Statistics Formulas for SSC CGL
The statistics formulas for SSC CGL you must know are: mean = sum of observations ÷ number of observations, median = middle value of the sorted data, mode = most frequent value, the empirical relation mode = 3 median − 2 mean, range = highest − lowest, variance = average of squared deviations from the mean, standard deviation = √variance, and mean deviation = average of absolute deviations. For grouped data, use the class-interval formulas for median and mode given in the tables below.
🔥 Key Exam Takeaway: Adding a constant to every value changes the mean, median and mode by that constant but leaves the range, mean deviation, variance and SD unchanged. Multiplying every value by k multiplies the mean and SD by k (SD by |k| if k is negative) and the variance by k².
Statistics in SSC CGL Syllabus 2026 (Tier 1, Tier 2 and JSO Paper)
The official SSC CGL 2026 notice (ssc.gov.in, para 13.11) places statistics in three places:
| Paper | What the official syllabus says | What to prepare from this note |
|---|---|---|
| Tier 1, Quantitative Aptitude | Histogram, frequency polygon, bar diagram and pie chart | Reading tables and graphs |
| Tier 2, Paper I, Mathematical Abilities (all posts) | "Statistics and probability: Use of Tables and Graphs: Histogram, Frequency polygon, Bar-diagram, Pie-chart; Measures of central tendency: mean, median, mode, standard deviation; calculation of simple probabilities" | Mean, median, mode, SD, empirical relation |
| Tier 2, Paper II, Statistics (only JSO and Statistical Investigator Grade-II) | Central tendency incl. quartiles, deciles, percentiles; dispersion: range, quartile deviation, mean deviation, standard deviation, relative dispersion; plus moments, correlation, regression, probability and more | Everything here, plus the wider JSO syllabus |
So for most CGL aspirants, the scoring formulas are the central tendency and standard deviation ones. JSO aspirants need the dispersion section too, and should study the full Paper II syllabus beyond this note.
Mean Formula for SSC CGL (Ungrouped, Frequency and Grouped Data)
| Data type | Mean formula | Notes |
|---|---|---|
| Raw (ungrouped) data | x̄ = (x₁ + x₂ + … + xₙ) ÷ n = Σx ÷ n | Basic average |
| Discrete frequency data | x̄ = Σfx ÷ Σf | f = frequency of each x |
| Grouped (class intervals) | x̄ = Σf·m ÷ Σf | m = class mark = (lower + upper) ÷ 2 |
| Assumed mean method | x̄ = A + Σfd ÷ Σf, where d = m − A | Saves time with big numbers |
| Step-deviation method | x̄ = A + (Σfu ÷ Σf) × h, where u = (m − A) ÷ h | h = class width |
| Combined mean of two groups | x̄ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂) | Weighted average |
| Corrected mean | New mean = (old sum − wrong value + correct value) ÷ n | Misread observation questions |
| First n natural numbers | Mean = (n + 1) ÷ 2 | 1 to 10 → 5.5 |
Property: the sum of deviations from the mean is always zero: Σ(x − x̄) = 0.
Median Formula for SSC CGL (Odd, Even and Grouped Data)
| Case | Median formula |
|---|---|
| n odd (sorted data) | Value at position (n + 1) ÷ 2 |
| n even (sorted data) | Average of the values at positions n ÷ 2 and (n ÷ 2) + 1 |
| Discrete frequency data | Use cumulative frequency (cf) to find the (N ÷ 2)th value |
| Grouped data | Median = L + [(N/2 − cf) ÷ f] × h |
In the grouped formula: L = lower limit of the median class, N = total frequency, cf = cumulative frequency of the class before the median class, f = frequency of the median class, h = class width. The median class is the first class whose cumulative frequency reaches N/2.
Trap: always sort the data first. Many wrong answers in CGL come from taking the middle of unsorted numbers.
