_n^{due} = A \left[ \frac{(1 + i)^n - 1}{i} \right] (1 + i)
Calculates the future value (FVA_n^due) of a series of equal payments (A) made at the beginning of each period for 'n' periods, compounded at an interest rate 'i' per period. This is applicable when payments are made at the start of the period, like rent.

Present Value of an Ordinary Annuity

Topic: Time Value of MoneyChapter: Chapter 14
PVAn=A[1−(1+i)−ni]PVA_n = A \left[ \frac{1 - (1 + i)^{-n}}{i} \right]
Calculates the present value (PVA_n) of a series of equal payments (A) received at the end of each period for 'n' periods, discounted at an interest rate 'i' per period. This is useful for valuing loans or investments with regular end-of-period payouts.

Present Value of an Annuity Due

Topic: Time Value of MoneyChapter: Chapter 14
PVAndue=A[1−(1+i)−ni](1+i)PVA_n^{due} = A \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)
Calculates the present value (PVA_n^due) of a series of equal payments (A) received at the beginning of each period for 'n' periods, discounted at an interest rate 'i' per period. This applies to situations where payments occur at the start of the period.

Present Value of a Perpetuity

Topic: Time Value of MoneyChapter: Chapter 14
PVperpetuity=AiPV_{perpetuity} = \frac{A}{i}
Calculates the present value of an infinite series of equal payments (A) received at the end of each period, discounted at an interest rate 'i' per period. This is used for valuing assets that provide a constant stream of income indefinitely.

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

Topic: Time Value of MoneyChapter: Chapter 14
EAR=(1+inomm)m−1EAR = \left(1 + \frac{i_{nom}}{m}\right)^m - 1
Calculates the effective annual rate (EAR) when interest is compounded 'm' times per year at a nominal annual rate (i_nom). This rate reflects the true annual return or cost of an investment or loan, considering the effect of compounding frequency.

Simple Interest (Interest Amount)

Topic: Time Value of MoneyChapter: Chapter 14
I=P×R×TI = P \times R \times T
Calculates the simple interest earned or paid. 'I' is the simple interest, 'P' is the principal amount, 'R' is the annual interest rate (as a decimal), and 'T' is the time period in years.

Simple Interest (Total Amount)

Topic: Time Value of MoneyChapter: Chapter 14
A=P(1+R×T)A = P(1 + R \times T)
Calculates the total amount (principal + simple interest) after a given period. 'A' is the total amount, 'P' is the principal amount, 'R' is the annual interest rate (as a decimal), and 'T' is the time period in years.

Compound Interest (Future Value / Amount)

Topic: Time Value of MoneyChapter: Chapter 14
A=P(1+i)nA = P(1 + i)^n
Calculates the future value or total amount after compounding. 'A' is the future value/amount, 'P' is the principal amount, 'i' is the interest rate per compounding period (annual rate / number of compounding periods per year), and 'n' is the total number of compounding periods (years \times number of compounding periods per year).

Compound Interest (Interest Amount)

Topic: Time Value of MoneyChapter: Chapter 14
CI=P[(1+i)n−1]CI = P[(1 + i)^n - 1]
Calculates the compound interest earned or paid. 'CI' is the compound interest, 'P' is the principal amount, 'i' is the interest rate per compounding period, and 'n' is the total number of compounding periods.

Present Value (Compound Interest)

Topic: Time Value of MoneyChapter: Chapter 14
P=A(1+i)−nP = A(1 + i)^{-n}
Calculates the present value of a future amount. 'P' is the present value, 'A' is the future value, 'i' is the interest rate per compounding period, and 'n' is the total number of compounding periods.

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 14
E=(1+rm)m−1E = (1 + \frac{r}{m})^m - 1
Calculates the effective annual rate of interest, considering the effect of compounding. 'E' is the effective annual rate, 'r' is the nominal annual interest rate (as a decimal), and 'm' is the number of compounding periods per year.

Future Value of Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 14
FVA=PMT×[(1+i)n−1]iFV_A = PMT \times \frac{[(1 + i)^n - 1]}{i}
Calculates the future value of a series of equal payments made at the end of each period. 'FV_A' is the future value of the annuity, 'PMT' is the payment per period, 'i' is the interest rate per period, and 'n' is the total number of payments/periods.

