Class 10th Statistics One Shot Class 10 Maths Chapter 13 Shobhit Nirwan

Statistics: Mean, Median, and Mode of Grouped Data

Statistics is the science of collecting, organising, analysing, and interpreting numerical data. In CBSE Class 10 Mathematics, Chapter 13 focuses on calculating the three measures of central tendency — mean, median, and mode — specifically for grouped (continuous) data presented in frequency distribution tables. While in earlier classes students dealt with ungrouped (raw) data, Class 10 introduces the more realistic scenario where large datasets are organised into class intervals, requiring specialised formulas and methods for accurate calculation.

The mean (average) of grouped data can be calculated using three methods. The direct method uses the formula: Mean = Σfᵢxᵢ / Σfᵢ, where xᵢ is the class mark (midpoint) of each class interval and fᵢ is its frequency. The class mark is calculated as (upper limit + lower limit)/2. When the class marks and frequencies involve large numbers, the assumed mean method is more efficient: Mean = a + Σfᵢdᵢ / Σfᵢ, where a is the assumed mean (usually the class mark of the middle class), dᵢ = xᵢ − a is the deviation of each class mark from a. The step-deviation (or coded) method further simplifies calculations by dividing each deviation by the class width h: Mean = a + h × (Σfᵢuᵢ / Σfᵢ), where uᵢ = (xᵢ − a)/h. This method is particularly useful when the class widths are uniform and the values are large. Students should verify which method gives the most convenient calculation for a given dataset.

The mode is the value that occurs most frequently in a dataset. For grouped data, we first identify the modal class — the class interval with the highest frequency — and then use the formula: Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class, f₁ is the frequency of the modal class, f₀ is the frequency of the class preceding the modal class, f₂ is the frequency of the class succeeding the modal class, and h is the class width. The median is the middle value when data is arranged in order. For grouped data, we first find the cumulative frequency of each class and identify the median class using N/2 (where N = Σfᵢ). The formula is: Median = l + [(N/2 − cf) / f] × h, where l is the lower limit of the median class, N is the total frequency, cf is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class width. An important empirical relationship connects all three measures: 3 × Median = Mode + 2 × Mean. This relationship is useful for estimating one measure when the other two are known. Ogive curves (cumulative frequency curves) provide a graphical method for finding the median — the intersection point of the "less than" and "more than" ogives gives the median value on the x-axis.

• Mean of grouped data: use direct method (Σfᵢxᵢ/Σfᵢ), assumed mean method, or step-deviation method depending on data size.
• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h — requires identifying the modal class with highest frequency.
• Median = l + [(N/2 − cf)/f] × h — requires cumulative frequencies to identify the median class.
• Empirical relationship: 3 Median = Mode + 2 Mean — useful for estimating one measure from the other two.
• Ogive curves (cumulative frequency graphs) visually determine the median at the intersection of "less than" and "more than" curves.