Mode Formula for SSC CGL (Grouped Data and Empirical Relation)
| Case | Mode formula |
|---|---|
| Raw data | Value that occurs most often (data can have no mode, one mode or more) |
| Grouped data | Mode = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h |
| Empirical relation (moderately skewed data) | Mode = 3 Median − 2 Mean |
| Same relation rearranged | Mean − Mode = 3 (Mean − Median) |
In the grouped formula: L = lower limit of the modal class (the class with the highest frequency), f₁ = its frequency, f₀ = frequency of the class before it, f₂ = frequency of the class after it, h = class width.
| Shape of data | Order of mean, median and mode |
|---|---|
| Symmetric | Mean = Median = Mode |
| Positively skewed (long tail on the right) | Mode < Median < Mean |
| Negatively skewed (long tail on the left) | Mean < Median < Mode |
Range, Mean Deviation, Variance and Standard Deviation Formulas for SSC CGL
| Measure | Formula | Key fact |
|---|---|---|
| Range | Highest value − Lowest value | Simplest measure of dispersion |
| Coefficient of range | (H − L) ÷ (H + L) | Relative measure, no unit |
| Mean deviation (about A) | MD = Σ∣x − A∣ ÷ n, where A is the mean or the median | Use absolute values, ignore signs |
| Mean deviation (frequency data) | MD = Σf∣x − A∣ ÷ Σf | Same idea with weights |
| Coefficient of mean deviation | MD ÷ (mean or median used) | Relative measure |
| Variance (σ²) | Σ(x − x̄)² ÷ n | Squared unit |
| Variance, shortcut | Σx² ÷ n − (x̄)² | Mean of squares minus square of mean |
| Variance (frequency data) | Σf(x − x̄)² ÷ Σf = Σfx² ÷ Σf − (x̄)² | Grouped: use class marks |
| Standard deviation (σ) | σ = √variance | Same unit as the data |
| SD of first n natural numbers | σ = √[(n² − 1) ÷ 12] | 1 to 10 → √8.25 ≈ 2.87 |
| Coefficient of variation | CV = (σ ÷ x̄) × 100 | Lower CV = more consistent |
| Quartile deviation (JSO) | QD = (Q₃ − Q₁) ÷ 2 | Semi-interquartile range |
Facts examiners like:
- Mean deviation is least when taken about the median.
- Variance and SD are never negative. SD = 0 only when all values are equal.
- This note uses the population formula (divide by n), which is the standard in SSC questions. A sample variance divides by n − 1, so for 2, 4, 4, 4, 5, 5, 7, 9 the population variance is 4 but the sample variance is 32/7 ≈ 4.57. If options show both, read the question wording.
Effect of Changing Data on Mean, Median, Mode and SD (SSC CGL Shortcut)
| Change to every value | Mean, median, mode | Range, MD, SD | Variance |
|---|---|---|---|
| Add or subtract k | Change by +k or −k | No change | No change |
| Multiply by k | Multiplied by k | Multiplied by ∣k∣ | Multiplied by k² |
| Divide by k (k ≠ 0) | Divided by k | Divided by ∣k∣ | Divided by k² |
| Multiply by a, then add b | a × old + b | ∣a∣ × old | a² × old |
Check: data 2, 4, 4, 4, 5, 5, 7, 9 has mean 5 and SD 2. After "×3 then +5" it becomes 11, 17, 17, 17, 20, 20, 26, 32, with mean 20 and SD 6, as the table predicts.
Statistics Solved Examples for SSC CGL Tier 2
Example 1: Mean, median and mode of raw data
Find the mean, median and mode of 3, 5, 5, 7, 8, 9, 12.
Solution: Sum = 49, n = 7 → mean = 7. The data is sorted and n is odd, so the median is the 4th value = 7. 5 occurs twice → mode = 5.
Example 2: Median when n is even
Find the median of 12, 5, 8, 20, 15, 9.
Solution: Sorted: 5, 8, 9, 12, 15, 20. Median = (9 + 12) ÷ 2 = 10.5.
Example 3: Mean of a frequency distribution
| x | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| f | 3 | 5 | 8 | 4 |
Solution: Σfx = 30 + 100 + 240 + 160 = 530; Σf = 20. Mean = 530 ÷ 20 = 26.5.
Example 4: Mean, median and mode of grouped data
| Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 12 | 10 | 5 |
| Cumulative frequency | 5 | 13 | 25 | 35 | 40 |
Mean: class marks 5, 15, 25, 35, 45. Σfm = 25 + 120 + 300 + 350 + 225 = 1,020; N = 40. Mean = 25.5.
