Statistics
1. Why Dispersion? - Beyond Average
Average tells you middle, but not spread. Two classes both average 70 marks. Class A has 68,70,72 - all close. Class B has 40,70,100 - very spread. Same mean, different story. Dispersion measures how far data is from average. If dispersion small, data consistent. If big, data scattered. That is why we need range, mean deviation, variance, standard deviation.
Measures of central tendency (mean) not enough. Need dispersion = spread from centre. Same mean can have different spread. Dispersion shows consistency. Low dispersion = consistent, high = scattered.
• Mean tells middle, not spread
• Example: Class A 68,70,72 vs Class B 40,70,100 both mean 70
• Dispersion = how far data from average
• Low dispersion = consistent, High = scattered
• Need: Range, Mean Deviation, Variance, SD
2. Range - Quickest Idea
Range is simplest. Max minus Min. Example marks: 40,55,70,85,100 → Range = 100-40=60. It tells total spread, but uses only two values, ignores middle data. If one student gets 0 by mistake, range becomes big even if rest close. So range is quick but not reliable.
Range = Max - Min. Simplest measure, uses only extremes. Affected by outliers. Range = L - S. Quick idea of total spread but not robust.
• Formula: Range = Maximum - Minimum
• Example: 40 to 100 → Range 60
• Pros: Very quick to calculate
• Cons: Uses only 2 values, ignores middle, affected by outlier
• Use: Rough idea of spread
3. Mean Deviation - Average Distance from Centre
Mean deviation asks: on average, how far is each point from mean or median? Take absolute value, because distance cannot be negative. Formula: average of |x - mean|. Example data 2,4,6. Mean 4. Deviations |2-4|=2, |4-4|=0, |6-4|=2. Sum 4, divide by 3 → 1.33. So on average, points are 1.33 away from mean.
Mean Deviation = (1/n) Σ|x_i - A| where A=mean or median, | | absolute, ignore sign. Average absolute distance from centre. Smaller MD = data closer to centre.
• Definition: Average of absolute deviations from mean or median
• Formula: MD = (1/n) Σ|x_i - A|, A=mean/median
• Absolute: | | makes all positive, direction ignored
• Meaning: Average distance from centre
• Small MD = close to centre, consistent
4. Mean Deviation About Mean
When A is mean. For discrete data without frequency: MD(mean)= Σ|x - mean| / n. For frequency data: MD = Σ f|x - mean| / Σ f. First find mean, then deviation, then multiply by frequency, then average. Example x:5(f2),7(f3),9(f4),10(f1). Mean =77/10=7.7. Deviations: 2.7,0.7,1.3,2.3. f|x-mean|:5.4+2.1+5.2+2.3=15. MD=15/10=1.5. So average distance from mean 1.5.
MD about mean: Ungrouped MD(mean)=Σ|x_i - x̄|/n. Grouped discrete MD=Σ f_i|x_i - x̄|/Σ f_i. Continuous use mid values. First find mean, then |x-x̄|, then weighted average.
• Ungrouped: MD = Σ|x_i - x̄| / n
• Grouped: MD = Σ f|x - x̄| / Σ f
• Steps: Find mean → |x-mean| → f×|dev| → divide by Σf
• Example: Mean 7.7 MD 1.5 → average 1.5 away from mean
• Use: When mean is central tendency
5. Mean Deviation About Median
Median is middle value after sorting. For odd n, middle. For even, average of two middle. Then same process: |x - median| average. Why median? When data has outlier, median better than mean. Example 1,2,3,4,100. Mean 22, median 3. Mean deviation about mean will be large due to 100, but about median shows typical deviation better. So if outlier present, use median.
MD about median: M = median. Ungrouped MD(M)=Σ|x_i - M|/n. Grouped MD(M)=Σ f|x_i - M|/Σ f. Median = middle value. For odd (n+1)/2 th, even average of n/2 and n/2+1. Better when outliers present as median not affected.
• Median M = middle after sorting
• Odd: (n+1)/2 th value, Even: avg of n/2 and n/2+1
• MD(M)=Σ|x_i - M|/n or Σ f|x-M|/Σ f
• When to use: Outlier present, median better than mean
• Example: 1,2,3,4,100 median 3 better than mean 22
6. Variance and Standard Deviation - The Main Players
Mean deviation uses absolute value, which is not easy in calculus. So we square deviations. Variance = average of squared deviations from mean. σ² = Σ(x - mean)² / n. Standard deviation σ = √variance. SD is in same unit as data, variance in square unit. Example data 2,4,4,4,5,5,7,9. Mean 5. Squared deviations:9,1,1,1,0,0,4,16 sum 32 divide by 8 → variance 4, SD 2. SD 2 means typical distance about 2 from mean.
