CBSE • Class 10 • Mathematics • Chapter 13 (Statistics) • Exercise 13.2

Exercise 13.2: Statistics — NCERT Solutions

Mode of grouped data using the standard formula.

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What this exercise covers

Modal classMode formulaComparing mean and mode

Step-by-step solutions — Exercise 13.2

3 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 13.2 Q1 • 3 marks

The following data gives the number of goals scored by teams in a tournament: 0–10 (f=7), 10–20 (f=14), 20–30 (f=13), 30–40 (f=12), 40–50 (f=20), 50–60 (f=11). Find the mode of the distribution.
Hint (Socratic — try this first)▾
Which class has the highest frequency, and how do its neighbouring frequencies affect the mode?
Step-by-step solution▾

Mode formula: Mode=l+(f1−f02f1−f0−f2)×h\text{Mode} = l + \left(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)\times h.

The modal class is the one with maximum frequency = 20, i.e. 40–50.

  • l=40l = 40 (lower boundary of modal class)
  • f1=20f_1 = 20 (frequency of modal class)
  • f0=12f_0 = 12 (frequency of preceding class)
  • f2=11f_2 = 11 (frequency of succeeding class)
  • h=10h = 10

Mode=40+(20−122(20)−12−11)×10=40+817×10\text{Mode} = 40 + \left(\frac{20-12}{2(20)-12-11}\right)\times 10 = 40 + \frac{8}{17}\times 10 =40+4.71=44.71= 40 + 4.71 = 44.71

The mode is approximately 44.71 goals.

Common mistake:
Choosing the wrong modal class (e.g. picking the middle class) or swapping f0f_0 and f2f_2.
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Exercise 13.2 Q2 • 3 marks

The ages (in years) of patients admitted in a hospital on a certain day are: 5–15 (f=6), 15–25 (f=11), 25–35 (f=21), 35–45 (f=23), 45–55 (f=14), 55–65 (f=5). Find the modal age.
Hint (Socratic — try this first)▾
After identifying the class with the greatest frequency, what are its lower boundary and the frequencies just before and after it?
Step-by-step solution▾

The maximum frequency is 23, so the modal class is 35–45.

  • l=35l = 35, f1=23f_1 = 23, f0=21f_0 = 21, f2=14f_2 = 14, h=10h = 10

Mode=35+(23−212(23)−21−14)×10\text{Mode} = 35 + \left(\frac{23-21}{2(23)-21-14}\right)\times 10 =35+246−35×10=35+211×10= 35 + \frac{2}{46-35}\times 10 = 35 + \frac{2}{11}\times 10 =35+1.82=36.82= 35 + 1.82 = 36.82

The modal age is approximately 36.82 years.

Common mistake:
Miscalculating the denominator 2f1−f0−f22f_1 - f_0 - f_2 by adding instead of subtracting one of the frequencies.
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Exercise 13.2 Q3 • 2 marks

For a distribution, the mean is 53.5 and the median is 52.4. Using the empirical relationship, estimate the mode.
Hint (Socratic — try this first)▾
Do you recall the approximate relation connecting the three measures of central tendency?
Step-by-step solution▾

Empirical relationship: Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Substitute Mean =53.5= 53.5 and Median =52.4= 52.4: Mode=3(52.4)−2(53.5)\text{Mode} = 3(52.4) - 2(53.5) =157.2−107=50.2= 157.2 - 107 = 50.2

The mode is approximately 50.2.

Common mistake:
Writing the formula wrongly as 2 Median−3 Mean2\,\text{Median} - 3\,\text{Mean} instead of 3 Median−2 Mean3\,\text{Median} - 2\,\text{Mean}.
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How to approach Exercise 13.2

  1. Re-read the chapter summary first. Open Statistics and refresh the key concepts: Grouped data, Mean, Median, Mode.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Statistics

  1. Exercise 13.1 — Mean of grouped data using the direct, assumed-mean and step-deviation methods.
  2. Exercise 13.2 — Mode of grouped data using the standard formula.
  3. Exercise 13.3 — Median of grouped data and cumulative-frequency interpretation.

← Exercise 13.1

Mean of grouped data using the direct, assumed-mean and step-deviation methods.

Exercise 13.3 →

Median of grouped data and cumulative-frequency interpretation.

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