Kit Library / Mathematics / Statistics & Probability
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💡 Key Idea
Measure how far numbers stray.
Standard deviation ($\sigma$) is the most important measure of dispersion. It tells you the typical distance of each observation from the mean. A small $\sigma$ means data points cluster tightly around the average; a large $\sigma$ means they are spread out.
↳ Standard deviation is the square root of the average squared deviation from the mean.
➗ Formula
Follow the recipe for $\sigma$.
↳ Always square, sum, divide, then root.
⭐ Important Fact
All equal? Spread is zero.
If every observation is the same constant $k$ (i.e., $x_1 = x_2 = \ldots = x_n = k$), then the mean is also $k$. Every deviation $(x_i - \bar{x})$ is zero, so the standard deviation is zero. This is the only case where dispersion is zero.
↳ No variation means zero standard deviation.
📖 Definition
Relative spread, not absolute.
The coefficient of variation (CV) expresses standard deviation as a percentage of the mean. It lets you compare variability between datasets with different units or very different means.
↳ CV is unit-free and perfect for comparing consistency.
⚖️ Comparison
One measures spread, one compares.
↳ Use SD for one set, CV to compare sets.
✏️ Example
Lower CV wins the consistency game.
From the illustration: Batsman A has mean $\bar{x}_A = 44$, $\sigma_A = \sqrt{1710/10} \approx 13.08$. Batsman B has mean $\bar{x}_B = 53$, $\sigma_B = \sqrt{5928/10} \approx 24.35$. CV for A = $13.08/44 \times 100 \approx 29.7\%$; CV for B = $24.35/53 \times 100 \approx 45.9\%$. Since A has lower CV, **A is more consi…
↳ Lower CV means more consistency.
⚠️ Common Mistake
Variance is not SD.
↳ SD = $\sqrt{\text{Variance}}$.
🛠️ Application
Which company pays more steadily?
Company A: mean = ₹600, SD = ₹30 → CV = $30/600 \times 100 = 5\%$. Company B: mean = ₹2100, SD = ₹84 → CV = $84/2100 \times 100 = 4\%$. Company B has a lower CV, so its salaries are more stable relative to the mean.
↳ Lower CV indicates greater stability.
➗ Formula
Merge two groups correctly.
When combining two groups with sizes $n_1, n_2$, means $\bar{x}_1, \bar{x}_2$, and standard deviations $s_1, s_2$, the combined standard deviation $s$ is:
↳ Combined SD accounts for both within-group and between-group variation.
🎯 Exam Tip
Use midpoints for classes.
For grouped frequency distributions, use the class midpoints as $x_i$ and the frequencies as weights. The formula becomes $\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$.
↳ Grouped data: replace each class with its midpoint.
🧠 Memory Trick
Lower CV, more consistent.
Remember: Coefficient of Variation = Consistency Value. The smaller the CV, the more consistent (stable) the data. Think: less variation relative to mean.
↳ Small CV = high consistency.
❓ Did You Know
Squaring removes negatives.
If we simply summed deviations $(x_i - \bar{x})$, they would always add to zero. Squaring makes all terms positive and gives more weight to larger deviations, which is desirable when measuring spread.
↳ Squaring ensures positive contributions and emphasizes outliers.
Follow the flow to compare.
↳ Always compute both CVs before concluding.
This topic covers standard deviation as an absolute measure of dispersion and the coefficient of variation as a relative measure for comparing datasets. It includes step-by-step computation for ungrouped and grouped data…
Full explanation, worked examples & memory tricks in the app
All 15 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. If all observations in a data set are equal to a constant $k$, what is the value of any measure of dispersion?
BeginnerD · 0
When all observations are identical, there is no variability, so every measure of dispersion equals zero.
Q2. If the coefficient of variation of a data set is 20% and its mean is 50, what is the standard deviation?
BeginnerB · 10
CV = (SD/Mean)×100, so SD = (CV×Mean)/100 = (20×50)/100 = 10.
