Statistics
Class 11 Maths • Chapter 13 • Comprehensive Interactive Notes
1. Measures of Dispersion
Dispersion measures how "spread out" the data is. The mean tells us the center, but
dispersion tells us about the variation.
- Range: Max Value - Min Value.
- Mean Deviation: Average of absolute differences from Mean/Median.
- Variance (\( \sigma^2 \)): Average of squared differences from Mean.
- Standard Deviation (\( \sigma \)): Square root of Variance.
Dispersion Dashboard
Enter numbers separated by commas (e.g., 6, 7, 10, 12, 13, 4, 8, 12)
Mean (\(\bar{x}\)): -
Range: -
Mean Dev (\(\bar{x}\)): -
Variance (\(\sigma^2\)): -
Std Dev (\(\sigma\)): -
Deviation Visualizer
Green Line = Mean, Red Lines =
Deviations
📌 Why Standard Deviation is the Best Measure
- ✔ Based on all observations
- ✔ Algebraically manageable
- ✔ Less affected by extreme values
- ✔ Used in advanced statistics
CBSE Favourite:
“Why is standard deviation better than mean deviation?”
2. Formula Matrix
| Measure |
Formula |
| Mean Deviation (\(\bar{x}\)) |
\( \frac{\sum |x_i - \bar{x}|}{n} \) |
| Mean Deviation (M) |
\( \frac{\sum |x_i - M|}{n} \) |
| Variance (\(\sigma^2\)) |
\( \frac{\sum (x_i - \bar{x})^2}{n} \) |
| Measure |
Formula |
| Mean Deviation (\(\bar{x}\)) |
\( \frac{\sum f_i |x_i - \bar{x}|}{N} \) |
| Variance (\(\sigma^2\)) |
\( \frac{1}{N^2}[N \sum f_i x_i^2 - (\sum f_i x_i)^2] \) |
| Standard Deviation |
\( \sqrt{\text{Variance}} \) |
🧮 Step-wise Method: Standard Deviation
Steps for Ungrouped Data:
- Find mean \( \bar{x} \)
- Find deviations \( x_i - \bar{x} \)
- Square deviations
- Find average of squared deviations
- Take square root
Formula:
\[
\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}
\]
CBSE Tip:
Always show steps — even if calculation is simple.
3. Analysis of Frequency Distributions
To compare the variability of two series with different units, we use the Coefficient of
Variation (C.V.).
\( C.V. = \frac{\sigma}{\bar{x}} \times 100 \)
Rule: Lower C.V. = More Consistent.
Consistency Checker
Compare two datasets (e.g., Players A & B).
Concept Mastery Quiz
1. Which measure of dispersion is based on squared deviations?
2. If Standard Deviation is 4, Variance is:
3. A lower Coefficient of Variation indicates:
4. Mean deviation can be calculated about:
5. The variance of 5, 5, 5, 5, 5 is:
🧠 One-Page Revision Checklist
- ✔ Meaning of dispersion understood
- ✔ Formula of range remembered
- ✔ Mean deviation (mean & median) revised
- ✔ Variance & standard deviation formulas memorised
- ✔ Difference between MD and SD clear
- ✔ CV formula revised
- ✔ Rule: lower CV → more consistency
Self-Test:
- ❓ Why SD is preferred over MD?
- ❓ If SD = 6, what is variance?
- ❓ Which series is more consistent?