Central tendency measures; wage distribution

Measures of central tendency

A measure of central tendency is a single value that represents the centre of a data set. The main measures are the arithmetic mean, median, mode, geometric mean and harmonic mean.

Favourable points

• Arithmetic mean: easy to understand; uses every observation; suitable for further algebraic work.
• Median: not affected by extreme values; can be found even with open-ended classes; good for skewed data.
• Mode: the most typical value; easy to locate; usable for qualitative data.
• Geometric mean: suitable for ratios, rates of growth and index numbers.
• Harmonic mean: suitable for averaging rates such as speed.

Calculation

The first and last classes are open-ended, so the mean cannot be found. The approximate (positional) averages, namely the median, quartiles and mode, are used instead. They only need the middle classes.

Make the classes continuous (35-37 becomes 34.5-37.5, and so on):

N = 220, class width h = 3.

Median: N/2 = 110, so the median class is 37.5-40.5.

Median = 37.5 + (110 − 96)/99 × 3 = 37.5 + 0.424 = Rs 37.92

Mode: modal class 37.5-40.5 (f₁ = 99, f₀ = 82, f₂ = 18)

Mode = 37.5 + (99 − 82) / (2×99 − 82 − 18) × 3
     = 37.5 + 17/98 × 3
     = Rs 38.02

Quartiles:

Q1 = 34.5 + (55 − 14)/82 × 3 = Rs 36.00
Q3 = 37.5 + (165 − 96)/99 × 3 = Rs 39.59

Answer: median ≈ Rs 37.92, mode ≈ Rs 38.02, Q₁ = Rs 36.00 and Q₃ ≈ Rs 39.59 per week.