JEE JEE Main and JEE Advanced · Mathematics
Summarise grouped or ungrouped data correctly with central-tendency and dispersion measures, use the denominator convention required by the JEE/NCERT descriptive-data model, and connect random-variable mean and variance to valid probability distributions where officially applicable.
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In short
Statistics describes the centre and spread of data. For JEE school-level descriptive statistics, the NCERT convention divides the sum of squared deviations by the number of observations, or by total frequency, rather than automatically using the n-1 unbiased sample-variance denominator.
The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.
Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.
Sources: JEE Main 2026 syllabus and JEE Advanced 2026 syllabus, both linked in the sources section below.
The formula set depends entirely on the data model.
Decide whether data is raw, discrete frequency or continuous grouped, and compute total count N or total frequency before applying any formula.
Grouped continuous data uses the class mark as the representative value.
For grouped continuous data, use the correct class boundaries and class marks. Using class limits instead of class marks introduces error into mean and variance calculations.
This page does not silently replace N by N-1.
State the denominator convention explicitly. If a question explicitly asks for an unbiased sample variance under another convention, follow the question's definition rather than this descriptive-data formula.
Standard deviation is the nonnegative square root of variance, in the same units as the data.
Keep variance nonnegative and standard deviation in the same units as the data. When comparing same-mean distributions, larger variance means greater squared spread around that common mean.
For a random variable, validate probabilities before computing mean or variance.
For a random variable, validate probabilities before mean or variance. A distribution table must satisfy pᵢ≥0 and Σpᵢ=1 before any expectation calculation is meaningful.
x bar equals one over N times the sum of x sub i.
Mean of N raw observations.
Use when — Raw ungrouped data.
Common trap — Dividing by the wrong count.
Sigma squared equals one over N times the sum of x sub i minus x bar, squared.
Variance under the NCERT/JEE descriptive-data convention.
Use when — Raw ungrouped data, descriptive convention on this page.
Common trap — Using N-1 instead of N.
Sigma squared equals one over N times the sum of x sub i squared, minus x bar squared.
Algebraically equivalent shortcut variance formula.
Use when — Same convention as the direct variance formula.
Common trap — Mixing this with a different denominator convention.
Sigma equals the square root of sigma squared.
Standard deviation as the principal nonnegative square root of variance.
Use when — After variance is computed.
Common trap — Taking a negative square root.
Mean deviation about A equals one over N times the sum of the absolute value of x sub i minus A.
Mean deviation about a stated central value A.
Use when — Raw ungrouped data.
Common trap — Forgetting absolute values.
x bar equals the sum of f sub i times x sub i, divided by N.
Mean of discrete frequency data.
Use when — Discrete frequency data.
Common trap — Dividing by number of classes instead of total frequency.
Sigma squared equals the sum of f sub i times x sub i minus x bar, squared, divided by N.
Variance of discrete frequency data.
Use when — Discrete frequency data.
Common trap — Dividing by number of classes instead of total frequency.
Mean deviation about A equals the sum of f sub i times the absolute value of x sub i minus A, divided by N.
Mean deviation about A for discrete frequency data.
Use when — Discrete frequency data.
Common trap — Forgetting absolute values.
Median equals l plus the quantity N over two minus c f, divided by f, times h.
Median estimate for continuous grouped frequency data.
Use when — Continuous grouped frequency table.
Common trap — Using the grouped median formula without identifying the median class.
Mode equals l plus the quantity f one minus f zero, divided by two f one minus f zero minus f two, times h.
Mode estimate for continuous grouped frequency data.
Use when — Standard continuous grouped-data setting with a clear modal class.
Common trap — Forcing the formula when the denominator 2f₁-f₀-f₂ equals zero or the modal class is ambiguous.
Mu equals E of X equals the sum of x sub i times p sub i.
Mean of a finite discrete random variable.
Use when — Valid finite discrete probability distribution.
Common trap — Averaging x-values without probabilities.
Variance of X equals the sum of x sub i minus mu, squared, times p sub i, which must be nonnegative.
Variance of a finite discrete random variable.
Use when — Valid finite discrete probability distribution.
Common trap — Producing a negative variance from an arithmetic error.
Answer: Mean 4, median 4, mode 4, variance 2, standard deviation √2, mean deviation 1.
N=4. Mean: x̄=(2+4+4+6)/4=4.
Median: ordered data are already 2,4,4,6, so median is (4+4)/2=4. Mode: 4 occurs most often, so mode is 4.
Variance: σ²=[(2-4)²+(4-4)²+(4-4)²+(6-4)²]/4=(4+0+0+4)/4=2. Standard deviation: σ=√2.
Mean deviation about the mean: MD(x̄)=(|2-4|+|4-4|+|4-4|+|6-4|)/4=(2+0+0+2)/4=1.
Validity checks: variance is nonnegative, standard deviation is nonnegative, all centre measures equal 4 for this dataset, and symmetric deviations around 4 make the computed centre plausible.
Using n-1 automatically when the JEE/NCERT descriptive formula expects n or total frequency.
Knowledge gapWhy it happens
The descriptive-data convention on this page divides by N, not N-1, unless the question states otherwise.
How it is corrected
State the denominator convention explicitly before computing variance.
Dividing frequency formulas by the number of classes instead of total frequency.
Execution errorWhy it happens
The denominator in frequency formulas is N, the total frequency, not the number of distinct classes.
How it is corrected
Compute total frequency N before applying frequency-weighted formulas.
Using class limits instead of appropriate class marks in grouped mean/variance calculations.
Execution errorWhy it happens
Grouped formulas use the class mark as the representative value, not the class boundary.
How it is corrected
Compute the class mark as the midpoint of each class interval before using it in a formula.
Using the grouped median formula without identifying the median class.
Decision / selection errorWhy it happens
The formula requires the correct lower boundary, cumulative frequency and frequency of the specific median class.
How it is corrected
Locate the median class using cumulative frequency before substituting into the formula.
Forcing the grouped mode formula when its denominator is zero or the modal class is ambiguous.
Needs reviewWhy it happens
The formula is invalid when 2f₁-f₀-f₂=0 or the modal class is not well defined.
How it is corrected
Check the denominator condition and modal-class clarity before applying the formula.
Forgetting absolute values in mean deviation.
Execution errorWhy it happens
Mean deviation is defined using the absolute value of each deviation from A.
How it is corrected
Apply absolute value to each deviation before averaging.
Taking a negative square root for standard deviation.
Execution errorWhy it happens
Standard deviation is defined as the principal nonnegative square root of variance.
How it is corrected
Take only the nonnegative square root when computing standard deviation.
Confusing data variance with random-variable variance without stating the model.
Needs reviewWhy it happens
Data variance and random-variable variance use different formula structures and require stating which model applies.
How it is corrected
State explicitly whether the question is about a dataset or a random variable before applying a variance formula.
Official syllabus scope verified against NTA JEE Main 2026 and JEE Advanced 2026 documents. No weightage, frequency or PYQ-count claim is made.
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