How do you find the median of grouped data?

Step 1: Find n/2 (half of total frequency). Step 2: Build cumulative frequency column. Step 3: Find the median class — the class whose cumulative frequency first exceeds n/2. Step 4: Apply formula: Median = l + [(n/2 − cf)/f] × h, where l = lower boundary of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

What is the empirical relationship between mean, median, and mode?

The empirical (approximate) relationship is: Mode = 3 Median − 2 Mean. This can be rearranged to: Mean − Mode = 3(Mean − Median). This relationship holds for moderately skewed (asymmetric) distributions. It is useful in exams to find one measure when the other two are known. For a perfectly symmetric distribution, Mean = Median = Mode.

What is the assumed mean method and why is it used?

The assumed mean method simplifies the calculation of mean for grouped data with large class marks. Choose a convenient assumed mean 'a' (usually the middle class mark). Calculate deviations dᵢ = xᵢ − a for each class. Then Mean = a + (Σfᵢdᵢ/Σfᵢ). The step deviation method further simplifies by dividing deviations by class width h: uᵢ = dᵢ/h; Mean = a + (Σfᵢuᵢ/Σfᵢ) × h. These methods reduce arithmetic errors with large numbers.

Mean, Median, and Mode of Grouped Data

Statistics • Class 10 Mathematics • NCERT • CBSE

For grouped data: Mean by direct method = Σfᵢxᵢ / Σfᵢ. Median = l + [(n/2 − cf)/f] × h. Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h. Empirical relationship: Mode ≈ 3 Median − 2 Mean.

Key Formulas

Mean (direct) = Σfᵢxᵢ / Σfᵢ
Mean (assumed mean) = a + Σfᵢdᵢ / Σfᵢ
Median = l + [(n/2 − cf)/f] × h
Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
Mode = 3 Median − 2 Mean

Frequently Asked Questions

How do you find the median of grouped data?: Step 1: Find n/2 (half of total frequency). Step 2: Build cumulative frequency column. Step 3: Find the median class — the class whose cumulative frequency first exceeds n/2. Step 4: Apply formula: Median = l + [(n/2 − cf)/f] × h, where l = lower boundary of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.
What is the empirical relationship between mean, median, and mode?: The empirical (approximate) relationship is: Mode = 3 Median − 2 Mean. This can be rearranged to: Mean − Mode = 3(Mean − Median). This relationship holds for moderately skewed (asymmetric) distributions. It is useful in exams to find one measure when the other two are known. For a perfectly symmetric distribution, Mean = Median = Mode.
What is the assumed mean method and why is it used?: The assumed mean method simplifies the calculation of mean for grouped data with large class marks. Choose a convenient assumed mean 'a' (usually the middle class mark). Calculate deviations dᵢ = xᵢ − a for each class. Then Mean = a + (Σfᵢdᵢ/Σfᵢ). The step deviation method further simplifies by dividing deviations by class width h: uᵢ = dᵢ/h; Mean = a + (Σfᵢuᵢ/Σfᵢ) × h. These methods reduce arithmetic errors with large numbers.

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