Present Value of Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 14
PVA=PMT×[1−(1+i)−n]iPV_A = PMT \times \frac{[1 - (1 + i)^{-n}]}{i}
Calculates the present value of a series of equal payments made at the end of each period. 'PV_A' is the present value of the annuity, 'PMT' is the payment per period, 'i' is the interest rate per period, and 'n' is the total number of payments/periods.

Future Value of Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 14
FVAD=PMT×[(1+i)n−1]i×(1+i)FV_{AD} = PMT \times \frac{[(1 + i)^n - 1]}{i} \times (1 + i)
Calculates the future value of a series of equal payments made at the beginning of each period. 'FV_{AD}' is the future value of the annuity due, 'PMT' is the payment per period, 'i' is the interest rate per period, and 'n' is the total number of payments/periods.

Present Value of Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 14
PVAD=PMT×[1−(1+i)−n]i×(1+i)PV_{AD} = PMT \times \frac{[1 - (1 + i)^{-n}]}{i} \times (1 + i)
Calculates the present value of a series of equal payments made at the beginning of each period. 'PV_{AD}' is the present value of the annuity due, 'PMT' is the payment per period, 'i' is the interest rate per period, and 'n' is the total number of payments/periods.

Present Value of Perpetuity

Topic: Time Value of MoneyChapter: Chapter 14
PVPerpetuity=PMTiPV_{Perpetuity} = \frac{PMT}{i}
Calculates the present value of an infinite series of equal payments. 'PV_{Perpetuity}' is the present value of the perpetuity, 'PMT' is the payment per period, and 'i' is the interest rate per period.

Arithmetic Mean (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=∑Xn\bar{X} = \frac{\sum X}{n}
Calculates the average of a simple series of observations. 'Xˉ\bar{X}' is the mean, '∑X\sum X' is the sum of all observations, and 'n' is the number of observations.

Arithmetic Mean (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=∑fX∑f\bar{X} = \frac{\sum fX}{\sum f}
Calculates the weighted average for frequency distributions. 'f' is the frequency of each observation/class, 'X' is the observation value (or mid-point for continuous series), and '∑f\sum f' is the total number of observations (N).

Arithmetic Mean (Assumed Mean Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=A+∑fd∑f\bar{X} = A + \frac{\sum fd}{\sum f}
A shortcut method for calculating the mean, especially useful for large datasets. 'A' is the assumed mean, 'd' is the deviation of each observation from the assumed mean (X-A), and 'f' is the frequency. For individual series, '∑f\sum f' becomes 'n'.

Arithmetic Mean (Step Deviation Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=A+∑fd′∑f×c\bar{X} = A + \frac{\sum fd'}{\sum f} \times c
An extension of the assumed mean method, used when class intervals are uniform. 'A' is the assumed mean, 'd'' is the step deviation (d/c), 'c' is the common class interval, and 'f' is the frequency.

Median (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
M=size of (n+12)th itemM = \text{size of } (\frac{n+1}{2})^{\text{th}} \text{ item}
The middle value of a dataset when arranged in ascending or descending order. 'n' is the number of observations. If 'n' is even, the median is the average of the two middle values.

Median (Discrete Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
M=size of (N+12)th itemM = \text{size of } (\frac{N+1}{2})^{\text{th}} \text{ item}
The median for discrete frequency distribution. 'N' is the total frequency (∑f\sum f). The value corresponding to the cumulative frequency just greater than or equal to the calculated item position is the median.

Median (Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
M=L+N2−c.f.f×iM = L + \frac{\frac{N}{2} - c.f.}{f} \times i
The median for continuous frequency distribution. 'L' is the lower limit of the median class, 'N' is total frequency, 'c.f.' is the cumulative frequency of the class preceding the median class, 'f' is the frequency of the median class, and 'i' is the class interval of the median class.

Mode (Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Z=L+f1−f02f1−f0−f2×iZ = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i
The value that appears most frequently in a dataset. For continuous series, 'L' is the lower limit of the modal class, 'f1f_1' is the frequency of the modal class, 'f0f_0' is the frequency of the class preceding the modal class, 'f2f_2' is the frequency of the class succeeding the modal class, and 'i' is the class interval.

