Skip to main content

Ideal central tendency; correct mean and SD

Short answer

Ideal measure: rigidly defined, easy to understand/calculate, based on all observations, not unduly affected by extremes. Corrected mean = 20.56, corrected SD = 5.76 (using n=16, wrong score 23→32).

Write any four characteristics of ideal measure of central tendency. For a group of 16 candidates, the mean and standard deviation were found to be 20 and 5 respectively. Later it is discovered that the score 32 was measured as 23. Find the correct mean and correct standard deviation.

Characteristics of an ideal measure of central tendency

  1. It should be rigidly defined, so that everyone gets the same value from the same data.
  2. It should be easy to understand and calculate.
  3. It should be based on all observations.
  4. It should be not unduly affected by extreme values (sampling stability).
  5. It should be suitable for further algebraic treatment.

Corrected mean and SD

Given n = 16, x̄ = 20, σ = 5. The wrong score 23 should be 32. The SD is taken with divisor n.

Σx  = 16 × 20 = 320
Σx² = 16 (σ² + x̄²) = 16 (25 + 400) = 6800

Corrected totals:

Σx  = 320 − 23 + 32 = 329
Σx² = 6800 − 23² + 32² = 6800 − 529 + 1024 = 7295

Corrected mean and SD:

Mean = 329 / 16 = 20.5625
σ²   = 7295/16 − (20.5625)² = 455.9375 − 422.8164 = 33.1211
σ    = √33.1211 = 5.755

Answer: correct mean ≈ 20.56 and correct SD ≈ 5.76.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 1 questions →