Median: N/2 = 20, which first falls in 20–30 (cf 25). L = 20, cf = 13, f = 12, h = 10. Median = 20 + (20 − 13) ÷ 12 × 10 = 20 + 70/12 = 25 5/6 ≈ 25.83.
Mode: modal class 20–30, f₁ = 12, f₀ = 8, f₂ = 10. Mode = 20 + (12 − 8) ÷ (24 − 8 − 10) × 10 = 20 + 4/6 × 10 = 26 2/3 ≈ 26.67.
Example 5: Empirical relation (most asked)
For a moderately skewed distribution, the mean is 30 and the median is 28. Find the mode.
Solution: Mode = 3 × 28 − 2 × 30 = 84 − 60 = 24.
Example 6: Variance and standard deviation
Find the variance and SD of 2, 4, 4, 4, 5, 5, 7, 9.
Solution: Mean = 40 ÷ 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32. Variance = 32 ÷ 8 = 4, SD = 2. Shortcut check: Σx² = 232, 232 ÷ 8 − 5² = 29 − 25 = 4.
Example 7: Mean deviation about the mean
Find the mean deviation of 3, 7, 9, 5, 6 about the mean.
Solution: Mean = 30 ÷ 5 = 6. Absolute deviations: 3, 1, 3, 1, 0 → sum 8. MD = 8 ÷ 5 = 1.6.
Example 8: SD of the first 10 natural numbers
Solution: σ = √[(10² − 1) ÷ 12] = √(99 ÷ 12) = √8.25 ≈ 2.87.
Example 9: Effect of change of scale and origin
The mean of a data set is 20 and its SD is 4. Each value is multiplied by 3 and then 5 is added. Find the new mean and SD.
Solution: New mean = 3 × 20 + 5 = 65. New SD = 3 × 4 = 12 (adding 5 does not change the SD).
Example 10: Combined mean
The average marks of 40 students are 60 and of another 60 students are 70. Find the average of all 100.
Solution: (40 × 60 + 60 × 70) ÷ 100 = (2,400 + 4,200) ÷ 100 = 66.
Example 11: Corrected mean
The mean of 25 observations is 40. One observation, 52, was wrongly taken as 25. Find the correct mean.
Solution: Wrong sum = 1,000. Correct sum = 1,000 − 25 + 52 = 1,027. Correct mean = 1,027 ÷ 25 = 41.08.
Example 12: Range and coefficient of range
For 12, 18, 25, 40, 31: range = 40 − 12 = 28; coefficient of range = 28 ÷ 52 = 7/13 ≈ 0.54.
Common Mistakes in SSC CGL Statistics Questions
| Trap | Correct way |
|---|---|
| Median of unsorted data | Sort first, then pick the middle |
| Using the median class frequency for cf | cf is of the class before the median class |
| Using 2f₁ − f₀ + f₂ in the mode formula | The denominator is 2f₁ − f₀ − f₂ |
| Writing Mode = 3 Mean − 2 Median | It is Mode = 3 Median − 2 Mean |
| Thinking adding a constant changes SD | SD, variance, range and MD do not change |
| Multiplying variance by k | Variance is multiplied by k² |
| Forgetting the square root | SD = √variance; check which one the question asks for |
Statistics Formulas for SSC CGL: Quick Revision Table
| Measure | One-line formula |
|---|---|
| Mean | Σx ÷ n, or Σfx ÷ Σf |
| Median (grouped) | L + [(N/2 − cf) ÷ f] × h |
| Mode (grouped) | L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h |
| Empirical relation | Mode = 3 Median − 2 Mean |
| Range | H − L |
| Mean deviation | Σ∣x − A∣ ÷ n |
| Variance | Σx² ÷ n − (x̄)² |
| SD | √variance |
| SD of 1 to n | √[(n² − 1) ÷ 12] |
| Coefficient of variation | (σ ÷ x̄) × 100 |
Statistics Questions for SSC CGL with Answers (Practice Set)
Q1. Find the mean of 6, 8, 10, 12, 14. Answer: 10. Sum 50 ÷ 5.
Q2. Find the median of 13, 7, 21, 9, 17, 11, 15. Answer: 13. Sorted: 7, 9, 11, 13, 15, 17, 21; the 4th value is 13.
Q3. In a moderately skewed distribution, the median is 22 and the mean is 25. Find the mode. Answer: 16. 3 × 22 − 2 × 25 = 66 − 50 = 16.