Empirical Relationship between Mean, Median, Mode

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}
An approximate relationship that holds true for moderately skewed distributions. It allows estimation of one measure if the other two are known.

Geometric Mean (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
GM=(X1×X2×...×Xn)1norlog⁡GM=∑log⁡XnGM = (X_1 \times X_2 \times ... \times X_n)^{\frac{1}{n}} \quad \text{or} \quad \log GM = \frac{\sum \log X}{n}
Used for data that grows exponentially, such as growth rates or average percentages. It is the nth root of the product of 'n' observations. Logarithmic form simplifies calculation.

Geometric Mean (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
log⁡GM=∑flog⁡XN\log GM = \frac{\sum f \log X}{N}
Calculates the geometric mean for frequency distributions. 'f' is the frequency, 'X' is the observation/mid-point, and 'N' is the total frequency. After calculating log GM, take the antilog to find GM.

Harmonic Mean (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
HM=n∑1XHM = \frac{n}{\sum \frac{1}{X}}
Used for averaging rates, speeds, or ratios, especially when the unit of measurement is different. 'n' is the number of observations, and 'X' represents the individual observations.

Harmonic Mean (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
HM=N∑fXHM = \frac{N}{\sum \frac{f}{X}}
Calculates the harmonic mean for frequency distributions. 'N' is the total frequency, 'f' is the frequency, and 'X' is the observation/mid-point.

Range

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
R=L−SR = L - S
The simplest measure of dispersion, representing the difference between the largest (L) and smallest (S) values in a dataset.

Coefficient of Range

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Coefficient of Range=L−SL+S\text{Coefficient of Range} = \frac{L - S}{L + S}
A relative measure of dispersion, useful for comparing variability between datasets with different units or magnitudes. 'L' is the largest value and 'S' is the smallest value.

Quartile Deviation (QD)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
QD=Q3−Q12QD = \frac{Q_3 - Q_1}{2}
Also known as the semi-interquartile range, it measures the spread of the middle 50% of the data. 'Q1Q_1' is the first quartile and 'Q3Q_3' is the third quartile.

Coefficient of Quartile Deviation

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Coefficient of QD=Q3−Q1Q3+Q1\text{Coefficient of QD} = \frac{Q_3 - Q_1}{Q_3 + Q_1}
A relative measure of dispersion based on quartiles, useful for comparison across different datasets. 'Q1Q_1' is the first quartile and 'Q3Q_3' is the third quartile.

Mean Deviation (from Mean or Median)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
MD=∑∣X−A∣norMD=∑f∣X−A∣NMD = \frac{\sum |X - A|}{n} \quad \text{or} \quad MD = \frac{\sum f|X - A|}{N}
The average of the absolute deviations of observations from a central value (Mean or Median, denoted by 'A'). 'n' is the number of observations (or 'N' for frequency distribution), and 'f' is the frequency.

Coefficient of Mean Deviation

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Coefficient of MD=MDMean or Median\text{Coefficient of MD} = \frac{\text{MD}}{\text{Mean or Median}}
A relative measure of mean deviation, calculated by dividing the mean deviation by the central value from which it was calculated (Mean or Median).

Standard Deviation (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑(X−Xˉ)2nor∑X2n−(Xˉ)2\sigma = \sqrt{\frac{\sum (X - \bar{X})^2}{n}} \quad \text{or} \quad \sqrt{\frac{\sum X^2}{n} - (\bar{X})^2}
The most widely used measure of dispersion, indicating how much variation or dispersion exists from the average (mean). 'σ\sigma' is standard deviation, 'X' is observation, 'Xˉ\bar{X}' is mean, 'n' is number of observations.

Standard Deviation (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑f(X−Xˉ)2∑for∑fX2∑f−(Xˉ)2\sigma = \sqrt{\frac{\sum f(X - \bar{X})^2}{\sum f}} \quad \text{or} \quad \sqrt{\frac{\sum fX^2}{\sum f} - (\bar{X})^2}
Calculates standard deviation for frequency distributions. 'f' is frequency, 'X' is observation/mid-point, 'Xˉ\bar{X}' is mean, and '∑f\sum f' is total frequency (N).

