Find the median from the following discrete series (the same data as Worked Example 2):
| Marks (x) | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| Number of students (f) | 4 | 6 | 10 | 7 | 3 |
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Median
What is the Median?
Imagine you and four friends are comparing pocket money. The amounts are: ₹20, ₹50, ₹30, ₹100, ₹40. If you line these up from smallest to largest — ₹20, ₹30, ₹40, ₹50, ₹100 — the middle value is ₹40. That's the median: the number that splits the data exactly in half.
The median answers a simple question: "What is the typical value, if we ignore extremes?" In the example above, the average (mean) is ₹48, but one friend gets ₹100, which pulls the average up. The median stays at ₹40, which better represents what most of you actually get.
The median is a positional average. It doesn't care how big or small the numbers are — only where they sit in the sorted order.
The Precise Definition
For ungrouped data (a list of numbers):
- Arrange all values in ascending order (smallest to largest).
- Let n be the number of observations.
- If n is odd: the median is the (2n+1)th value.
- If n is even: the median is the average of the (2n)th and (2n+1)th values.
Example 1 (odd count):
Data: 2, 5, 1, 8, 4
Sorted: 1, 2, 4, 5, 8
n=5, position = 25+1=3
Median = 4
Example 2 (even count):
Data: 3, 7, 2, 9, 1, 6
Sorted: 1, 2, 3, 6, 7, 9
n=6, positions = 26=3 and 26+1=4
Values at those positions: 3 and 6
Median = 23+6=4.5
A common mistake: for even n, students sometimes pick the 2nth value alone. You must average the two middle numbers.
Why the Median Matters …
For a discrete series, the median is located by building a running (cumulative) total of the frequencies and finding where the halfway point of the class falls.
Median = the x-value where the cumulative frequency first reaches N/2 = 15. …
| x | f | Cumulative frequency (cf) |
|---|---|---|
| 10 | 4 | 4 |
| 20 | 6 | 10 |
| 30 | 10 | 20 |
| 40 | 7 | 27 |
| 50 | 3 | 30 |
N = 30, so 2N=15. …
Cross-check by descending cumulative frequency. Building the cf from the highest value downward: 50 (cf 3), 40 (cf 3+7=10), 30 (cf 10+10=20), 20 (cf 20+6=26), 10 (cf 26+4=30). The first descending-cf value reachin …
A common error is stopping at the frequency of 10 (the highest single frequency in the table) and mistaking it for the median value itself, instead of continuing …
- CBSE 2024Set ANNUAL1 markMCQQ.From the following, the one which is not equal to median, is :(a) Second quartile(b) Sixth decile(c) Fiftieth percentile(d) Fifth decile
›Reveal solutionSolution
Median =Q2=D5=P50; the sixth decile D6 is the odd one out.
The median divides the data at the 50% point. The partition values that also fall at 50% are:
- Second quartile Q2 (2 of 4 parts =50%),
- Fifth decile D5 (5 of 10 parts =50%),
- Fiftieth percentile P50 (50%). …
- CBSE 2024Set ANNUAL1 markQ.Answer each of the following question in one sentence each : What do you mean by median of a distribution ?
›Reveal solutionSolution
Median = the central value splitting ordered data into two equal halves.
When the observations are arranged in ascending (or descending) order, the median is the value of the middle item; exactly 50% of the observations fall below it and 50% above it. It is a positional average unaffected by …
- CBSE 2024Set ANNUAL1 markQ.Fill in the blanks : The median of 25, 27, 30, 35, 38, 32 and 40 is ________.
›Reveal solutionSolution
Median of the 7 ordered values is the 4th one =32.
…
- CBSE 2023Set ANNUAL1 markMCQQ.The median of 5, 15, 12, 7, 14 and 18 is equal to :(a) 12(b) 14(c) 13(d) 15
›Reveal solutionSolution
Order the data and average the two central values because n=6 is even; the median is 13.
This is a very-short-answer item from the Measures of Central Tendency chapter of the CHSE Odisha +2 Class-12 Business Mathematics & Statistics syllabus (aligned with the NCERT/CBSE commerce curriculum).
Step 1 — arrange the values in ascending order:
5, 7, 12, 14, 15, 18
Step 2 — count the observations: n=6 (even).
…
- CBSE 2023Set ANNUAL1 markQ.Fill in the blanks : Median is a ______ average.
›Reveal solutionSolution
The correct word is positional.
The median is the value of the middle item when the data are arranged in order. Its value is fixed by the position of the central observation, not by the numerical magnitude of every value — so it is called a positional average. (By contrast, the arithmeti …
- CBSE 2023Set ANNUAL1 markQ.Rectify the underlined portions of the following sentences : Second quartile and mean of a series are equal.
›Reveal solutionSolution
Replace mean with median: Q2 equals the median.
The quartiles divide an ordered series into four equal parts. The second quartile Q2 lies exactly at the half-way point of the distribution — which is precisely the definition of the median. Hence the second quartile and the …
- CBSE 2022Set ANNUAL1 markMCQQ.The median of 2,4,6,8,10,12,14,16,18,20 and 22 is :(a) Fifth value(b) Sixth value(c) Eighth value(d) Tenth value
›Reveal solutionSolution
Median position =2n+1=212=6th value.
The eleven values 2,4,6,…,22 are already arranged in ascending order. For an odd number of observations n=11, the median is the middle item at position …
- CBSE 2022Set ANNUAL1 markMCQQ.The median of 98.13, 98.01, 98.1, 98.42 and 98.08 is :(a) 98.08(b) 98.13(c) 98.01(d) 98.1
›Reveal solutionSolution
Ordered data: 98.01,98.08,98.1,98.13,98.42; median is the middle value 98.1.
Arrange in ascending order:
98.01,98.08,98.1,98.13,98.42.
…
- CBSE 2019Set ANNUAL1 markQ.Rectify the underlined portions of the following sentences:(i) Median is a <u>mathematical</u> average.
›Reveal solutionSolution
The correct word is positional — the median is a positional average, not a mathematical one.
Averages are of two broad kinds:
- Mathematical averages — computed using all the values through arithmetic operations: Arithmetic Mean, Geometric Mean, Harmonic Mean.
- Positional averages — located by their position in an arranged (ascending/descending) series: Median and Mode. …
- CBSE 2019Set ANNUAL1 markQ.Fill in the blanks:(viii) The median of 198,205,179,146,210,186 and 190 is __________.
›Reveal solutionSolution
Median =190.
Data: 198,205,179,146,210,186,190. Arrange in ascending order:
146,179,186,190,198,205,210
…
- CBSE 2019Set ANNUAL1 markMCQQ.(b) Median of 72,78,81,88,92,83 and 100 is(a) 88(b) 78(c) 81(d) 83
›Reveal solutionSolution
Correct option: (d) 83.
Data: 72,78,81,88,92,83,100. Arrange in ascending order:
72,78,81,83,88,92,100.
…
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