Q4. Find the variance and SD of 1, 2, 3, 4, 5. Answer: variance 2, SD √2 ≈ 1.41. Mean 3; squared deviations 4, 1, 0, 1, 4 = 10; 10 ÷ 5 = 2. (Also √[(25 − 1) ÷ 12] = √2.)
Q5. Find the mean deviation about the median of 4, 6, 9, 12, 14. Answer: 3.2. Median 9; absolute deviations 5, 3, 0, 3, 5 = 16; 16 ÷ 5 = 3.2.
Q6. The SD of a data set is 6. Every value is doubled. Find the new SD and the new variance. Answer: SD 12, variance 144. SD × 2; variance × 4 (36 × 4).
Q7. The average weight of 30 boys is 52 kg and of 20 girls is 47 kg. Find the average weight of the whole class. Answer: 50 kg. (1,560 + 940) ÷ 50 = 2,500 ÷ 50.
Q8. Find the mode of this grouped data.
| Class | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|
| Frequency | 4 | 9 | 15 | 11 | 6 |
Answer: 36. Modal class 30–40; 30 + (15 − 9) ÷ (30 − 9 − 11) × 10 = 30 + 6 = 36.
Q9. Find the median of this distribution.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| f | 4 | 6 | 10 | 7 | 3 |
Answer: 3. N = 30, so the median is the average of the 15th and 16th values. Cumulative frequencies are 4, 10, 20, 27, 30; both the 15th and 16th values are 3.
Q10. Batsman A has mean 40 runs and SD 8; batsman B has mean 60 runs and SD 9. Who is more consistent? Answer: B. CV of A = 8 ÷ 40 × 100 = 20%; CV of B = 9 ÷ 60 × 100 = 15%. Lower CV means more consistent.
Statistics for SSC CGL FAQs
Is statistics in the SSC CGL Tier 2 syllabus?
Yes. The SSC CGL 2026 notice lists "Measures of central tendency: mean, median, mode, standard deviation" and simple probability in Tier 2 Paper I (Mathematical Abilities), which is compulsory for all posts. A separate full Statistics paper (Paper II) is only for JSO and Statistical Investigator Grade-II candidates.
What is the relation between mean, median and mode?
For a moderately skewed distribution, Mode = 3 Median − 2 Mean. For a symmetric distribution all three are equal.
What is the formula of standard deviation for SSC CGL?
σ = √[Σ(x − x̄)² ÷ n], or the shortcut σ = √[Σx² ÷ n − (x̄)²]. For the first n natural numbers, σ = √[(n² − 1) ÷ 12].
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean, so its unit is squared. Standard deviation is the square root of the variance, so it has the same unit as the data.
Does adding a number to every observation change the standard deviation?
No. Adding or subtracting a constant shifts the mean, median and mode, but the range, mean deviation, variance and standard deviation stay the same.
About which value is mean deviation minimum?
Mean deviation is smallest when it is calculated about the median.
What is the formula for median of grouped data?
Median = L + [(N/2 − cf) ÷ f] × h, where L is the lower limit of the median class, cf is the cumulative frequency before it, f is its frequency and h is the class width.
Related topics: revise percentage tricks for SSC CGL, algebra formulas for SSC CGL, mensuration formulas for SSC CGL and physics for RRB Group D next.
Practice Next
- Topic test: SSC CGL Statistics Mock Test 04.
- New SSC CGL tests (Tier 1 level): SSC CGL Data Interpretation Mock Test 01.
- Practise central-tendency basics in the SSC CGL Average Mock Test 01 and Average Mock Test 02.
- Take a full SSC CGL Quantitative Aptitude Sectional Mock Test 01.
- New SSC topic test: SSC CGL Statistics Mock Test 01: Mean, Median and Mode Questions (15 Questions).
- More practice: SSC CGL Statistics Mock Test 02.
- More SSC CGL formula sheets: Pipe and Cistern Formula for SSC CGL, CI and SI Difference Formula for 2, 3 and 4 Years and Unit Digit Tricks for SSC CGL.
- For reasoning, read the Calendar Odd Days Trick.
- Check the exam dates in the SSC CGL 2026 notification.
- Browse all study notes.
Related: average tricks for SSC CGL.