Statistics
Statistics deals with summarising data using a single representative value. In Class 10 you learn to find the mean, median and mode of grouped data, and the empirical relationship that links the three. The formulas look long but follow a fixed recipe, so practice makes this a dependable scoring chapter.
Learning objectives
- Find the mean of grouped data by the direct and assumed-mean methods.
- Identify the modal class and compute the mode.
- Identify the median class and compute the median.
- Use the empirical relationship between mean, median and mode.
- Interpret what each measure tells you about the data.
Key concepts
Mean of grouped data
Use the class marks (midpoints) xᵢ with frequencies fᵢ. Direct method: mean = Σfᵢxᵢ / Σfᵢ. The assumed-mean method, mean = a + Σfᵢdᵢ/Σfᵢ (with dᵢ = xᵢ − a), simplifies the arithmetic.
Mode of grouped data
The modal class is the class with the highest frequency. Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class, f₁ its frequency, f₀ and f₂ the frequencies of the classes before and after, and h the class size.
Median of grouped data
The median class is the one in which the cumulative frequency first reaches n/2. Median = l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency, and h the class size.
Empirical relationship
For moderately skewed data, the three measures are linked by: Mode = 3 Median − 2 Mean. It lets you find any one of them when the other two are known.
Important formulas
Mean (direct)
Σfᵢxᵢ / Σfᵢ
Mean (assumed)
a + Σfᵢdᵢ / Σfᵢ
Mode
l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
Median
l + [(n/2 − cf)/f] × h
Empirical relation
Mode = 3 Median − 2 Mean
Key definitions
- Class mark
- The midpoint of a class interval, (lower limit + upper limit)/2.
- Modal class
- The class interval with the highest frequency.
- Cumulative frequency
- The running total of frequencies up to and including a class.
Solved examples
Q1. Find the mean of the data: classes 0–10, 10–20, 20–30 with frequencies 4, 6, 10.
Solution: Class marks: 5, 15, 25. Σfᵢxᵢ = 4(5) + 6(15) + 10(25) = 20 + 90 + 250 = 360. Σfᵢ = 20. Mean = 360/20 = 18.
Q2. In a data set, the mean is 30 and the median is 28. Estimate the mode.
Solution: Mode = 3 Median − 2 Mean = 3(28) − 2(30) = 84 − 60 = 24.
Q3. What is the class mark of the interval 20–30?
Solution: Class mark = (20 + 30)/2 = 25.
Common mistakes to avoid
- Using class limits instead of class marks (midpoints) for the mean.
- Picking the modal class by cumulative frequency instead of by the highest frequency.
- Forgetting to use n/2 (not the highest cumulative frequency) to find the median class.
- Misremembering the empirical relation — it is Mode = 3 Median − 2 Mean.
Statistics — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The mean of grouped data by the direct method is:
Practice questions
Short answer
Write the formula for the median of grouped data.
Median = l + [(n/2 − cf)/f] × h.
How do you find the class mark?
Class mark = (lower limit + upper limit)/2.
If mode = 24 and mean = 30, find the median.
From Mode = 3 Median − 2 Mean: 24 = 3M − 60 ⇒ 3M = 84 ⇒ M = 28.
Long answer
Find the mean of: 0–20 (f=7), 20–40 (f=10), 40–60 (f=8) using the direct method.
Class marks 10, 30, 50. Σfx = 7(10)+10(30)+8(50)=70+300+400=770. Σf = 25. Mean = 770/25 = 30.8.
The frequencies of classes 0–10, 10–20, 20–30, 30–40 are 6, 10, 8, 6. Find the modal class and explain how you would compute the mode.
The highest frequency is 10 in the class 10–20, so that is the modal class. Then apply Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]h with l = 10, f₁ = 10, f₀ = 6, f₂ = 8, h = 10, giving Mode = 10 + [(10 − 6)/(20 − 6 − 8)]×10 = 10 + (4/6)×10 ≈ 16.67.
HOTS (Higher Order Thinking)
Can the mean, median and mode of a data set all be equal? When?
Yes — for a perfectly symmetric distribution (such as a normal-shaped data set), the mean, median and mode coincide.
Why is the median sometimes preferred over the mean?
The median is not affected by extreme values (outliers), so it represents skewed data better than the mean.
Quick revision
Revision notes
- Mean (direct) = Σfᵢxᵢ / Σfᵢ; use class marks.
- Mode uses the modal class (highest frequency).
- Median uses the median class (where cf reaches n/2).
- Empirical: Mode = 3 Median − 2 Mean.
Key takeaways
- Always use class marks, not class limits, for the mean.
- Build a cumulative-frequency column for the median.
- The empirical relation links all three measures.
Frequently asked questions
What is the difference between mean, median and mode?
The mean is the average, the median is the middle value, and the mode is the most frequent value.
Which method is easiest for the mean?
The assumed-mean method reduces large multiplications and is often the quickest for grouped data.
How do I choose the median class?
Find n/2, then locate the class where the cumulative frequency first reaches or exceeds n/2.