Mean, Median, Mode and Standard Deviation — When to Use Which
Mean, median and mode all try to answer the same question — "what is a typical value in this data?" — but they answer it differently, and choosing the wrong one gives a technically correct number that describes the data badly. Standard deviation then answers the second question: "how spread out is the data around that typical value?" This article works one data set completely, including every step of the standard deviation, and makes the population-versus-sample difference explicit, because that is where most marks are lost.
The four formulas
Median sort the data first, then:
• n odd → the middle value, at position (n + 1)/2
• n even → the average of the two middle values, positions n/2 and n/2 + 1
Mode the value that occurs most often
Population standard deviation σ = √[ Σ(x − μ)² / N ]
Sample standard deviation s = √[ Σ(x − x̄)² / (n − 1) ]
| Symbol | Meaning | Unit |
|---|---|---|
| x | Each individual data value | same as the data |
| n or N | Number of values (n for a sample, N for a whole population) | none |
| x̄ or μ | Mean of the sample (x̄) or of the population (μ) | same as the data |
| Σ(x − x̄)² | Sum of squared deviations from the mean | data unit squared |
| σ² or s² | Variance | data unit squared |
| σ or s | Standard deviation | same as the data |
That last row is why reports quote standard deviation rather than variance: if the data is in marks, the variance is in "marks squared", which means nothing to a reader.
Which one should you use?
| Measure | Use it when | Weakness |
|---|---|---|
| Mean | Data is roughly symmetric and has no extreme values; you need a value for further algebra | Dragged badly by outliers |
| Median | Data is skewed or has outliers — incomes, house prices, reaction times | Ignores the actual sizes of most values |
| Mode | Data is categories or repeated whole values — shoe sizes, most common blood group | May not exist, or there may be several |
| Standard deviation | You need to describe consistency or spread around the mean | Also sensitive to outliers, since deviations are squared |
Worked example 1 — the full calculation
Data (test marks of 8 students): 12, 15, 15, 18, 20, 22, 25, 33. The values are already in ascending order, and n = 8.
Step 1 — Mean.
Σx = 12 + 15 + 15 + 18 + 20 + 22 + 25 + 33
= 12 + 15 = 27; 27 + 15 = 42; 42 + 18 = 60; 60 + 20 = 80; 80 + 22 = 102;
102 + 25 = 127; 127 + 33 = 160
x̄ = 160 / 8 = 20
Step 2 — Median. n = 8 is even, so average the 4th and 5th values:
4th = 18, 5th = 20 → median = (18 + 20) / 2 = 19
Step 3 — Mode. 15 appears twice; every other value appears once → mode = 15
Step 4 — Deviations and their squares.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 12 | −8 | 64 |
| 15 | −5 | 25 |
| 15 | −5 | 25 |
| 18 | −2 | 4 |
| 20 | 0 | 0 |
| 22 | 2 | 4 |
| 25 | 5 | 25 |
| 33 | 13 | 169 |
| Σ | 0 | 316 |
Adding the squares: 64 + 25 = 89; 89 + 25 = 114; 114 + 4 = 118; 118 + 0 = 118; 118 + 4 = 122; 122 + 25 = 147; 147 + 169 = 316.
The middle column summing to exactly 0 is not a coincidence — the deviations from the mean always cancel. That is a free check on your mean, and it is also why we square them before adding.
Step 5 — Population standard deviation (treating these 8 students as
the entire group of interest):
σ² = 316 / 8 = 39.5
σ = √39.5 ≈ 6.28 marks
Step 6 — Sample standard deviation (treating these 8 as a sample drawn
from a larger class):
s² = 316 / (8 − 1) = 316 / 7 ≈ 45.14
s = √45.14 ≈ 6.72 marks
Independent check using the computational formula
σ² = (Σx²)/N − x̄²:
Σx² = 144 + 225 + 225 + 324 + 400 + 484 + 625 + 1089 = 3516
(3516 / 8) − 20² = 439.5 − 400 = 39.5 ✔ — matches Step 5 exactly.
