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CBSE · Class 10 · Mathematics

Statistics 13.3: Mode of Grouped Data

5 concepts · 40 practice questions with answers · Free to read

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Modal class and the symbols in the mode formula

From daily life

The most popular shoe size in a shop is the mode. In grouped data we first find the most crowded class, then place the mode inside it using the neighbours.

Simple definition

The modal class is the class with the greatest frequency. The mode is l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h, where l is the lower limit of the modal class, h its width, f₁ its frequency, f₀ the frequency of the class before it and f₂ that of the class after it.

Formula: mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h

34106501020304050modal class20–30l = 20, h = 10mode =l + ((f₁ − f₀) ÷(2f₁ − f₀ − f₂)) × h

In the figure: Frequencies 3, 4, 10, 6, 5: the modal class is 20–30.

Rule 1: Meaning of each symbol — Three frequencies and two numbers from the modal class.

  • Easy: f₁ = 10 is the frequency of the modal class.
  • Medium: f₀ = 4 comes from the class 10–20 before it.
  • Exam-type: f₂ = 6 comes from the class 30–40 after it.

Rule 2: Why the neighbours matter — The mode leans towards the busier neighbour.

  • Easy: If f₀ is bigger than f₂, the mode moves toward the lower end.
  • Medium: If f₂ is bigger, it moves toward the upper end.
  • Exam-type: If f₀ = f₂, the mode is exactly the middle of the class.
Remember

Check that the mode lies inside the modal class.

Common mistake

Writing f₀ and f₂ the wrong way round.

Memory clue

"Before, biggest, after: f-zero, f-one, f-two."

Practice questions

Easy
  1. 1The modal class of a distribution is 60–70. State its class width h.
  2. 2The modal class of a distribution is 90–100. State its lower limit l.
  3. 3The modal class of a distribution is 165–180. State its upper limit.
Medium
  1. 4In a grouped distribution the modal class has l = 150, h = 15, f₁ = 31, f₀ = 7 and f₂ = 10. Find the mode.
  2. 5In a grouped distribution the modal class has l = 180, h = 15, f₁ = 28, f₀ = 2 and f₂ = 4. Find the mode.
  3. 6For a modal class, f₁ = 13, f₀ = 2 and f₂ = 3. Find the denominator 2f₁ − f₀ − f₂ of the mode formula.
Challenging
  1. 7For a modal class, f₁ = 29, f₀ = 22 and f₂ = 2. Find the denominator 2f₁ − f₀ − f₂ of the mode formula.
  2. 8For a modal class, f₁ = 16, f₀ = 13 and f₂ = 9. Find the denominator 2f₁ − f₀ − f₂ of the mode formula.
Show answers (try them yourself first)
  1. 110
  2. 290
  3. 3180
  4. 4158
  5. 5187.8
  6. 621
  7. 734
  8. 810

Finding the modal class, f₀, f₁, f₂

From daily life

Look down the frequency column for the tallest bar; its left neighbour is f₀ and its right neighbour is f₂.

Simple definition

Find the largest f; that class is the modal class and its frequency is f₁. The frequency just above it in the table is f₀ and just below it is f₂. If the modal class is the first class, f₀ = 0; if it is the last, f₂ = 0.

341065f₀f₁f₂f₁ = largest = 10f₀ = before = 4f₂ = after = 6modal class = 20–30

In the figure: f₁ = 10 (class 20–30), f₀ = 4 (class 10–20), f₂ = 6 (class 30–40).

Rule 1: Pick the three frequencies — Start from the largest frequency.

  • Easy: f = 3, 4, 10, 6, 5: the largest is 10, so f₁ = 10.
  • Medium: The class above has 4, so f₀ = 4.
  • Exam-type: The class below has 6, so f₂ = 6.

Rule 2: Edge cases — Missing neighbours count as zero.

  • Easy: f = 9, 6, 4, 2: modal class is the first, f₀ = 0.
  • Medium: f = 2, 5, 8: modal class is the last, f₂ = 0.
  • Exam-type: Two equal biggest frequencies mean two modal classes; the question will give more help.
Remember

Underline the largest frequency first, then circle the two neighbours.

Common mistake

Picking f₁ as the class with the largest class mark instead of the largest frequency.

Memory clue

"Tallest bar, its two neighbours."