Standard Deviation (Step Deviation Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑fd′2∑f−(∑fd′∑f)2×c\sigma = \sqrt{\frac{\sum fd'^2}{\sum f} - (\frac{\sum fd'}{\sum f})^2} \times c
A shortcut method for standard deviation, especially useful when class intervals are uniform. 'd'' is step deviation (d/c), 'c' is common class interval, 'f' is frequency.

Variance

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ2=Standard Deviation2\sigma^2 = \text{Standard Deviation}^2
The square of the standard deviation. It measures the average of the squared differences from the mean. Variance is often used in theoretical statistics because of its additive properties.

Coefficient of Variation (CV)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
CV=σXˉ×100CV = \frac{\sigma}{\bar{X}} \times 100
A relative measure of dispersion, expressed as a percentage. It is used to compare the relative variability or consistency of two or more datasets, especially when they have different units or means. 'σ\sigma' is standard deviation and 'Xˉ\bar{X}' is the mean.

Arithmetic Mean (Discrete Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=∑fX∑f\bar{X} = \frac{\sum fX}{\sum f}
Calculates the arithmetic mean for a discrete frequency distribution. ∑fX\sum fX is the sum of products of observations and their frequencies, and ∑f\sum f is the total frequency.

Arithmetic Mean (Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=∑fm∑f\bar{X} = \frac{\sum fm}{\sum f}
Calculates the arithmetic mean for a continuous frequency distribution. mm is the mid-point of each class interval, ff is the frequency of each class, and ∑f\sum f is the total frequency.

Arithmetic Mean (Step-Deviation Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ=A+∑fd′N×c\bar{X} = A + \frac{\sum fd'}{N} \times c
Calculates the arithmetic mean using an assumed mean (AA) and step deviation. d′=m−Acd' = \frac{m - A}{c}, where cc is the common class width, ff is frequency, and N=∑fN = \sum f.

Combined Mean

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉ12=n1Xˉ1+n2Xˉ2n1+n2\bar{X}_{12} = \frac{n_1 \bar{X}_1 + n_2 \bar{X}_2}{n_1 + n_2}
Calculates the mean of two or more groups combined. n1,n2n_1, n_2 are the number of observations in group 1 and 2, and Xˉ1,Xˉ2\bar{X}_1, \bar{X}_2 are their respective means. Can be extended for more groups.

Weighted Mean

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Xˉw=∑WX∑W\bar{X}_w = \frac{\sum WX}{\sum W}
Calculates the mean where each observation (XX) is assigned a weight (WW) reflecting its importance.

Median Position (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Median Position=(n+12)thobservation\text{Median Position} = \left(\frac{n+1}{2}\right)^{\text{th}} \text{observation}
Determines the position of the median in an ordered individual series. nn is the number of observations. The median is the value at this position.

Median Position (Discrete Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Median Position=(N+12)thobservation\text{Median Position} = \left(\frac{N+1}{2}\right)^{\text{th}} \text{observation}
Determines the position of the median in a discrete frequency distribution. N=∑fN = \sum f is the total frequency. The median is the value corresponding to the first cumulative frequency greater than or equal to this position.

Empirical Relationship (Mean, Median, Mode)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}
An empirical relationship used for moderately skewed distributions to estimate one measure if the other two are known.

Mean Deviation about Mean (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
MDXˉ=∑∣X−Xˉ∣nMD_{\bar{X}} = \frac{\sum |X - \bar{X}|}{n}
The average of the absolute deviations of observations (XX) from their arithmetic mean (Xˉ\bar{X}). nn is the number of observations.

Mean Deviation about Mean (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
MDXˉ=∑f∣X−Xˉ∣∑fMD_{\bar{X}} = \frac{\sum f|X - \bar{X}|}{\sum f}
The average of the absolute deviations of observations (XX or mid-points mm) from their arithmetic mean (Xˉ\bar{X}), weighted by their frequencies (ff). ∑f\sum f is the total frequency.

