Measures of Central Tendency: Mean, Median, Mode, GM and HM with Formulas for SSC
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A measure of central tendency is a single value that represents the centre or the typical value of a data set. The main measures are the mean, median and mode, along with the geometric and harmonic means. They form the base of SSC CGL Tier 2 statistics (Paper III) and the Statistical Investigator exam. This note gives each formula for raw and grouped data, the important relations, solved examples, practice questions and FAQs.
Types of Averages
| Measure | Meaning | Formula (raw data) |
|---|---|---|
| Arithmetic mean (AM) | Sum divided by the number of items | Σx / n |
| Weighted mean | Mean where items have different importance | Σwx / Σw |
| Median | Middle value of the ordered data | Odd n: ((n + 1)/2)th item; even n: average of (n/2)th and (n/2 + 1)th |
| Mode | Most frequent value | Value with the highest frequency |
| Geometric mean (GM) | nth root of the product | (x1 × x2 × … × xn)1/n |
| Harmonic mean (HM) | Reciprocal of the mean of reciprocals | n / Σ(1/x) |
Formulas for Grouped Data
| Measure | Formula |
|---|---|
| Mean (direct) | Σfx / Σf |
| Mean (step deviation) | A + (Σfu / Σf) × h, where u = (x − A)/h |
| Median | L + [(n/2 − cf) / f] × h |
| Mode | L + [(f1 − f0) / (2f1 − f0 − f2)] × h |
- L = lower limit of the median or modal class; h = class width; f = frequency of that class.
- cf = cumulative frequency of the class before the median class.
- f1 = frequency of the modal class; f0 and f2 = frequencies of the classes just before and after it.
- The median class is the first class whose cumulative frequency is at least n/2.
Important Relations
- Empirical relation: Mode = 3 Median − 2 Mean.
- For positive values: AM ≥ GM ≥ HM. They are equal only when all values are the same.
- For two numbers: GM² = AM × HM.
- Combined mean of two groups: (n1·m1 + n2·m2) / (n1 + n2).
- The sum of deviations of items from their mean is zero.
- If a constant is added to every item, the mean, median and mode increase by that constant. If every item is multiplied by a constant, they are multiplied by it too.
- The mean is affected by extreme values; the median and mode are not.
When to Use Which Measure
| Situation | Best measure |
|---|---|
| Data is symmetric with no extreme values | Mean |
| Data has extreme values (income, house prices) | Median |
| Most common size or category is needed (shoe size) | Mode |
| Average rate of growth, ratios, index numbers | Geometric mean |
| Average speed over equal distances, price per unit | Harmonic mean |
Solved Examples
| Question | Solution |
|---|---|
| Mean of 4, 8, 12, 16, 20 | 60 / 5 = 12 |
| Median of 3, 7, 9, 15, 21 | Middle value = 9 |
| Median of 2, 4, 6, 8 | (4 + 6)/2 = 5 |
| Mode of 2, 3, 3, 4, 5, 3 | 3 appears most often, so 3 |
| Weighted mean of 80 (weight 2) and 90 (weight 3) | (160 + 270)/5 = 86 |
| Combined mean: 20 students with mean 40 and 30 students with mean 60 | (800 + 1800)/50 = 52 |
| Mean 24 and median 22. Find the mode. | 3 × 22 − 2 × 24 = 18 |
| GM of 2 and 8 | √16 = 4 |
| HM of 2 and 3 | 2 × 2 × 3 / (2 + 3) = 2.4 |
Grouped data example
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 12 | 15 | 10 |
| Cumulative frequency | 5 | 13 | 25 | 40 | 50 |
- n = 50, so n/2 = 25. The first class with cf ≥ 25 is 20–30. Median = 20 + [(25 − 13)/12] × 10 = 30.
- The modal class is 30–40 (highest frequency 15). Mode = 30 + [(15 − 12)/(30 − 12 − 10)] × 10 = 30 + 3.75 = 33.75.
Quick Tips
- Arrange data in order before finding the median.
- For grouped data, check the class limits (use true limits for inclusive classes).
- Use AM ≥ GM ≥ HM to check your answers.
- Combined mean questions are best solved with total sums.
Quick Revision Points
- Mode = 3 Median − 2 Mean.
- AM ≥ GM ≥ HM; GM² = AM × HM for two numbers.
- Median class has cf ≥ n/2.
- Sum of deviations from the mean is zero.
- Mean is affected by extreme values; median is not.
Practice Questions With Answers
- Mean of 5, 10, 15?
Answer: 10 - Median of 1, 3, 5, 7, 9?
Answer: 5 - Median of 10, 20, 30, 40?
Answer: 25 - Mode of 4, 5, 5, 6, 7?
Answer: 5 - Mean = 15, median = 14. Find the mode.
Answer: 12 - GM of 4 and 9?
Answer: 6 - Which average is affected most by extreme values?
Answer: Mean - What is the sum of the deviations of items from their mean?
Answer: Zero - If every item is increased by 5, the mean increases by?
Answer: 5 - For two numbers, GM² equals?
Answer: AM × HM
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Measures of Central Tendency notes page.
Frequently Asked Questions
What is central tendency?
It is the tendency of the data to cluster around a central value, which is described by the mean, median or mode.
What is the arithmetic mean?
The sum of all values divided by the number of values.
What is the median?
The middle value of the data after arranging it in order.
What is the mode?
The value that occurs most often in the data.
What is the empirical relation between mean, median and mode?
Mode = 3 Median − 2 Mean, for a moderately skewed distribution.
What is the relation between AM, GM and HM?
AM ≥ GM ≥ HM for positive numbers.
When do we use the geometric mean?
For growth rates, ratios and index numbers.
When do we use the harmonic mean?
For averages of rates such as speed over equal distances.
What is a weighted mean?
A mean in which each value is multiplied by its weight, divided by the sum of the weights.
Which measure is best for skewed data?
The median.
How do I find the combined mean of two groups?
(n1 m1 + n2 m2)/(n1 + n2).
Is this topic important in SSC?
Yes, it is a core topic for SSC CGL Tier 2 statistics and the Statistical Investigator papers.
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Rahul Kumar
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