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Class 11 Maths Notes

Statistics Class 11 Notes

Complete, exam-ready notes on statistics: measures of dispersion including range and mean deviation, variance and standard deviation for ungrouped and grouped data, the short-cut and step-deviation methods, combined mean and variance, and the coefficient of variation — written for CBSE boards and JEE/NEET revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is statistics about in Class 11?

Statistics in Class 11 covers measures of dispersion — range, mean deviation, variance and standard deviation — that quantify how spread out data is, plus the coefficient of variation for comparing distributions.

Range and Measures of Dispersion

Range

Range is the simplest measure of dispersion: the difference between the maximum and minimum values in a dataset. For ungrouped data, range = x_max − x_min. It is easy to compute but depends on extreme values and does not reflect the spread of the bulk of the data.

  • Measures of dispersion: range, mean deviation, variance and standard deviation.
  • Range is useful for quick comparisons but is unreliable for data with outliers.
  • Mean deviation and standard deviation use every observation and are more robust.

Mean Deviation for Ungrouped Data

Mean deviation about the mean

M.D.(xˉ)=∑∣xi−xˉ∣nM.D.(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}

Mean deviation is the average of the absolute deviations from a central value — either the mean or the median. It is less sensitive to outliers than the standard deviation because absolute values do not amplify large deviations.

  • Mean deviation about the median:
  • M.D.(median)=∑∣xi−median∣nM.D.(\text{median}) = \frac{\sum |x_i - \text{median}|}{n}
  • .
  • For symmetric data, M.D. about the mean and M.D. about the median are approximately equal.
  • Mean deviation is easier to compute but less used in further statistics than variance.

Mean Deviation for Grouped Data

  • Discrete frequency distribution:
  • M.D.(xˉ)=∑fi∣xi−xˉ∣∑fiM.D.(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}
  • .
  • For continuous data, use the class mark (midpoint) of each interval as x_i.
  • The procedure is the same — compute the mean, find absolute deviations, weight by frequencies, and average.

Quick check

The sum of absolute deviations about the median is always minimum. If an exam asks for the best central tendency for a skewed distribution, choose the median.

Variance and Standard Deviation

Variance and standard deviation

Variance is the average of the squared deviations from the mean. For ungrouped data, σ2=∑(xi−xˉ)2n\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}. The standard deviation σ is the square root of variance and has the same units as the original data.

  • For grouped data:
  • σ2=∑fi(xi−xˉ)2∑fi\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}
  • .
  • Variance amplifies large deviations more than mean deviation because it squares the differences.
  • Standard deviation is the most widely used measure of spread in statistics.
σ2=∑fi(xi−xˉ)2∑fi,σ=σ2\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i},\quad \sigma = \sqrt{\sigma^2}
Grouped variance and standard deviation

Short-Cut and Step-Deviation Method

  • Short-cut method: choose an assumed mean A and compute deviations d_i = x_i − A. Then variance =
  • ∑fidi2∑fi−(∑fidi∑fi)2\frac{\sum f_i d_i^2}{\sum f_i} - \left(\frac{\sum f_i d_i}{\sum f_i}\right)^2
  • .
  • Step-deviation method: set
  • ui=xi−Ahu_i = \frac{x_i - A}{h}
  • where h is the class width. This further simplifies calculations for large datasets.
  • Both methods give the same result as the direct method but with smaller numbers.
σ2=∑fidi2∑fi−(∑fidi∑fi)2\sigma^2 = \frac{\sum f_i d_i^2}{\sum f_i} - \left(\frac{\sum f_i d_i}{\sum f_i}\right)^2
Short-cut variance formula

Combined Mean, Variance and Coefficient of Variation

  • Combined mean of two groups:
  • xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}
  • .
  • Combined variance requires means, variances and sizes of both groups — a formula-heavy topic often tested in JEE.
  • The coefficient of variation (CV) compares consistency across distributions:
  • CV=σxˉ×100CV = \frac{\sigma}{\bar{x}} \times 100
  • .
  • The distribution with the lower CV is more consistent (less variable relative to its mean).

Exam favourite

JEE and board exams often ask: 'Which of two groups is more consistent?' Always compute CV for both and compare — do not just compare standard deviations.

Comparing Two Distributions

  • To compare two distributions, examine their means (central tendency) and their standard deviations or CVs (spread).
  • A higher standard deviation does not always mean more variability — use the CV when means differ significantly.
  • When comparing marks of two students or performance of two teams, CV is the standard tool.

Coefficient of variation

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

A dimensionless measure of relative variability. Lower CV means the data points tend to be closer to the mean — the distribution is more consistent.

Solved Examples

Example: Find the variance of the data set {2, 4, 6, 8, 10}.

Solution: Mean xˉ=(2+4+6+8+10)/5=6\bar{x} = (2+4+6+8+10)/5 = 6. Deviations from mean: −4, −2, 0, 2, 4. Squared deviations: 16, 4, 0, 4, 16. Variance σ2=40/5=8\sigma^2 = 40/5 = 8, so σ=22≈2.83\sigma = 2\sqrt{2} \approx 2.83.

Example: Group A has mean 60, standard deviation 5; Group B has mean 75, standard deviation 6. Which is more consistent?

Solution: CV of A = (5/60) × 100 = 8.33%. CV of B = (6/75) × 100 = 8.00%. Group B has the lower CV, so it is slightly more consistent despite having a higher standard deviation.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Mean deviation about mean (ungrouped)

M.D.(xˉ)=∑∣xi−xˉ∣nM.D.(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}

Mean deviation about mean (grouped)

M.D.(xˉ)=∑fi∣xi−xˉ∣∑fiM.D.(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}

Variance (ungrouped)

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

Variance (grouped)

σ2=∑fi(xi−xˉ)2∑fi\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}

Standard deviation

σ=σ2\sigma = \sqrt{\sigma^2}

Short-cut variance

σ2=∑xi2n−xˉ2\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2

Coefficient of variation

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

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Range = max − min; quick but sensitive to outliers.
  • Mean deviation about the median is always less than or equal to that about the mean.
  • Variance uses squared deviations — it amplifies outliers more than mean deviation.
  • Short-cut and step-deviation methods give the same variance as the direct method.
  • Use CV to compare consistency — not just standard deviation — when means differ.
  • For the combined variance formula, JEE tests the full expression; know it cold.
  • Standard deviation has the same units as the data; variance has squared units.
  • Always check whether the data is ungrouped (direct formula) or grouped (weighted formula).

FAQ

Common questions

What is the difference between mean deviation and standard deviation?

Mean deviation averages absolute deviations from the mean or median; standard deviation averages squared deviations. Standard deviation is more sensitive to outliers but is used in further statistical analysis.

Why use the short-cut method for variance?

The short-cut method subtracts an assumed mean from every value, reducing the size of numbers and making manual calculations faster and less error-prone.

What does a low coefficient of variation mean?

A low CV indicates that the data points are clustered close to the mean relative to the mean itself — the distribution is more consistent or stable.

When should I use the step-deviation method?

Use step-deviation when the data values or class marks are large and have a common factor (the class width h). Dividing by h further simplifies the arithmetic.

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