Mean Deviation about Median (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
MDM=∑∣X−M∣nMD_M = \frac{\sum |X - M|}{n}
The average of the absolute deviations of observations (XX) from their median (MM). nn is the number of observations.

Mean Deviation about Median (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
MDM=∑f∣X−M∣∑fMD_M = \frac{\sum f|X - M|}{\sum f}
The average of the absolute deviations of observations (XX or mid-points mm) from their median (MM), weighted by their frequencies (ff). ∑f\sum f is the total frequency.

Variance (Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ2=∑(X−Xˉ)2n\sigma^2 = \frac{\sum (X - \bar{X})^2}{n}
The average of the squared deviations of observations (XX) from their arithmetic mean (Xˉ\bar{X}). nn is the number of observations.

Variance (Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ2=∑f(X−Xˉ)2∑f\sigma^2 = \frac{\sum f(X - \bar{X})^2}{\sum f}
The average of the squared deviations of observations (XX or mid-points mm) from their arithmetic mean (Xˉ\bar{X}), weighted by their frequencies (ff). ∑f\sum f is the total frequency.

Standard Deviation (Alternative Formula - Individual Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑X2n−(∑Xn)2\sigma = \sqrt{\frac{\sum X^2}{n} - \left(\frac{\sum X}{n}\right)^2}
An alternative formula for calculating standard deviation, often computationally easier.

Standard Deviation (Alternative Formula - Discrete/Continuous Series)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑fX2∑f−(∑fX∑f)2\sigma = \sqrt{\frac{\sum fX^2}{\sum f} - \left(\frac{\sum fX}{\sum f}\right)^2}
An alternative formula for calculating standard deviation for frequency distributions, often computationally easier.

Standard Deviation (Assumed Mean Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑fd2N−(∑fdN)2\sigma = \sqrt{\frac{\sum fd^2}{N} - \left(\frac{\sum fd}{N}\right)^2}
Calculates standard deviation using an assumed mean (AA). d=X−Ad = X - A (or m−Am-A), ff is frequency, and N=∑fN = \sum f.

Standard Deviation (Step-Deviation Method)

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ=∑fd′2N−(∑fd′N)2×c\sigma = \sqrt{\frac{\sum fd'^2}{N} - \left(\frac{\sum fd'}{N}\right)^2} \times c
Calculates standard deviation using an assumed mean (AA) and step deviation. d′=X−Acd' = \frac{X - A}{c} (or m−Ac\frac{m-A}{c}), where cc is the common class width, ff is frequency, and N=∑fN = \sum f.

Combined Standard Deviation

Topic: Mean, Median, Mode and DispersionChapter: Chapter 14
σ12=n1(σ12+d12)+n2(σ22+d22)n1+n2\sigma_{12} = \sqrt{\frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}}
Calculates the standard deviation of two or more groups combined. n1,n2n_1, n_2 are the number of observations, σ1,σ2\sigma_1, \sigma_2 are their standard deviations, and d1=Xˉ1−Xˉ12d_1 = \bar{X}_1 - \bar{X}_{12}, d2=Xˉ2−Xˉ12d_2 = \bar{X}_2 - \bar{X}_{12} where Xˉ12\bar{X}_{12} is the combined mean.

Arithmetic Mean (Ungrouped Data)

Topic: Logical ReasoningChapter: Chapter 14
Xˉ=∑Xn\bar{X} = \frac{\sum X}{n}
Represents the sum of all observations (X) divided by the number of observations (n). It is the most common measure of central tendency.

Arithmetic Mean (Grouped Data - Direct Method)

Topic: Logical ReasoningChapter: Chapter 14
Xˉ=∑fX∑f\bar{X} = \frac{\sum fX}{\sum f}
Calculated by summing the product of each class midpoint (X) and its frequency (f), then dividing by the total frequency (sum of f). Used for frequency distributions.

Combined Mean

Topic: Logical ReasoningChapter: Chapter 14
Xˉ12=n1Xˉ1+n2Xˉ2n1+n2\bar{X}_{12} = \frac{n_1\bar{X}_1 + n_2\bar{X}_2}{n_1 + n_2}
Used to find the mean of a combined group when the means (\bar{X}_1, \bar{X}_2) and sizes (n_1, n_2) of individual groups are known. Can be extended for more than two groups.