Practice questions

Easy
  1. 1The following distribution of daily wages of workers (in ₹) is given: class intervals 300–320, 320–340, 340–360, 360–380, 380–400 with frequencies 6, 7, 6, 8, 5 respectively. Find the modal class.
  2. 2The following distribution of daily wages of workers (in ₹) is given: class intervals 300–310, 310–320, 320–330, 330–340, 340–350 with frequencies 3, 23, 13, 5, 6 respectively. Find f₀ (frequency of the class before the modal class).
  3. 3The following distribution of marks of students is given: class intervals 20–25, 25–30, 30–35, 35–40, 40–45 with frequencies 2, 9, 5, 1, 3 respectively. Find f₀ (frequency of the class before the modal class).
Medium
  1. 4The following distribution of ages of persons (in years) is given: class intervals 10–15, 15–20, 20–25, 25–30, 30–35 with frequencies 1, 2, 1, 14, 2 respectively. Find f₂ (frequency of the class after the modal class).
  2. 5The following distribution of ages of persons (in years) is given: class intervals 20–30, 30–40, 40–50, 50–60, 60–70 with frequencies 2, 11, 4, 1, 2 respectively. Find the modal class.
  3. 6The following distribution of monthly electricity consumption of households (in units) is given: class intervals 100–120, 120–140, 140–160, 160–180 with frequencies 1, 5, 1, 1 respectively. Find f₁ (frequency of the modal class).
Challenging
  1. 7The following distribution of daily wages of workers (in ₹) is given: class intervals 200–220, 220–240, 240–260, 260–280, 280–300, 300–320 with frequencies 13, 21, 11, 40, 10, 5 respectively. Find the modal class.
  2. 8The following distribution of marks of students is given: class intervals 0–5, 5–10, 10–15, 15–20, 20–25 with frequencies 1, 2, 4, 2, 1 respectively. Find f₁ (frequency of the modal class).
Show answers (try them yourself first)
  1. 1360–380
  2. 23
  3. 32
  4. 42
  5. 530–40
  6. 65
  7. 7260–280
  8. 84

Computing the mode from a table

From daily life

Once the three frequencies are known, the mode comes from one subtraction on top, one on the bottom and a multiplication by the class width.

Simple definition

Put l, h, f₁, f₀, f₂ in mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h. The denominator is the sum of the two differences, f₁ − f₀ and f₁ − f₂.

Formula: mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h

modal class 20–30: l = 20, h = 10f₁ = 10, f₀ = 4, f₂ = 6mode = 20 + 6 ÷ (20 − 4 − 6) × 10mode = 26

In the figure: mode = 20 + 6 ÷ (20 − 4 − 6) × 10 = 26.

Live figure

Computing the mode from a table

In the app you can move this figure with your finger.

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Rule 1: Worked example 1 — Classes 0–10 … 40–50 with f = 3, 4, 10, 6, 5.

  • Easy: l = 20, h = 10, f₁ = 10, f₀ = 4, f₂ = 6.
  • Medium: Numerator 10 − 4 = 6; denominator 20 − 4 − 6 = 10.
  • Exam-type: mode = 20 + (6/10) × 10 = 26.

Rule 2: Worked example 2 — Classes 10–20 … 40–50 with f = 5, 12, 8, 3.

  • Easy: Modal class 20–30: l = 20, f₁ = 12, f₀ = 5, f₂ = 8.
  • Medium: mode = 20 + 7/(24 − 5 − 8) × 10 = 20 + 70/11.
  • Exam-type: mode ≈ 20 + 6.36 = 26.36.
Remember

Write numerator and denominator separately before dividing.

Common mistake

Using f₁ − f₀ − f₂ instead of 2f₁ − f₀ − f₂ in the denominator.

Memory clue

"Gap above over total gap, times width."

Practice questions

Easy
  1. 1The following distribution of weights of students (in kg) is given: class intervals 40–50, 50–60, 60–70, 70–80 with frequencies 15, 5, 20, 10 respectively. Find the mode of the data.
  2. 2The following distribution of weights of students (in kg) is given: class intervals 40–50, 50–60, 60–70, 70–80 with frequencies 7, 10, 1, 2 respectively. Find the mode of the data.
  3. 3The following distribution of monthly electricity consumption of households (in units) is given: class intervals 60–100, 100–140, 140–180, 180–220 with frequencies 2, 4, 16, 3 respectively. Find the mode of the data.
Medium
  1. 4The following distribution of marks of students is given: class intervals 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 8, 1, 1, 9, 1 respectively. Find the mode of the data.
  2. 5The following distribution of daily wages of workers (in ₹) is given: class intervals 200–250, 250–300, 300–350, 350–400, 400–450 with frequencies 11, 1, 11, 12, 5 respectively. Find the mode of the data.
  3. 6The following distribution of monthly electricity consumption of households (in units) is given: class intervals 60–100, 100–140, 140–180, 180–220 with frequencies 2, 9, 13, 1 respectively. Find the mode of the data.
Challenging
  1. 7The following distribution of daily wages of workers (in ₹) is given: class intervals 200–210, 210–220, 220–230, 230–240 with frequencies 2, 9, 6, 3 respectively. Find the mode of the data.
  2. 8The following distribution of monthly electricity consumption of households (in units) is given: class intervals 100–140, 140–180, 180–220, 220–260 with frequencies 7, 1, 11, 1 respectively. Find the mode of the data.
Show answers (try them yourself first)
  1. 166
  2. 252.5
  3. 3159.2
  4. 435
  5. 5356.25
  6. 6150
  7. 7217
  8. 8200

Working backwards with the mode formula

From daily life

If the mode is known and one neighbour’s frequency is lost, the equation tells us what was lost.

Simple definition

Write the unknown frequency as x, put it in the mode formula, set the result equal to the given mode, and solve the equation. The answer must be a whole number and must keep the modal class unchanged.