Population or sample? The difference that costs marks
The only difference between the two formulas is the divisor: N versus n − 1. The reason is that a sample's own mean is calculated from that same sample, so the data sits slightly closer to it than it does to the true population mean. Dividing by n − 1 instead of n makes the answer slightly larger and removes that bias. The quantity n − 1 is called the number of degrees of freedom.
| Situation | Divide by | Symbol |
|---|---|---|
| The data is the complete group you care about (marks of all 8 students in a class of 8) | N | σ |
| The data is a sample used to estimate a bigger group (8 titration readings, 8 students out of 400) | n − 1 | s |
In our example the two answers differ by about 7% — 6.28 against 6.72 — which is more than enough to be marked wrong. In CBSE Class 11 statistics, questions almost always treat the given list as the complete data and use N. In a chemistry or physics practical, repeated readings are a sample, so n − 1 is correct. On a scientific calculator these usually appear as σx (or σn) and sx (or σn−1) — check which key you are pressing.
Worked example 2 — why the median exists
Data (monthly incomes of 5 people, in thousands of rupees): 2, 3, 3, 4, 88
Mean = (2 + 3 + 3 + 4 + 88) / 5 = 100 / 5 = 20
Median = middle value of the sorted list, position (5 + 1)/2 = 3rd = 3
Mode = 3
The mean says a typical income is 20 thousand, but four of the five people earn 4 or less. One extreme value has pulled the mean far away from everybody. Here the median is the honest summary. This is exactly why national income and house-price figures are reported as medians.
Worked example 3 — from a frequency table
Data: 2 students scored 10, 3 students scored 20, 5 students scored 30.
Mean = Σfx / Σf
Σfx = (2 × 10) + (3 × 20) + (5 × 30) = 20 + 60 + 150 = 230
Σf = 2 + 3 + 5 = 10
x̄ = 230 / 10 = 23
Median. Written out, the 10 values are
10, 10, 20, 20, 20, 30, 30, 30, 30, 30. With n = 10 (even), average the 5th and 6th:
5th = 20, 6th = 30 → median = (20 + 30) / 2 = 25
Mode = the value with the highest frequency = 30 (f = 5)
Track the cumulative counts (2, then 5, then 10) before picking the 5th and 6th values — skipping that step is the usual source of a wrong median here.
Common mistakes that cost marks
- Not sorting before finding the median. The median of an unsorted list read straight through is simply wrong. Sort first, every time.
- Using n instead of n − 1, or the reverse. Decide first whether the data is a whole population or a sample, and say so in your answer.
- Reporting variance as standard deviation. The final step is the square root. Forgetting it gave 39.5 instead of 6.28 in example 1.
- Rounding the mean early. If x̄ were rounded to 20.0 from something like 19.87, every one of the eight deviations would be slightly wrong and the error would compound through the squares. Keep full precision until the last line.
- Saying "the mode is 0" when no value repeats. If every value occurs once, the data has no mode. If two values tie for the highest frequency, it is bimodal — both are the mode.
- Forgetting to weight by frequency. In example 3 the mean is not (10 + 20 + 30)/3 = 20; it is Σfx/Σf = 23.
- Quoting the mean for badly skewed data without mentioning the outlier — technically correct, but it misdescribes the data, and application questions award marks for choosing the right measure.
Where this appears in exams
| Exam | Typical use |
|---|---|
| CBSE/ICSE Class 9–10 | Mean, median and mode of ungrouped and grouped data; cumulative frequency curves |
| CBSE/ICSE Class 11 | Measures of dispersion — range, mean deviation, variance and standard deviation |
| CBSE/ICSE Class 12 | Probability distributions, mean and variance of a random variable |
| JEE Main | Direct one-mark questions on variance and standard deviation of small data sets |
| Chemistry & physics practicals | Repeated titration or measurement readings — sample standard deviation as the precision of a result |
| CSIR-NET / GATE | Error analysis, least-squares fitting and reproducibility of measurements |
One useful extra: coefficient of variation
To compare the spread of two data sets measured in different units or with very different means, use the coefficient of variation, CV = (σ / x̄) × 100. For example 1 that is (6.28 / 20) × 100 ≈ 31.4%. A smaller CV means a more consistent set of results — which is exactly how the precision of repeated laboratory measurements is judged.
Check the whole set in one go. Paste your data into the Statistics tool and it returns the mean, median, mode, variance and both standard deviations — so you can confirm your hand calculation and see immediately whether you used the population or the sample formula.
Open the Statistics Calculator →Building up Class 11–12 maths and chemistry together? ABC Chemistry runs Class 11–12 coaching at the Gurugram centre and online classes across India, with home tuition available in the Delhi-NCR area — details at abcchemistry.in.