Questions
3 questions in typical board papers
Difficulty
Easy
Importance
Core — never skip
Overview
Statistics is the study of collecting, organizing, and interpreting numerical data to derive meaningful insights. For board exams, mastering measures of central tendency for grouped data is essential as it consistently forms high-scoring long-answer questions. The core objective is to move from raw data interpretation to accurate calculation of Mean, Median, and Mode using structured frequency distributions.
Mean of Grouped Data
Calculating the mean for grouped data involves handling class intervals rather than discrete values. Students must choose the most efficient method based on the magnitude of the class marks and frequencies.
- Direct Method: Mean = Σ(fi * xi) / Σfi
- Assumed Mean Method: Mean = a + Σ(fi * di) / Σfi
- Step-Deviation Method: Mean = a + [Σ(fi * ui) / Σfi] * h
- Class mark (xi) = (Upper Limit + Lower Limit) / 2
- Deviation (di) = xi - a
- Step-deviation (ui) = (xi - a) / h
Median of Grouped Data
The median represents the middle value of a dataset. Calculating it requires the cumulative frequency (cf) to identify the specific median class within the frequency table.
- Median Formula: l + [(N/2 - cf) / f] * h
- l is the lower limit of the median class
- N is the total frequency (Σfi)
- cf is the cumulative frequency of the class preceding the median class
- f is the frequency of the median class
- h is the class size
Mode and Empirical Relationship
The mode is the value that appears most frequently in a distribution. For grouped data, it is calculated using the modal class, while the empirical relationship allows for verification across all three measures.
- Mode Formula: l + [(f1 - f0) / (2f1 - f0 - f2)] * h
- f1 is frequency of the modal class
- f0 is frequency of the class preceding the modal class
- f2 is frequency of the class succeeding the modal class
- Empirical Relationship: 3 Median = Mode + 2 Mean
- Modal class is the class interval with the highest frequency
Formula Sheet
Mean = a + [Σ(fi * ui) / Σfi] * h
Median = l + [(N/2 - cf) / f] * h
Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] * h
3 Median = Mode + 2 Mean
Exam Tip
Always verify your calculations using the empirical formula (3 Median = Mode + 2 Mean) if you have time, as it quickly highlights arithmetic errors.
Common Mistakes
- Mistaking the cumulative frequency (cf) of the median class for the cf of the preceding class in the median formula.
- Forgetting to multiply the final result by the class size (h) when using the Step-Deviation method for Mean.
- Failing to arrange data in inclusive form; always convert exclusive intervals before calculating cumulative frequency.
More Revision Notes
Ready to test yourself?
Play topic-wise Statistics questions in Aspirant Arcade — gamified MCQ practice.
Download Free