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Mean, Median, Mode & Standard Deviation

Short Notes

These are measures of central tendency and measures of dispersion that summarize data distributions.

  • Mean (μ): Average value — sensitive to outliers
  • Median: Middle value — robust to outliers
  • Mode: Most frequent value — can be multiple or none
  • Standard Deviation (σ): Measure of spread around the mean

Key Theories & Formulas

1. Mean (Arithmetic Mean)

For raw data: $\(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)$

For frequency distribution: $\(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)$

For grouped data (using class marks): $\(\bar{x} = \frac{\sum f_i m_i}{\sum f_i}\)$ where \(m_i\) = class mark

Properties:

  • \(\sum(x_i - \bar{x}) = 0\)
  • If every value increases by k, mean increases by k
  • If every value multiplied by k, mean multiplied by k

2. Median

For ungrouped data:

  • Arrange in order
  • If n is odd: Median = \(x_{(n+1)/2}\)
  • If n is even: Median = \(\frac{x_{n/2} + x_{n/2+1}}{2}\)

For grouped data: $\(Median = L + \frac{(N/2 - CF)}{f} \times h\)$

where:

  • L = lower boundary of median class
  • N = total frequency
  • CF = cumulative frequency before median class
  • f = frequency of median class
  • h = class width

3. Mode

For ungrouped data: Most frequently occurring value

For grouped data: $\(Mode = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\)$

where:

  • L = lower boundary of modal class
  • f₁ = frequency of modal class
  • f₀ = frequency of class before modal class
  • f₂ = frequency of class after modal class
  • h = class width

4. Relationship Between Mean, Median, Mode

For moderately skewed distributions: $\(Mode \approx 3 \times Median - 2 \times Mean\)$

Skewness Relationship
Symmetric Mean = Median = Mode
Right skewed Mode < Median < Mean
Left skewed Mean < Median < Mode

5. Variance and Standard Deviation

Population Variance: $\(\sigma^2 = \frac{\sum(x_i - \mu)^2}{N} = \frac{\sum x_i^2}{N} - \mu^2\)$

Sample Variance: $\(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\)$ (Bessel's correction)

Standard Deviation: \(\sigma = \sqrt{Var}\)

6. Other Measures of Dispersion

Measure Formula
Range max - min
Coefficient of Variation \(CV = \frac{\sigma}{\mu} \times 100\%\)
Interquartile Range (IQR) Q₃ - Q₁

7. Combined Mean & Variance

For two groups with means \(\bar{x}_1, \bar{x}_2\), sizes \(n_1, n_2\):

Combined Mean: $\(\bar{x}_{combined} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}\)$

Combined Variance: $\(\sigma_{combined}^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}\)$

where \(d_1 = \bar{x}_1 - \bar{x}_{combined}\), \(d_2 = \bar{x}_2 - \bar{x}_{combined}\)

Example Problems

Problem 1: Find mean, median, mode: 2, 3, 3, 4, 5, 5, 5, 6, 7

Solution:

  • Mean: (2+3+3+4+5+5+5+6+7)/9 = 40/9 ≈ 4.44
  • Median: 9 values → 5th value = 5
  • Mode: 5 appears most (3 times) → 5

Problem 2: Data: 10, 20, 30, 40, 50. Find variance and standard deviation.

Solution:

  1. Mean = 150/5 = 30
  2. \(\sum(x_i - 30)^2 = 400 + 100 + 0 + 100 + 400 = 1000\)
  3. Variance = 1000/5 = 200
  4. Standard Deviation = \(\sqrt{200}\) = 14.14

Problem 3: If the mean of 5 numbers is 8 and one number is removed, the mean becomes 7. Find the removed number.

Solution:

  1. Sum of 5 numbers = 5 × 8 = 40
  2. Sum of 4 numbers = 4 × 7 = 28
  3. Removed number = 40 - 28 = 12

Hardest GATE Questions

Topic: Combined Statistics

Question (GATE 2019 Style): Group A: 50 students, mean = 60, standard deviation = 8 Group B: 100 students, mean = 75, standard deviation = 10 Find the combined standard deviation.

Solution:

  1. Combined mean: \(\bar{x}_c = \frac{50(60) + 100(75)}{150} = \frac{10500}{150} = 70\)
  2. \(d_1 = 60 - 70 = -10\), \(d_2 = 75 - 70 = 5\)

  3. Combined variance: \(\sigma_c^2 = \frac{50(64 + 100) + 100(100 + 25)}{150} = \frac{50(164) + 100(125)}{150}\) \(= \frac{8200 + 12500}{150} = \frac{20700}{150} = 138\)

  4. Combined SD = \(\sqrt{138}\) ≈ 11.75

Topic: Effect of Linear Transformation

Question (GATE 2016 Variant): If the mean and variance of a dataset are 20 and 16, what are the mean and variance if each value is transformed by Y = 3X - 5?

Solution:

  1. E[Y] = E[3X - 5] = 3E[X] - 5 = 3(20) - 5 = 55
  2. Var(Y) = Var(3X - 5) = 9·Var(X) = 9(16) = 144
  3. New SD = 12

Topic: Coefficient of Variation

Question: Stock A has mean return 15% with SD 6%. Stock B has mean return 20% with SD 10%. Which stock is more consistent (less risky)?

Solution:

  1. CV(A) = 6/15 × 100 = 40%
  2. CV(B) = 10/20 × 100 = 50%
  3. Stock A is more consistent (lower CV)

Topic: Median from Grouped Data

Question: Find the median from: | Class | 0-10 | 10-20 | 20-30 | 30-40 | |-------|------|-------|-------|-------| | Freq | 5 | 15 | 20 | 10 |

Solution:

  1. N = 50, N/2 = 25
  2. Cumulative frequencies: 5, 20, 40, 50
  3. Median class: 20-30 (CF just exceeds 25)
  4. L = 20, CF = 20, f = 20, h = 10
  5. Median = 20 + \(\frac{25-20}{20}\) × 10 = 20 + 2.5 = 22.5

References