Skip to main content

Central tendency; mean, median, SD of marks

Short answer

Using midpoints: mean = Σfx / 100 = 52 marks. Median class 40–60, median = 40 + (18/27)×20 ≈ 53.33 marks. Standard deviation = √(636) ≈ 25.22 marks.

Describe the various measures of central tendency and its application. The following table represents the marks of 100 students.

Marks0-2020-4040-6060-8080-100
No. of students1418272615
Find the mean, median and standard deviation of all 100 students.

Measures of central tendency and their use

  • Arithmetic mean: the sum divided by the number of values. Used for general averages such as average marks or income, and in further statistical work.
  • Median: the middle value of the ordered data. Used for skewed data and open-ended classes (for example, median income).
  • Mode: the most frequent value. Used when the most common item matters, such as the most popular shoe size.
  • Geometric mean: used for growth rates, ratios and index numbers.
  • Harmonic mean: used for averages of rates such as speed over equal distances.

Calculation

Class width h = 20 and N = 100.

MarksMid xffxf(x − 52)²cf
0-20101414024,69614
20-4030185408,71232
40-605027135010859
60-80702618208,42485
80-1009015135021,660100
Total100520063,600

Mean

x̄ = Σfx / N = 5200 / 100 = 52

Median: N/2 = 50, class 40-60 (cf before = 32, f = 27)

Median = 40 + (50 − 32)/27 × 20 = 40 + 13.33 = 53.33

Standard deviation

σ² = Σf(x − x̄)² / N = 63,600 / 100 = 636
σ  = √636 = 25.22

(With divisor N − 1 the SD would be 25.35.)

Answer: mean = 52, median ≈ 53.33, standard deviation ≈ 25.22 marks.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 1 questions →