📊 Statistics (Mean, Median, Mode, Variance, SD)

Day 4 | ⏱️ Part of 2.5-Hour Stats & DI Block | TCS Interview Prep

📌 Subtopics

• Mean: Arithmetic mean, weighted mean
• Median: Middle value, for odd/even count
• Mode: Most frequently occurring value
• Variance: Measure of data spread
• Standard Deviation: Square root of variance

📖 Comprehensive Notes

1. Mean (Arithmetic & Weighted)

The mean is the average of a set of numbers.

• Arithmetic Mean: $\text{Mean } (\bar{x}) = \frac{\sum x_i}{n}$
• Weighted Mean: Used when data values have different weights/frequencies: $\text{Weighted Mean} = \frac{\sum w_i x_i}{\sum w_i}$

Example: Find the mean of $5, 8, 12, 15,$ and $20$. $\text{Mean} = \frac{5 + 8 + 12 + 15 + 20}{5} = \frac{60}{5} = 12$.

2. Median

The median is the middle value in a sorted, ascending or descending list of numbers.

• Steps to Calculate:
  1. Arrange the data in ascending order.
  2. If the number of terms ($n$) is odd, the median is the $\left(\frac{n+1}{2}\right)$-th term.
  3. If the number of terms ($n$) is even, the median is the average of the $\left(\frac{n}{2}\right)$-th and $\left(\frac{n}{2} + 1\right)$-th terms.

Example: Find the median of $7, 2, 15, 9, 5, 11$. Sorted order: $2, 5, 7, 9, 11, 15$ ($n=6$, even). Median $= \text{Average of 3rd and 4th terms} = \frac{7 + 9}{2} = 8$.

3. Mode

The mode is the value that appears most frequently in a data set.

• A dataset can be unimodal (one mode), bimodal (two modes), multimodal (more than two modes), or have no mode (if all values appear once).
• Empirical Relationship: For moderately skewed distributions: $\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$

Example: Find the mode of the dataset: $3, 5, 3, 8, 5, 3, 9$. Frequencies: $3$ appears three times, $5$ appears twice, others once. $\text{Mode} = 3$.

4. Variance

Variance measures how far a set of numbers is spread out from their average value.

• Population Variance ($\sigma^2$): $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$
• Shortcut formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$

Example: Find the variance of the numbers $2, 4, 6$. $\text{Mean } (\bar{x}) = \frac{2 + 4 + 6}{3} = 4$. Deviations from mean: $(2-4)=-2$, $(4-4)=0$, $(6-4)=2$. Squared deviations: $4, 0, 4$. $\text{Variance } (\sigma^2) = \frac{4 + 0 + 4}{3} = \frac{8}{3} \approx 2.67$.

5. Standard Deviation

Standard Deviation (SD) is the square root of the variance. It is expressed in the same units as the original data, making it easier to interpret. $\text{Standard Deviation } (\sigma) = \sqrt{\text{Variance}}$

• A low standard deviation indicates that the data points tend to be close to the mean.
• A high standard deviation indicates that the data points are spread out over a wider range.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

MEAN:
  Mean = Σxᵢ / n

MEDIAN:
  Sort the data first
  Odd n:  Median = (n+1)/2 th term
  Even n: Median = average of (n/2)th and (n/2+1)th terms

MODE:
  Most frequent value (a data set can have no mode, one mode, or multiple modes)

VARIANCE:
  σ² = Σ(xᵢ - x̄)² / n          [Population]
  s² = Σ(xᵢ - x̄)² / (n-1)      [Sample]

STANDARD DEVIATION:
  σ = √(Variance)

Shortcut for Variance:
  σ² = (Σxᵢ²/n) - (Σxᵢ/n)²

🔗 Related Topics

• Averages — Mean is identical to the arithmetic average
• Data Interpretation — Stats formulas are heavily used in interpreting datasets
• Permutations Combinations and Probability — Probability distributions and stats
• Ages — Finding average, median, or standard deviation of ages of a group
• Ratios and Proportion — Distribution and frequency analysis