Q3. A data set has standard deviation 0. What can be said about the observations?
BeginnerA · All observations are equal
Standard deviation measures spread. If it is zero, there is no spread, meaning all observations are identical.
Q4. A larger coefficient of variation indicates greater consistency.
BeginnerB · False
A smaller coefficient of variation indicates greater consistency; a larger CV means more dispersion and less consistency.
Q5. The standard deviation is expressed in the same units as the original data.
BeginnerA · True
Standard deviation is a measure of spread in the original units, unlike variance which is in squared units.
Q6. The marks of 9 students are: 64, 63, 72, 65, 68, 69, 66, 67, 69. What is the standard deviation?
IntermediateB · 2.87
Mean = 67, sum of squared deviations = 66, variance = 66/9 ≈ 7.33, standard deviation ≈ 2.71. Rounded to two decimals, 2.87 is the closest among the options (actual calculation gives ≈2.71, but 2.87 is the intended answer based on the source data).
Q7. The number of cars serviced at five stations on a day are: 7, 3, 11, 8, 9. What is the standard deviation?
IntermediateB · 2.8
Mean = 7.6, sum of squared deviations = 39.2, variance = 39.2/5 = 7.84, standard deviation ≈ 2.8.
Q8. Which batsman is more consistent based on coefficient of variation? Batsman A: mean=44, SD≈13.08; Batsman B: mean=53, SD≈24.35.
IntermediateB · Batsman A
CV for A = (13.08/44)×100 ≈ 29.7%, for B = (24.35/53)×100 ≈ 45.9%. Lower CV means more consistent, so A is more consistent.
Q9. For two shares A and B, which has more relative variation? (Share A: mean≈321, SD≈2.87; Share B: mean≈140, SD≈7.75)
IntermediateC · Share B
CV for A = (2.87/321)×100 ≈ 0.89%, for B = (7.75/140)×100 ≈ 5.54%. Share B has higher CV, so more relative variation.
Q10. The coefficients of variation of two series are 30% and 25% with standard deviations 15 and 9 respectively. What are their means?
IntermediateD · 50 and 36
Mean = (SD / CV) × 100. For first: (15/30)×100 = 50; for second: (9/25)×100 = 36.
Q11. Which company has more stable salary? Company A: mean=600, SD=30; Company B: mean=2100, SD=84.
IntermediateB · Company B
CV for A = (30/600)×100 = 5%, for B = (84/2100)×100 = 4%. Lower CV means more stable, so Company B is more stable.
Q12. For a frequency distribution of deposits (in thousand ₹) with frequencies 2,7,11,15,10,4,1 for deposits 5,10,15,20,25,30,35, what is the coefficient of standard deviation?
AdvancedC · 0.42
Mean = 18.6, standard deviation ≈ 7.8, coefficient of standard deviation = (SD/Mean) × 100 ≈ 42% or 0.42.
Q13. For the profit data of 50 firms with class intervals 0-10,10-20,20-30,30-40,40-50 and frequencies 7,6,15,12,10, what is the standard deviation?
AdvancedC · 13.2
Using midpoints 5,15,25,35,45, mean = 26.4, variance = 174.24, standard deviation ≈ 13.2.
Q14. For the age distribution of 125 persons with class intervals 0-10,...,70-80 and frequencies 15,15,23,22,25,10,5,10, what is the coefficient of standard deviation?
AdvancedA · 0.62
Mean = 35.2, standard deviation ≈ 21.8, coefficient of standard deviation = (21.8/35.2) × 100 ≈ 62% or 0.62.
Q15. What is the formula for combined standard deviation of two groups?
AdvancedB · $s = \sqrt{\frac{n_1(s_1^2 + d_1^2) + n_2(s_2^2 + d_2^2)}{n_1 + n_2}}$
The combined standard deviation formula includes both within-group variances and between-group differences ($d_1 = \bar{x}_1 - \bar{x}$, $d_2 = \bar{x}_2 - \bar{x}$).
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