Mean of Grouped Data MCQs Quiz | Class 10
Dive into the world of Statistics with this quiz on the Mean of Grouped Data, a key topic from Class X Mathematics (Code 041), Unit VII: Statistics & Probability. This quiz covers the Direct, Assumed Mean, and Step Deviation approaches, along with essential numerical problems. Test your knowledge and understanding of these methods. After attempting all 10 multiple-choice questions, submit your answers to see your score and review the correct solutions. You can also download a personalized PDF answer sheet for future reference.
Understanding the Mean of Grouped Data
In statistics, when data is presented in the form of a frequency distribution, it’s called grouped data. To find the mean of such data, we cannot use the simple formula for ungrouped data (sum of observations / number of observations) because individual observations are not known. Instead, we estimate the mean using the mid-points (class marks) of the class intervals.
There are three primary methods to calculate the mean of grouped data:
1. Direct Method
This is the simplest method and is preferred when the class marks (xi) and frequencies (fi) are small. The formula is:
Mean (x-bar) = sum(fi * xi) / sum(fi)
- xi (Class Mark): The mid-point of each class interval. Calculated as (Lower Limit + Upper Limit) / 2.
- fi (Frequency): The number of observations in each class interval.
Steps:
- Find the class mark (xi) for each class interval.
- Calculate the product of fi and xi for each class.
- Find the sum of all (fi * xi).
- Find the sum of all frequencies (sum(fi)).
- Divide sum(fi * xi) by sum(fi).
2. Assumed Mean Method (Shortcut Method)
This method is useful when the class marks (xi) and frequencies (fi) are large, making the calculations in the Direct Method tedious. It simplifies the calculation by assuming a mean and then adjusting it.
Mean (x-bar) = A + sum(fi * di) / sum(fi)
- A (Assumed Mean): A value chosen from the class marks (xi), preferably near the middle, to minimize calculations.
- di (Deviation): The difference between the class mark and the assumed mean (di = xi – A).
Steps:
- Find the class mark (xi) for each class interval.
- Choose an assumed mean (A) from the xi values.
- Calculate the deviation (di = xi – A) for each class.
- Calculate the product of fi and di for each class.
- Find the sum of all (fi * di).
- Find the sum of all frequencies (sum(fi)).
- Apply the formula: Mean = A + sum(fi * di) / sum(fi).
3. Step Deviation Method
This method is a further simplification of the Assumed Mean Method and is highly recommended when the class intervals are of uniform size (equal class width, ‘h’). It further reduces the numerical values involved in calculations.
Mean (x-bar) = A + (sum(fi * ui) / sum(fi)) * h
- A (Assumed Mean): Same as in the Assumed Mean Method.
- h (Class Size/Width): The uniform difference between the upper and lower limits of consecutive classes (or between consecutive class marks).
- ui (Step Deviation): Calculated as di / h, or (xi – A) / h.
Steps:
- Find the class mark (xi) for each class interval.
- Choose an assumed mean (A) from the xi values.
- Calculate the deviation (di = xi – A) for each class.
- Calculate the step deviation (ui = di / h) for each class.
- Calculate the product of fi and ui for each class.
- Find the sum of all (fi * ui).
- Find the sum of all frequencies (sum(fi)).
- Apply the formula: Mean = A + (sum(fi * ui) / sum(fi)) * h.
Comparison of Methods
| Method | When to Use | Formula |
|---|---|---|
| Direct Method | When fi and xi are small. | sum(fi * xi) / sum(fi) |
| Assumed Mean Method | When fi and xi are large. | A + sum(fi * di) / sum(fi) |
| Step Deviation Method | When fi and xi are large AND class size (h) is uniform. | A + (sum(fi * ui) / sum(fi)) * h |
Quick Revision Points
- The mean calculated by all three methods for the same data set will always be the same.
- Class mark is the representative value of a class interval.
- Assumed mean (A) is typically chosen from the middle xi value to keep deviations small.
- Class size (h) is the difference between the upper limit and the lower limit of a class.
- Statistics problems involving mean of grouped data often require careful calculation and attention to signs (positive/negative deviations).
Extra Practice Questions (Solutions not provided here)
- Find the mean for the following distribution using the Direct Method:
Class: 0-10, 10-20, 20-30, 30-40, 40-50
Frequency: 5, 8, 12, 10, 5
- Using the Assumed Mean Method, find the mean of the marks obtained by 100 students:
Marks: 0-10, 10-20, 20-30, 30-40, 40-50
Number of students: 10, 20, 30, 25, 15
(Take Assumed Mean A = 25)
- Calculate the mean daily wages of the following data using the Step Deviation Method:
Daily Wages (Rs): 100-120, 120-140, 140-160, 160-180, 180-200
Number of Workers: 12, 14, 8, 6, 10
- What are the advantages of using the Step Deviation Method over the Direct Method?
- If the mean of the following distribution is 62.8, find the missing frequency ‘p’:
Class: 0-20, 20-40, 40-60, 60-80, 80-100, 100-120
Frequency: 5, 8, 12, p, 7, 8
Content created and reviewed by the CBSE Quiz Editorial Team based on the latest NCERT textbooks and CBSE syllabus. Our goal is to help students practice concepts clearly, confidently, and exam-ready through well-structured MCQs and revision content.