Statistics

Statistics is the branch of mathematics that deals with collection, analysis, interpretation, and presentation of data.

In aptitude exams, statistics questions usually revolve around:

  • Central Tendency: Mean, Median, Mode
  • Dispersion: Range, Variance, Standard Deviation

Think of statistics as tools to summarize and compare datasets effectively.

2. Key Formulas & Shortcuts

A. Mean (Average)

1. Arithmetic Mean (AM):

Mean=Sum of observationsNumber of observations\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}

2. Weighted Mean:

If observations have weights:

Weighted Mean=∑wixi∑wi\text{Weighted Mean} = \frac{\sum w_i x_i}{\sum w_i}

Where:

  • xix_i = value
  • wiw_i = corresponding weight

B. Median

The middle value when data is arranged in ascending or descending order.

  • For odd number of values:
Median=Middle value\text{Median} = \text{Middle value}
  • For even number of values:
Median=(n/2)th term + (n/2 + 1)th term2\text{Median} = \frac{\text{(n/2)th term + (n/2 + 1)th term}}{2}

C. Mode

The most frequently occurring value in the data set.

If no value repeats → No mode.
If multiple values have highest frequency → Multimodal.

D. Range

Range=Maximum value−Minimum value\text{Range} = \text{Maximum value} - \text{Minimum value}

E. Variance and Standard Deviation

1. Variance (σ2\sigma^2):

Measures how much the data points deviate from the mean.

σ2=∑(xi−xˉ)2n\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}

2. Standard Deviation (σ\sigma):

σ=Variance\sigma = \sqrt{\text{Variance}}

For aptitude, it's rarely asked in full derivation — mostly conceptually.

3. Conceptual Tips & Common Mistakes

  • Don’t confuse mean with median. Median is positional; mean is arithmetic.

  • In skewed distributions, mean ≠ median ≠ mode.

  • For grouped data, use assumed mean method for speed.

  • Weighted mean often appears in mixture or average speed problems in disguise.

  • Know when no mode or multiple modes occur.

  • Median is not affected by extreme values, mean is.

  • Understand the difference:

    • Mean: balancing point
    • Median: central point
    • Mode: most common value

4. Visual Explanation

A. Mean, Median, Mode (Symmetric vs Skewed)

Symmetric Distribution:

Mean=Median=Mode\text{Mean} = \text{Median} = \text{Mode}

Right-Skewed (Positive Skew):

Mode<Median<Mean\text{Mode} < \text{Median} < \text{Mean}

Left-Skewed (Negative Skew):

Mean<Median<Mode\text{Mean} < \text{Median} < \text{Mode}

B. Data Distribution Visual

Dataset: 2, 3, 4, 4, 4, 5, 6

Mean = 4  
Median = 4  
Mode = 4

Change to 2, 3, 4, 4, 4, 5, 100 →
Mean shifts right (to ~23), Median still 4, Mode still 4.

5. Solved Examples

Example 1: Mean (Average)

Q: The average age of 5 students is 20. Find their total age.

A:

Total=20×5=100\text{Total} = 20 \times 5 = 100

Example 2: Weighted Mean

Q: A student scores 60 in Maths (weight 4), 70 in English (weight 3), and 80 in Science (weight 3). Find weighted average.

Weighted Mean=60⋅4+70⋅3+80⋅34+3+3=240+210+24010=69\text{Weighted Mean} = \frac{60 \cdot 4 + 70 \cdot 3 + 80 \cdot 3}{4 + 3 + 3} = \frac{240 + 210 + 240}{10} = 69

Example 3: Median (Odd Count)

Q: Find the median of 3, 5, 7, 9, 11

A:
Middle value = 7

Example 4: Median (Even Count)

Q: Find the median of 10, 12, 15, 18

Median=12+152=13.5\text{Median} = \frac{12 + 15}{2} = 13.5

Example 5: Mode

Q: Find the mode of: 2, 4, 4, 4, 5, 5, 7

A:
Mode = 4 (highest frequency)

Example 6: Range

Q: Data = 10, 15, 20, 25, 40

Range=40−10=30\text{Range} = 40 - 10 = 30
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