What is the formula for mean of grouped data?

Mean = Σ(fᵢxᵢ) / Σfᵢ where fᵢ is frequency and xᵢ is class mark (midpoint). This is the Direct Method.

What is the median formula for grouped data?

Median = L + [(n/2 − cf) / f] × h where L = lower class boundary of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

What is the mode formula for grouped data?

Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h where L = lower boundary of modal class, f₁ = highest frequency, f₀ = frequency before modal class, f₂ = frequency after modal class, h = class width.

Class 10 Maths Chapter 6 Statistics Solutions | Maharashtra Board SSC

This page provides complete solutions for Maharashtra Board Class 10 Maths Chapter 6 — Statistics. Statistics in Class 10 covers measures of central tendency — Mean, Median, and Mode — for grouped data (data arranged in frequency tables). These three averages are essential for understanding data and are heavily tested in board exams.

📋 Table of Contents

Introduction to Statistics

Statistics is the branch of mathematics dealing with collecting, organising, and interpreting data. In Class 10, we focus on three measures of central tendency for grouped data (data given in class intervals with frequencies).

🔢 Statistics — All Key Formulas

Class Mark (midpoint): xᵢ = (upper limit + lower limit) / 2
MEAN (Direct Method): x̄ = Σ(fᵢxᵢ) / Σfᵢ
MEAN (Assumed Mean): x̄ = A + Σ(fᵢdᵢ)/Σfᵢ where dᵢ = xᵢ − A
MEAN (Step Deviation): x̄ = A + (Σfᵢuᵢ/Σfᵢ) × h where uᵢ = (xᵢ−A)/h
MEDIAN: M = L + [(n/2 − cf) / f] × h
MODE: Mo = L + [(f₁−f₀) / (2f₁−f₀−f₂)] × h
Empirical Relation: Mode = 3 Median − 2 Mean

Mean of Grouped Data

The mean (arithmetic average) for grouped data is calculated by multiplying each class mark by its frequency, adding all products, then dividing by total frequency.

Practice Set 6.1 — Mean Solutions

Q: Find the mean of the following data: Class: 0-10, 10-20, 20-30, 30-40, 40-50. Frequency: 5, 10, 25, 30, 10

Step 1: Find class marks (midpoints): x₁=5, x₂=15, x₃=25, x₄=35, x₅=45

Step 2: Calculate fᵢxᵢ: 5×5=25, 10×15=150, 25×25=625, 30×35=1050, 10×45=450

Step 3: Σfᵢxᵢ = 25+150+625+1050+450 = 2300

Step 4: Σfᵢ = 5+10+25+30+10 = 80

Step 5: Mean = Σfᵢxᵢ / Σfᵢ = 2300/80 = 28.75

✅ Answer: Mean = 28.75

Q: Find mean using Step Deviation method: Class: 10-20, 20-30, 30-40, 40-50, 50-60. Frequency: 4, 8, 14, 10, 4

Step 1: Class marks: 15, 25, 35, 45, 55. Assume A = 35, h = 10

Step 2: uᵢ = (xᵢ − 35)/10: −2, −1, 0, 1, 2

Step 3: fᵢuᵢ: 4×(−2)=−8, 8×(−1)=−8, 14×0=0, 10×1=10, 4×2=8

Step 4: Σfᵢuᵢ = −8−8+0+10+8 = 2

Step 5: Σfᵢ = 4+8+14+10+4 = 40

Step 6: Mean = A + (Σfᵢuᵢ/Σfᵢ)×h = 35 + (2/40)×10 = 35 + 0.5 = 35.5

✅ Answer: Mean = 35.5

Median of Grouped Data

The median is the middle value. For grouped data, first find the median class (where cumulative frequency ≥ n/2), then apply the formula: M = L + [(n/2 − cf) / f] × h

Practice Set 6.2 — Median Solutions

Q: Find the median: Marks: 0-10, 10-20, 20-30, 30-40, 40-50. Frequency: 3, 5, 7, 4, 1. Total n = 20

Step 1: Cumulative frequencies: 3, 8, 15, 19, 20

Step 2: n = 20, n/2 = 10

Step 3: Median class: cf just below 10 is 8 (0-10 class), so median class is 20-30 (where cf becomes 15 ≥ 10)

Step 4: L = 20, cf = 8 (cumulative before 20-30), f = 7, h = 10

Step 5: Median = 20 + [(10 − 8) / 7] × 10 = 20 + (2/7)×10 = 20 + 2.86 = 22.86

✅ Answer: Median ≈ 22.86

Q: Find the median: Class: 100-120, 120-140, 140-160, 160-180, 180-200. Frequency: 12, 14, 8, 6, 10. n = 50

