What is an average?
A measure of central tendency (an average) is a single value that represents a whole set of data. Most values crowd around it. The three common ones are the arithmetic mean, the median and the mode.
Words used: n or N = number of items, Σ (sigma) = "add all", x̄ = mean, f = frequency.
Arithmetic mean
Individual series
- Direct method: x̄ = Σx ÷ n.
- Assumed mean method: pick any value A, find d = x − A; x̄ = A + Σd ÷ n. Saves work with big numbers.
Discrete series
- Direct: x̄ = Σfx ÷ N.
- Assumed mean: x̄ = A + Σfd ÷ N.
Continuous series
Use class marks (mid-values) as x.
- Direct: x̄ = Σfm ÷ N.
- Assumed mean: x̄ = A + Σfd ÷ N, d = m − A.
- Step deviation: d′ = (m − A) ÷ c, where c is the common width; x̄ = A + (Σfd′ ÷ N) × c.
Two properties
- The sum of deviations from the mean is always zero: Σ(x − x̄) = 0. That is why it is the balance point.
- The sum of squared deviations from the mean is the smallest possible.
Weighted arithmetic mean
When items are not equally important, give each a weight w: x̄w = Σwx ÷ Σw. Example: in a price index, rice (bought a lot) gets more weight than salt.
Merits: easy, uses every value, good for further maths. Demerits: pulled a lot by extreme values; cannot be found for open-ended classes without assumptions.
Median
The median is the middle value when data is arranged in order. Half the items are below it and half above. It is a positional average.
Individual series
Arrange in order. Median = value of the (n + 1) ÷ 2th item. If n is even, take the mean of the two middle items.
Discrete series
Find cumulative frequencies (cf). Find (N + 1) ÷ 2. The first x whose cf is equal to or more than this is the median.
Continuous series
Find N ÷ 2. The class whose cf first reaches N ÷ 2 is the median class.
Median = l + ((N/2 − c.f.) ÷ f) × h, where l = lower limit of the median class, c.f. = cumulative frequency of the class before it, f = frequency of the median class, h = class width.
Merit: not affected by extreme values, so it is good for income and wealth data. Demerit: it ignores the size of other values.
Mode
The mode is the value that occurs most often.
- Individual and discrete series: find it by inspection: the value with the highest frequency. When frequencies are close, a grouping table helps.
- Continuous series: find the modal class (highest frequency). Then Mode = l + ((f1 − f0) ÷ (2f1 − f0 − f2)) × h, where f1 = frequency of the modal class, f0 = of the class before, f2 = of the class after.
Data can have one mode (unimodal), two (bimodal) or more. The mode can also be found from a histogram.
Relation between mean, median and mode
In a symmetrical distribution, mean = median = mode. For a moderately skewed distribution, an approximate relation is Mode ≈ 3 Median − 2 Mean. When a few very large values are present (as in incomes), mean > median > mode.
Economic interpretation: which average to use?
- Mean: per capita income, average yield per hectare, average cost. Good when total matters and there are no extreme values.
- Median: typical income or wage in a country with a few very rich people; median house price.
- Mode: most common shirt size, most popular price of a product, the family size most common in a village. Businesses use it for stock planning.
Example: incomes 2, 4, 6, 8, 20. Mean = 8, but four out of five families earn less than 8. The median, 6, describes the typical family better.
Try it: pocket money averages
Ask 7 friends their weekly pocket money. Find the mean, the median and the mode. Now imagine one friend gets ₹1,000. Recalculate. Which average changed the most? Then try the same thing with the sliders on the last 3D step.
Board exam pattern
This unit (with correlation and index numbers) carries about 25 marks. Expect a 6-mark numerical on mean by step deviation or median/mode of a continuous series, and short questions on which average suits a situation. Always write the formula, show the table (x, f, cf, d′, fd′) and give the answer with units.
Key formulas and definitions
- Mean (individual) = Σx ÷ n; assumed mean: A + Σd ÷ n
- Mean (discrete) = Σfx ÷ N; A + Σfd ÷ N
- Mean (continuous, step deviation) = A + (Σfd′ ÷ N) × c, d′ = (m − A) ÷ c
- Weighted mean = Σwx ÷ Σw
- Median (individual) = ((n + 1) ÷ 2)th item
- Median (continuous) = l + ((N/2 − c.f.) ÷ f) × h
- Mode (continuous) = l + ((f1 − f0) ÷ (2f1 − f0 − f2)) × h
- Mode ≈ 3 Median − 2 Mean
Worked examples
1. Find the mean of 12, 15, 18, 21, 24 by the assumed mean method (A = 18).
d = x − 18: −6, −3, 0, 3, 6. Σd = 0. Mean = 18 + 0 ÷ 5 = 18.
2. x: 1, 2, 3, 4, 5 (children per family), f: 4, 6, 8, 5, 2. Find the mean.
fx: 4, 12, 24, 20, 10; Σfx = 70; N = 25. Mean = 70 ÷ 25 = 2.8 children.
3. Classes 0–10, 10–20, 20–30, 30–40, 40–50 with f = 5, 8, 12, 10, 5. Find the mean by step deviation (A = 25, c = 10).
m: 5, 15, 25, 35, 45; d′: −2, −1, 0, 1, 2; fd′: −10, −8, 0, 10, 10; Σfd′ = 2; N = 40. Mean = 25 + (2 ÷ 40) × 10 = 25.5.
4. Find the median of 7, 3, 9, 12, 5, 10.
Order: 3, 5, 7, 9, 10, 12. n = 6, so the median is the mean of the 3rd and 4th items: (7 + 9) ÷ 2 = 8.
5. Same table as above (f = 5, 8, 12, 10, 5). Find the median.
cf: 5, 13, 25, 35, 40. N/2 = 20 → median class 20–30. l = 20, c.f. = 13, f = 12, h = 10. Median = 20 + (7 ÷ 12) × 10 = 25.83.
6. Same table. Find the mode.
Modal class 20–30: f1 = 12, f0 = 8, f2 = 10. Mode = 20 + (4 ÷ (24 − 18)) × 10 = 20 + 6.67 = 26.67.
7. A worker earns ₹300 a day for 4 days and ₹500 a day for 1 day. Find his weighted average daily wage.
Σwx = 300 × 4 + 500 × 1 = 1,700; Σw = 5. Weighted mean = ₹340.
8. Mean = 40, median = 38. Estimate the mode.
Mode ≈ 3 × 38 − 2 × 40 = 114 − 80 = 34.
Common mistakes
- Finding the median without arranging the data in order.
- Using N/2 in the individual series formula; use (n + 1) ÷ 2 there, and N/2 for continuous series.
- Taking c.f. of the median class itself instead of the class before it.
- Forgetting to multiply by c in the step deviation method.