Variance σ² = (1/n) Σ(x_i - x̄)² or Σ f(x_i - x̄)²/Σ f. Standard Deviation σ = √σ². Measures spread. Low σ = data clustered near mean, high σ = dispersed. Unit: variance square unit, SD same unit as data. Population formula /n, sample uses /(n-1) but in Class 11 use /n.
• Variance σ² = average of squared deviations
• Ungrouped: σ² = Σ(x_i - x̄)² / n
• Grouped: σ² = Σ f(x_i - x̄)² / Σ f
• SD σ = √variance, same unit as data
• Meaning: Low σ = clustered, High σ = spread
• Example: Data mean 5 variance 4 SD 2
7. Short Cut Method for SD
Finding mean then deviation squares is long if numbers big. Short cut: Assume mean A. Let d = x - A. Then variance = [Σd² / n] - (Σd / n)². Even better step deviation: If data has common factor h, let u=(x-A)/h, then σ = h × √[Σu²/n - (Σu/n)²]. This saves time in exams. Example x:10,20,30,40 mean 25, let A=30 d:-20,-10,0,10 Σd=-20 Σd²=600 variance =600/4 - (-5)²=150-25=125 SD=√125.
Direct formula long. Short cut: Let d_i = x_i - A (assumed mean). Then σ² = Σd_i²/n - (Σd_i/n)². Step deviation if h common: u_i=(x_i - A)/h, σ² = h²[Σu_i²/n - (Σu_i/n)²]. For grouped: σ² = h²[Σf u²/Σf - (Σf u/Σf)²]. Saves calculation.
• Assumed mean A: d=x-A
• Variance = Σd²/n - (Σd/n)²
• Step deviation: u=(x-A)/h, σ²=h²[Σu²/n - (Σu/n)²]
• Grouped: Replace Σ with Σf
• Use: When numbers big, choose A middle value, h common factor
• Saves time in exam
8. Coefficient of Variation - Compare Two Series
SD alone cannot compare two groups with different means. Example: Factory A average salary 5000 SD 500, Factory B average 10000 SD 600. Which has more variation? SD says B bigger (600>500) but relative to mean, A varies 10% (500/5000), B varies 6% (600/10000). So A more variable. Coefficient of Variation CV = (SD/Mean)×100 %. Lower CV = more consistent, higher CV = more variable.
CV = (σ / x̄) × 100 %. Relative measure, unitless %. Used to compare variability of two series with different means or units. Lower CV = more consistent, stable. Higher CV = more variable. SD compares absolute spread, CV compares relative spread.
• Formula: CV = (σ / x̄) ×100 %
• Why: SD alone fails when means differ
• Example: A mean 5000 SD500 CV10%, B mean10000 SD600 CV6% → A more variable
• Lower CV = more consistent, Higher CV = more variable
• Use: Compare two groups, stock risk, factory consistency
9. Real Life Usage - Where Stats Lives Around You
Statistics is everywhere. School: Teacher sees SD of marks - low SD means all students similar level, high SD means some top some weak. Factory: Quality control uses SD to check if products same size. Stock market: Higher SD means more risky stock. Weather: Mean deviation of temperature shows climate consistency. Sports: Cricketer average 50 but SD high means sometimes 0 sometimes 100, not consistent. CV used to compare two batsmen with different averages.
Real uses: Education consistency via SD, factory quality control tolerance via SD, finance risk via SD and CV, weather climate variability, sports consistency of player via CV, medical lab test variation, agriculture yield consistency.
• School: SD of marks → consistency of class
• Factory: SD of product size → quality control
• Finance: SD of stock returns → risk, CV to compare stocks
• Weather: SD of temperature → climate variability
• Sports: CV to compare batsmen consistency despite different averages
• Daily: Delivery time variation, phone battery life consistency
• Why dispersion: Mean same but spread different → need dispersion
• Range: Max-Min, quick but uses only extremes
• Mean Deviation: MD = Σ|x-A|/n or Σf|x-A|/Σf, A=mean/median, absolute distance
• MD about Mean: A=x̄, about Median: A=M (better with outlier)
• Variance: σ² = Σ(x-x̄)²/n or Σf(x-x̄)²/Σf, average squared deviation
• SD: σ = √variance, same unit as data, low=clustered high=spread
• Short cut: d=x-A, σ²=Σd²/n-(Σd/n)², step u=(x-A)/h, σ²=h²[Σu²/n-(Σu/n)²]
• CV: (σ/x̄)×100%, relative measure, lower CV = more consistent
• Real: School consistency, factory quality, stock risk, weather, sports via CV