Statistics Class 11 Maths Chapter Notes

Key Terms in Statistics

Dispersion

The measure of the extent to which individual items in a series vary or are scattered from a central value (like mean or median).

Range

The simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a data set. Range = Maximum Value - Minimum Value.

Mean Deviation (MD)

The arithmetic mean of the absolute deviations of the observations from a measure of central tendency (mean, median, or mode).

Variance (σ²)

The arithmetic mean of the squares of the deviations of the observations from their arithmetic mean. It measures how far a set of numbers is spread out from their average value.

Standard Deviation (σ)

The positive square root of the variance. It is the most common and important measure of dispersion, expressed in the same units as the data.

Coefficient of Variation (CV)

A relative measure of dispersion used to compare the variability of two or more series with different units or different means. CV = (σ / x̄) * 100, where σ is the standard deviation and x̄ is the mean.

Ungrouped Data

Data given as individual data points or observations, also known as raw data.

Grouped Data

Data that has been organized into groups or classes, usually presented in a frequency distribution table.

Understanding Measures of Dispersion

In previous classes, you focused on measures of central tendency—mean, median, and mode. These numbers give us a single value that represents the 'center' of a dataset. However, they don't tell the whole story. For example, two cricket batsmen might have the same average score, but one might be very consistent (scores are close to the average) while the other is very inconsistent (scores vary wildly). This is where measures of dispersion come in. They quantify the variability or spread of data points. A low dispersion indicates that the data points tend to be clustered closely around the central value (high consistency), while a high dispersion indicates that they are spread out over a wider range (low consistency). The main measures you'll study are Range, Mean Deviation, Variance, and Standard Deviation. While Range is easy to calculate, it's a crude measure as it only depends on two extreme values. Mean Deviation is better as it considers all data points, but ignoring the signs of deviations makes it algebraically inconvenient. This leads to the most important measures: Variance and Standard Deviation, which square the deviations to make them positive, providing a robust and widely used method for analyzing data spread.

Must-Remember Formulas for Dispersion

• Mean Deviation about Mean (Ungrouped): M.D.(x̄) = (1/n) * Σ |xᵢ - x̄|
• Mean Deviation about Median (Ungrouped): M.D.(M) = (1/n) * Σ |xᵢ - M|
• Mean Deviation about Mean (Grouped): M.D.(x̄) = (1/N) * Σ fᵢ |xᵢ - x̄|, where N = Σfᵢ
• Mean Deviation about Median (Grouped): M.D.(M) = (1/N) * Σ fᵢ |xᵢ - M|, where N = Σfᵢ
• Variance (σ²) for Ungrouped Data: σ² = (1/n) * Σ (xᵢ - x̄)²
• Standard Deviation (σ) for Ungrouped Data: σ = √[(1/n) * Σ (xᵢ - x̄)²]
• Variance (σ²) for Grouped Data (Discrete): σ² = (1/N) * Σ fᵢ (xᵢ - x̄)², where N = Σfᵢ
• Variance (σ²) for Grouped Data (Continuous): σ² = (1/N) * Σ fᵢ (xᵢ - x̄)² (xᵢ is the mid-point of the class)
• Shortcut Method for Variance (σ²): σ² = (1/N²) * [N Σfᵢxᵢ² - (Σfᵢxᵢ)²] or σ² = h²[(1/N) Σfᵢuᵢ² - ((1/N) Σfᵢuᵢ)²] where uᵢ = (xᵢ - A)/h
• Coefficient of Variation (C.V.): C.V. = (σ / |x̄|) * 100. A series with a higher C.V. is more variable.

How to Calculate Standard Deviation for Grouped Data (Continuous)

1. —
2. —
3. —
4. —
5. —
6. —
7. —
8. —
9. —

Mean Deviation vs. Standard Deviation

Worked Example

• Calculating Variance and Standard Deviation (Ungrouped Data) Find the variance and standard deviation for the data: 6, 8, 10, 12, 14. 1. Find the mean (x̄): x̄ = (6+8+10+12+14) / 5 = 50 / 5 = 10 2. Calculate squared deviations (xᵢ - x̄)²: (6-10)² = (-4)² = 16 (8-10)² = (-2)² = 4 (10-10)² = 0² = 0 (12-10)² = 2² = 4 (14-10)² = 4² = 16 3. Find the sum of squared deviations: Σ(xᵢ - x̄)² = 16 + 4 + 0 + 4 + 16 = 40 4. Calculate Variance (σ²): σ² = (1/n) Σ(xᵢ - x̄)² = (1/5) 40 = 8 5. Calculate Standard Deviation (σ): σ = √Variance = √8 = 2√2 ≈ 2.828 Answer: The variance is 8 and the standard deviation is approx. 2.828.

Exam Traps & Scoring Tips

Board Exam Traps to Avoid:

• Forgetting the Square Root: A very common mistake is to calculate the variance (σ²) and forget to take the square root to find the standard deviation (σ). Always double-check the question to see what is asked.
• Grouped vs. Ungrouped Formula: Using the formula for ungrouped data on a grouped data problem (or vice-versa) will lead to a completely wrong answer. Identify the data type first.
• Calculation Errors: The calculations, especially with the shortcut method, involve many steps. Be very careful with signs and squaring. A small error in calculating Σfᵢxᵢ or Σfᵢxᵢ² will cascade. Re-check your totals.
• Mean vs. Median for Mean Deviation: If the question asks for Mean Deviation, read carefully whether it's 'about the mean' or 'about the median'. The calculation process is different for each.
• Units: Standard Deviation (σ) has the same units as the original data, while Variance (σ²) has the units squared. While not always asked, being aware of this shows conceptual clarity.

Practice Questions with Solutions

• Q: What is the relationship between Standard Deviation (σ) and Variance (σ²)? A: Standard Deviation is the positive square root of the Variance. σ = √σ².
• Q: If the variance of a data set is 49, what is its standard deviation? A: The standard deviation is √49 = 7.
• Q: Why is the Coefficient of Variation (C.V.) used? A: It is a relative measure of dispersion used to compare the variability of two different series, especially when their means or units are different.
• Q: What happens to the standard deviation if each observation in a data set is increased by a constant, say 5? A: The standard deviation remains unchanged. It is not affected by a change of origin (addition or subtraction).

Frequently Asked Questions on Statistics

Frequently Asked Questions