Median (Ungrouped Data)

Topic: Logical ReasoningChapter: Chapter 14
Median=Value of (n+12)th observation\text{Median} = \text{Value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ observation}
The middle value of a dataset when arranged in ascending or descending order. If 'n' is even, it's the average of the two middle values.

Median (Grouped Data)

Topic: Logical ReasoningChapter: Chapter 14
Median=L+N2−Cf×h\text{Median} = L + \frac{\frac{N}{2} - C}{f} \times h
L = lower boundary of median class, N = total frequency, C = cumulative frequency of class preceding median class, f = frequency of median class, h = class width. Used for frequency distributions.

Mode (Grouped Data)

Topic: Logical ReasoningChapter: Chapter 14
Mode=L+f1−f02f1−f0−f2×h\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h
L = lower boundary of modal class, f1 = frequency of modal class, f0 = frequency of class preceding modal class, f2 = frequency of class succeeding modal class, h = class width. Used for frequency distributions to find the most frequent value.

Empirical Relationship between Mean, Median, Mode

Topic: Logical ReasoningChapter: Chapter 14
Mode≈3×Median−2×Mean\text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean}
An approximate relationship for moderately skewed distributions, useful for estimating one measure if the other two are known.

Range

Topic: Logical ReasoningChapter: Chapter 14
Range=L−S\text{Range} = L - S
L = largest value, S = smallest value. Simplest measure of dispersion, indicating the difference between extreme values.

Quartile Deviation (QD)

Topic: Logical ReasoningChapter: Chapter 14
QD=Q3−Q12QD = \frac{Q_3 - Q_1}{2}
Q3 = third quartile, Q1 = first quartile. Measures the average difference between the first and third quartiles, representing the spread of the middle 50% of the data.

Coefficient of Quartile Deviation

Topic: Logical ReasoningChapter: Chapter 14
Coefficient of QD=Q3−Q1Q3+Q1\text{Coefficient of QD} = \frac{Q_3 - Q_1}{Q_3 + Q_1}
A relative measure of dispersion, useful for comparing variability between datasets with different units or magnitudes.

Mean Deviation (about Mean - Ungrouped Data)

Topic: Logical ReasoningChapter: Chapter 14
MDXˉ=∑∣X−Xˉ∣nMD_{\bar{X}} = \frac{\sum |X - \bar{X}|}{n}
The average of the absolute differences between each observation (X) and the mean (\bar{X}). It measures the average absolute deviation from the mean.

Standard Deviation (SD - Ungrouped Data)

Topic: Logical ReasoningChapter: Chapter 14
σ=∑(X−Xˉ)2n\sigma = \sqrt{\frac{\sum (X - \bar{X})^2}{n}}
The square root of the average of the squared deviations from the mean (\bar{X}). It is the most widely used measure of dispersion.

Standard Deviation (SD - Grouped Data - Direct Method)

Topic: Logical ReasoningChapter: Chapter 14
σ=∑f(X−Xˉ)2∑f\sigma = \sqrt{\frac{\sum f(X - \bar{X})^2}{\sum f}}
The square root of the average of the squared deviations of class midpoints (X) from the mean (\bar{X}), weighted by their frequencies (f).

Variance

Topic: Logical ReasoningChapter: Chapter 14
σ2=∑(X−Xˉ)2n (ungrouped)orσ2=∑f(X−Xˉ)2∑f (grouped)\sigma^2 = \frac{\sum (X - \bar{X})^2}{n} \text{ (ungrouped)} \quad \text{or} \quad \sigma^2 = \frac{\sum f(X - \bar{X})^2}{\sum f} \text{ (grouped)}
The square of the standard deviation. It measures the average of the squared differences from the mean.

Coefficient of Variation (CV)

Topic: Logical ReasoningChapter: Chapter 14
CV=σXˉ×100CV = \frac{\sigma}{\bar{X}} \times 100
A relative measure of dispersion, expressed as a percentage. Used to compare the variability of two or more datasets, especially when they have different units or means.
Chapter 14 Formulas — CA Foundation | Kartavvya