Formula: solve: l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h = mode

f₀ = x, f₁ = 10, f₂ = 6, mode = 2620 + (10 − x) ÷ (14 − x) × 10 = 2610 − x = 0.6 (14 − x)x = 4

In the figure: Mode 26, f₁ = 10, f₂ = 6 give f₀ = 4.

Rule 1: Find f₀ — Modal class 20–30 with the mode 26 given.

  • Easy: (10 − x) ÷ (20 − x − 6) × 10 = 26 − 20 = 6.
  • Medium: 10 − x = 0.6 (14 − x) = 8.4 − 0.6x.
  • Exam-type: 1.6 = 0.4x, so x = 4.

Rule 2: Verify — Put the value back and recompute the mode.

  • Easy: With f₀ = 4: mode = 20 + 6/10 × 10 = 26.
  • Medium: The class 20–30 is still the modal class, since 10 is the largest.
  • Exam-type: The answer 4 is a whole number.
Remember

Always put the answer back to confirm the mode you were given.

Common mistake

Letting the unknown change the modal class without checking.

Memory clue

"Make the equation, solve, plug back."

Practice questions

Easy
  1. 1A distribution has class width 20, f₁ = 23, f₀ = 10, f₂ = 11 and mode 110.4. Find the lower limit l of the modal class.
  2. 2A distribution has class width 10, f₁ = 10, f₀ = 9, f₂ = 6 and mode 42. Find the lower limit l of the modal class.
  3. 3A distribution has class width 10, f₁ = 13, f₀ = 10, f₂ = 8 and mode 83.75. Find the lower limit l of the modal class.
Medium
  1. 4In a distribution the modal class is 50–60 with f₀ = 15 and f₂ = 9. If the mode is 53.75, find f₁.
  2. 5A distribution has class width 20, f₁ = 10, f₀ = 5, f₂ = 7 and mode 152.5. Find the lower limit l of the modal class.
  3. 6In a distribution the modal class is 180–200 with f₀ = 13 and f₂ = 13. If the mode is 190, find f₁.
Challenging
  1. 7In a modal class 60–80, the frequencies of the two neighbouring classes are equal (f₀ = f₂ = 14) and f₁ = 28. Find the mode.
  2. 8For class intervals 300–310, 310–320, 320–330, 330–340, 340–350 with frequencies 13, 11, x, 2, 4, the mode is 323.2 and the modal class has frequency x. Find x.
Show answers (try them yourself first)
  1. 1100
  2. 240
  3. 380
  4. 424
  5. 5140
  6. 619
  7. 770
  8. 819

Mean, median, mode relation and applications

From daily life

In a tidy bell-shaped data the three averages coincide; in lopsided data they spread out in a fixed order.

Simple definition

For a moderately skewed distribution the empirical relation is mode = 3 × median − 2 × mean. Use it to find any one of the three when the other two are known. The mode suits “most common” questions, the mean suits totals and the median suits data with extreme values.

Formula: mode = 3 median − 2 mean

1015202530mode 18median 22mean 24Mode = 3 Median − 2 Mean3 × 22 − 2 × 24 = 66 − 48 = 18

In the figure: Median 22 and mean 24 give mode = 66 − 48 = 18.

Rule 1: Use the relation — Rearrange for whichever is unknown.

  • Easy: Mean 24, median 22: mode = 3 × 22 − 2 × 24 = 18.
  • Medium: Mode 18, mean 24: median = (18 + 48) ÷ 3 = 22.
  • Exam-type: Mode 18, median 22: mean = (66 − 18) ÷ 2 = 24.

Rule 2: Which average where — Match the question to the average.

  • Easy: A shop orders the most-sold shoe size: use the mode.
  • Medium: Total marks of a class: use the mean.
  • Exam-type: Typical house price when a few are very costly: use the median.
Remember

Use the relation only as a check or when data is moderately skewed.

Common mistake

Swapping mean and median in 3 × median − 2 × mean.

Memory clue

"Mode: three medians, minus two means."

Practice questions

Easy
  1. 1For a moderately skewed distribution, the mean is 44 and the median is 40. Find the mode using the empirical relation.
  2. 2For a moderately skewed distribution, the mean is 79 and the median is 71. Find the mode using the empirical relation.
  3. 3For a moderately skewed distribution, the mean is 64 and the median is 56. Find the mode using the empirical relation.
Medium
  1. 4For a moderately skewed distribution, the mode is 34 and the median is 44. Find the mean.
  2. 5For a moderately skewed distribution, the mode is 64 and the median is 44. Find the mean.
  3. 6For a moderately skewed distribution, the mode is 79 and the mean is 43. Find the median.
Challenging
  1. 7For the test scores of a class, the mean is 29 and the median is 25. Estimate the mode using 3 Median = Mode + 2 Mean.
  2. 8For a moderately skewed distribution the mean is 74 and the median is 71. Find the sum mode + median + mean.
Show answers (try them yourself first)
  1. 132
  2. 255
  3. 340
  4. 449
  5. 534
  6. 655
  7. 717
  8. 8210