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Math · Statistics and Probability

Statistics formulas for JEE

Every Statistics formula you need for JEE, grouped by concept.

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QuestionState the formula — Mean (Ungrouped Data)
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Mean (Ungrouped Data)xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_iArithmetic mean of ungrouped observations.statisticsmeanungrouped
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All 25 Statistics formulas
01

Central Tendency

8 formulas

Mean (Ungrouped Data)

xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Arithmetic mean of ungrouped observations.

statisticsmeanungrouped

Combined Mean

xˉc=n1xˉ1+n2xˉ2n1+n2\bar{x}_c = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

Mean of two groups combined together.

statisticsmeancombinedjee-advanced

Mean (Grouped Data)

xˉ=1N∑i=1nfixi\bar{x} = \frac{1}{N} \sum_{i=1}^{n} f_i x_i

Arithmetic mean of a discrete or continuous frequency distribution.

applies whenN = \sum f_i. For continuous data, x_i is the class midpoint.
statisticsmeangrouped

Mean (Step-Deviation Method)

xˉ=a+∑i=1nfidiN×h\bar{x} = a + \frac{\sum_{i=1}^{n} f_i d_i}{N} \times h

Calculates mean using assumed mean 'a' and step deviation.

applies whend_i = (x_i - a)/h
statisticsmeanstep_deviation

Median (Continuous Frequency Distribution)

M=l+(N2−Cf)×hM = l + \left( \frac{\frac{N}{2} - C}{f} \right) \times h

Median formula for grouped continuous data.

applies whenl=lower limit of median class, C=c.f. of preceding class, f=frequency of median class, h=class width.
statisticsmediangrouped

Median (Ungrouped, Even)

M=(n2)th+(n2+1)th2M = \frac{\left(\frac{n}{2}\right)^{th} + \left(\frac{n}{2}+1\right)^{th}}{2}

Median of ungrouped data when the number of observations is even.

applies whenData must be arranged in ascending or descending order.
statisticsmedianungrouped

Median (Ungrouped, Odd)

M=(n+12)th observationM = \left(\frac{n+1}{2}\right)^{th} \text{ observation}

Median of ungrouped data when the number of observations is odd.

applies whenData must be arranged in ascending or descending order.
statisticsmedianungrouped

Sum of Squares of First n Natural Nums

∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

Identity often used to calculate variance of natural numbers.

statisticsidentitysum_of_squares
02

Measures of Dispersion

16 formulas

Mean Deviation about a General Value

M.D.(a)=1n∑i=1n∣xi−a∣M.D.(a) = \frac{1}{n} \sum_{i=1}^{n} |x_i - a|

Mean of the absolute deviations of observations from a fixed value 'a'.

statisticsmean_deviationungrouped

Mean Deviation about Mean

M.D.(xˉ)=1n∑i=1n∣xi−xˉ∣M.D.(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|

Mean of the absolute deviations of observations from their arithmetic mean.

statisticsmean_deviationmean

Mean Deviation about Mean (Grouped)

M.D.(xˉ)=1N∑i=1nfi∣xi−xˉ∣M.D.(\bar{x}) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - \bar{x}|

Mean deviation about the mean for a frequency distribution.

statisticsmean_deviationgrouped

Mean Deviation about Median

M.D.(M)=1n∑i=1n∣xi−M∣M.D.(M) = \frac{1}{n} \sum_{i=1}^{n} |x_i - M|

Mean of the absolute deviations of observations from their median.

statisticsmean_deviationmedian

Mean Deviation about Median (Grouped)

M.D.(M)=1N∑i=1nfi∣xi−M∣M.D.(M) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - M|

Mean deviation about the median for a frequency distribution.

statisticsmean_deviationgrouped

Standard Deviation (Ungrouped Data)

σ=1n∑i=1n(xi−xˉ)2\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}

Positive square root of the variance for ungrouped data.

statisticsstandard_deviationungrouped

Standard Deviation (Grouped Data)

σ=1N∑i=1nfi(xi−xˉ)2\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2}

Standard deviation for discrete or continuous frequency distributions.

statisticsstandard_deviationgrouped

Standard Deviation (Shortcut Method)

σ=1NN∑i=1nfixi2−(∑i=1nfixi)2\sigma = \frac{1}{N} \sqrt{ N \sum_{i=1}^{n} f_i x_i^2 - \left( \sum_{i=1}^{n} f_i x_i \right)^2 }

Formula for standard deviation to simplify manual calculations.

statisticsstandard_deviationshortcut

Standard Deviation (Step-Dev)

σ=hNN∑i=1nfiyi2−(∑i=1nfiyi)2\sigma = \frac{h}{N} \sqrt{ N \sum_{i=1}^{n} f_i y_i^2 - \left( \sum_{i=1}^{n} f_i y_i \right)^2 }

Standard deviation calculation utilizing step deviations.

applies wheny_i = (x_i - A)/h
statisticsstandard_deviationstep_deviation

Combined Variance

σc2=n1(σ12+d12)+n2(σ22+d22)n1+n2\sigma_c^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}

Variance of two combined groups.

applies whend_1 = \bar{x}_1 - \bar{x}_c, d_2 = \bar{x}_2 - \bar{x}_c
statisticsvariancecombinedjee-advanced

Variance (Ungrouped Data)

σ2=1n∑i=1n(xi−xˉ)2\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2

Mean of the squares of deviations from the mean.

statisticsvarianceungrouped

Variance (Grouped Data)

σ2=1N∑i=1nfi(xi−xˉ)2\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2

Variance for discrete or continuous frequency distributions.

statisticsvariancegrouped

Variance of first n natural numbers

σ2=n2−112\sigma^2 = \frac{n^2 - 1}{12}

Direct formula for the variance of the first n continuous natural numbers.

statisticsvariancenatural_numbersjee-advanced

Variance (Shortcut Method)

σ2=1N2[N∑i=1nfixi2−(∑i=1nfixi)2]\sigma^2 = \frac{1}{N^2} \left[ N \sum_{i=1}^{n} f_i x_i^2 - \left( \sum_{i=1}^{n} f_i x_i \right)^2 \right]

Formula for variance to simplify manual calculations avoiding decimals.

statisticsvarianceshortcut

Variance (Step-Deviation Method)

σ2=h2N2[N∑i=1nfiyi2−(∑i=1nfiyi)2]\sigma^2 = \frac{h^2}{N^2} \left[ N \sum_{i=1}^{n} f_i y_i^2 - \left( \sum_{i=1}^{n} f_i y_i \right)^2 \right]

Variance calculation utilizing step deviations.

applies wheny_i = (x_i - A)/h
statisticsvariancestep_deviation

Variance under Linear Transformation

yi=axi+b  ⟹  σy2=a2σx2y_i = a x_i + b \implies \sigma_y^2 = a^2 \sigma_x^2

Effect of scaling and shifting origin on the variance.

applies whenVariance is independent of change of origin (b) but depends on change of scale (a).
statisticsvariancetransformation
03

Analysis of Frequency Distributions

1 formula

Coefficient of Variation

CV=σxˉ×100CV = \frac{\sigma}{\bar{x}} \times 100

A relative measure of dispersion used to compare consistency of datasets.

applies when\bar{x} \neq 0
statisticsdispersionrelativejee-advanced
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