Measures of Central Tendency
A measure of central tendency is a single value that represents the centre of a data set. It gives us a typical or representative value around which the data values cluster.
The Three Measures
1. Arithmetic Mean (AM)
2. Median
3. Mode
Arithmetic Mean
Simple Arithmetic Mean: Sum of all values divided by the number of values.
Mean (X-bar) = Sum(X) / N
Mean for Grouped Data (Direct Method):
Mean = Sum(f × m) / Sum(f)
where f = class frequency, m = class midpoint, N = total frequency.
Short-cut Method: Mean = A + (Sum(f × d) / N), where d = m - A (deviation from assumed mean A).
Step-deviation Method: Mean = A + (Sum(f × d') / N) × h, where d' = (m - A) / h, h = class width.
Weighted Mean: Used when different values have different importance (weights).
Weighted Mean = Sum(W × X) / Sum(W)
- Properties of Mean:
- Sum of deviations from mean = 0
- Mean is affected by extreme values.
- Based on all observations.
Median
Median is the middle value when data is arranged in ascending or descending order.
- For ungrouped data:
- If N is odd: Median = value of ((N+1)/2)th item.
- If N is even: Median = average of (N/2)th and ((N/2)+1)th items.
For grouped data:
Median = L + ((N/2 - cf) / f) × h
where L = lower class limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.
Median is NOT affected by extreme values. It is a positional measure.
Mode
Mode is the value that occurs most frequently in the data.
For grouped data:
Mode = L + (f1 - f0) / (2f1 - f0 - f2) × h
where L = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of class before modal class, f2 = frequency of class after modal class, h = class width.
Empirical Relationship: Mode = 3 Median - 2 Mean (approximately, for moderately skewed data)
Worked Examples
Find the mean of 10, 20, 30, 40, 50.
Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30
Find the median of 7, 3, 5, 1, 9.
Arrange: 1, 3, 5, 7, 9. N = 5 (odd). Median = ((5+1)/2)th = 3rd item = 5
Find the mode of 4, 7, 7, 9, 7, 4, 2.
7 appears 3 times (most frequently). Mode = 7
For grouped data with modal class 30-40 (f1=12), previous class f0=8, next class f2=7, h=10, L=30:
Mode = 30 + (12 - 8) / (2×12 - 8 - 7) × 10 = 30 + 4/9 × 10 = 30 + 4.44 = 34.44
For grouped data with N=40, median class 20-30 (f=16), cf before median class=12, h=10, L=20:
Median = 20 + (20 - 12)/16 × 10 = 20 + 8/16 × 10 = 20 + 5 = 25
If Mean = 30 and Median = 28, find Mode using the empirical relation.
Mode = 3 Median - 2 Mean = 3(28) - 2(30) = 84 - 60 = 24
A weighted mean problem: Marks in three subjects are 60, 75, 85 with weights 2, 3, 5 respectively.
Weighted Mean = (2×60 + 3×75 + 5×85) / (2+3+5) = (120 + 225 + 425) / 10 = 770/10 = 77
Common mistakes
Common mistakes
Forgetting to arrange data in order before finding the median. Also, many students apply the mode formula to find the most common class rather than computing it correctly. Remember: median formula requires identifying the correct median class where cumulative frequency first reaches or exceeds N/2.
Summary
Mean uses all values and is best for symmetric data but is distorted by outliers. Median is the middle value, unaffected by extremes — ideal for skewed data or income data. Mode is the most common value, useful for categorical data. The empirical relation Mode ≈ 3 Median - 2 Mean links all three measures.