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Statistics (Mean, Median, Mode)

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Quantitative Aptitude - Statistics (Mean, Median, Mode)

Fundamentals First

Statistics involves the collection, analysis, and interpretation of data. Mean, Median, and Mode are the three primary measures of the 'central tendency' or 'average' of a data set.

  • Mean (Arithmetic Average): The most common measure of average. It is the sum of all values divided by the number of values.
  • Median: The middle value of a dataset that has been sorted in order of magnitude. It is a positional average.
  • Mode: The value that appears most frequently in a dataset. It is the most popular value.

Key Formulas & Concepts

  • Mean: \( \text{Mean} = \frac{\sum x}{n} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \)
  • Median:
    1. Arrange the data in ascending or descending order.
    2. If the number of observations (n) is odd, the Median is the \( (\frac{n+1}{2})^{th} \) term.
    3. If 'n' is even, the Median is the average of the \( (\frac{n}{2})^{th} \) and \( (\frac{n}{2} + 1)^{th} \) terms.
  • Mode: Find the number that repeats the maximum number of times.
  • Empirical Relationship (Important!): For a moderately skewed distribution, the relationship between the three is given by: Mode ≈ 3(Median) - 2(Mean).

⚡ Quick Solving Tips

  • Sorting is a must for Median: The most common mistake is forgetting to sort the data before finding the median.
  • Effect of Outliers: The Mean is very sensitive to extremely high or low values (outliers). The Median is much more resistant to outliers. The Mode is not affected by outliers at all.
  • Multiple Modes: A dataset can have more than one mode (bimodal, trimodal) or no mode at all if all values occur with the same frequency.

✍️ Suggestions for Examinations

  • The empirical formula (Mode ≈ 3 Median - 2 Mean) is a high-yield formula. It is frequently asked directly or as a way to find one measure when the other two are given.
  • Understand the concept behind each measure. Questions might ask 'Which measure is best for...' (e.g., for house prices, median is better than mean because of outlier mansions).
  • Double-check your counting for 'n' and be careful with the even/odd median calculation.

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