Step 1: Cumulative frequencies: 12, 26, 34, 40, 50

Step 2: n/2 = 25

Step 3: Cumulative frequency first exceeds 25 at 26 (class 120-140). Median class = 120-140

Step 4: L = 120, cf = 12, f = 14, h = 20

Step 5: Median = 120 + [(25−12)/14] × 20 = 120 + (13/14)×20 = 120 + 18.57 = 138.57

✅ Answer: Median ≈ 138.57

Mode of Grouped Data

The mode is the most frequently occurring value. For grouped data, the modal class is the class with the highest frequency. The mode formula gives the exact value within the modal class.

Practice Set 6.3 — Mode Solutions

Q: Find the mode: Age: 0-10, 10-20, 20-30, 30-40, 40-50. Frequency: 5, 8, 15, 10, 2

Step 1: Modal class = 20-30 (highest frequency = 15)

Step 2: L = 20, f₁ = 15, f₀ = 8 (frequency before), f₂ = 10 (frequency after), h = 10

Step 3: Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h

Step 4: = 20 + [(15−8)/(30−8−10)] × 10

Step 5: = 20 + [7/12] × 10

Step 6: = 20 + 5.83 = 25.83

✅ Answer: Mode ≈ 25.83

Q: Find mode: Wages (₹): 120-140, 140-160, 160-180, 180-200, 200-220. Workers: 6, 9, 17, 12, 4

Step 1: Modal class = 160-180 (highest frequency = 17)

Step 2: L = 160, f₁ = 17, f₀ = 9, f₂ = 12, h = 20

Step 3: Mode = 160 + [(17−9)/(34−9−12)] × 20

Step 4: = 160 + [8/13] × 20

Step 5: = 160 + 12.31 = 172.31

✅ Answer: Mode ≈ ₹172.31

Practice Set 6.4 — Cumulative Frequency and Ogive

A cumulative frequency table shows the running total of frequencies. An ogive (cumulative frequency curve) is the graph of cumulative frequencies against upper class boundaries. It is used to find the median graphically.

Q: Draw a less-than ogive for: Class: 0-5, 5-10, 10-15, 15-20, 20-25. Frequency: 3, 7, 10, 8, 2

Step 1: Less-than cumulative frequencies: below 5: 3, below 10: 10, below 15: 20, below 20: 28, below 25: 30

Step 2: Plot points: (5,3), (10,10), (15,20), (20,28), (25,30)

Step 3: Join the points with a smooth curve — this is the less-than ogive.

Step 4: To find median from ogive: Draw horizontal line at n/2 = 15, note the x-value — this is the median.

Step 5: From the ogive, median ≈ 13 (where cumulative frequency = 15)

✅ Answer: Ogive plotted; Median ≈ 13 (read from graph)

Relation Between Mean, Median and Mode

🔢 Empirical Relationship

Mode = 3 Median − 2 Mean
This relation holds approximately for moderately skewed distributions
If Mean = Median = Mode → data is symmetrically distributed
Use this formula to find one average when other two are known

Q: If Mean = 26 and Median = 23, find Mode using empirical relation.

Step 1: Mode = 3 Median − 2 Mean

Step 2: Mode = 3(23) − 2(26) = 69 − 52 = 17

✅ Answer: Mode = 17

Frequently Asked Questions

What is the difference between Mean, Median and Mode?

Mean is the arithmetic average of all values. Median is the middle value when data is arranged in order. Mode is the most frequently occurring value. All three are measures of central tendency but describe different aspects of the data.

When should I use Assumed Mean method instead of Direct Method?

Use the Assumed Mean (or Step Deviation) method when the class marks are large numbers. It reduces computation by working with smaller numbers (deviations from an assumed mean). Both methods give the same answer.

How many marks does Chapter 6 Statistics carry in SSC board exam?

Statistics carries 8 to 10 marks in the SSC Algebra paper — making it one of the highest-weightage chapters. Typically there is one full problem (5-6 marks) requiring a frequency table with all calculations for Mean, Median, and/or Mode.

What is an ogive and how is it used in exams?

An ogive is a cumulative frequency curve (S-shaped). To draw it: plot cumulative frequency vs upper class boundary, then join with smooth curve. In exams, you may be asked to find the median from the ogive, or draw both less-than and more-than ogives and find their intersection (which gives the median).