---
source: ncert
topic: statistics
class_level: class_11
chapter: ch15
difficulty: intermediate
---

Class 11 | Ch15: Statistics
MEASURES OF DISPERSION (how spread out data is):
Range = Maximum - Minimum.
MEAN DEVIATION:
Mean deviation about mean: MD(x_bar)=sum(|xi-x_bar|)/n.
Mean deviation about median: MD(M)=sum(|xi-M|)/n.
For grouped data: MD=sum(f|xi-x_bar|)/sum(f).
VARIANCE AND STANDARD DEVIATION:
Variance: sigma^2=sum((xi-x_bar)^2)/n (ungrouped).
For grouped: sigma^2=sum(f(xi-x_bar)^2)/sum(f).
Shortcut: sigma^2=(sum(xi^2)/n)-(x_bar)^2 = (sum(fi*xi^2)/sum(fi))-(x_bar)^2.
Standard Deviation sigma=sqrt(Variance).
COEFFICIENT OF VARIATION: CV=(sigma/x_bar)×100.
Used to compare variability of two distributions with different means.
Higher CV = more variable/inconsistent.
For comparing: data with higher CV is more variable.
ANALYSIS OF FREQUENCY DISTRIBUTIONS:
Two distributions compared by CV when means are different.
Two distributions compared directly by sigma when means are same.
SOLVED EXAMPLES:
Example 1: Find variance and SD of 6,8,10,12,14.
Mean=50/5=10.
Deviations from mean: -4,-2,0,2,4. Squared: 16,4,0,4,16.
Variance=40/5=8. SD=sqrt(8)=2sqrt(2)≈2.83.
Example 2: Two distributions: A has mean=25, SD=5. B has mean=10, SD=4.
CV(A)=(5/25)×100=20%. CV(B)=(4/10)×100=40%.
B is more variable (higher CV).
Example 3: Find MD about mean for 2,4,6,8,10.
Mean=6. |deviations|=4,2,0,2,4. MD=12/5=2.4.
COMMON MISTAKES:
Variance is always non-negative (it's sum of SQUARES divided by n).
SD=sqrt(Variance), NOT Variance=SD^2 (both are true, don't confuse direction).
CV is in PERCENTAGE: multiply by 100.