Mean, Median, Ogive and Mode is an important Statistics-based topic for WBBSE Madhyamik Mathematics. This chapter introduces different measures of central tendency and graphical representation of cumulative frequency data.
This revision article covers important MCQs, True or False questions, fill-in-the-blanks and numerical problems. The solutions are arranged step by step so that students can understand how each answer is obtained.
Mean, Median and Mode – Basic Concepts
Mean, Median and Mode are three important measures of central tendency. They help us represent a collection of observations using a suitable central value.
| Measure | Meaning |
|---|---|
| Mean | The sum of all observations divided by the total number of observations. |
| Median | The middle value after arranging the observations in ascending or descending order. |
| Mode | The value or class having the highest frequency. |
| Ogive | A graph representing cumulative frequency. |
Important Formulae
Multiple Choice Questions – 1 Mark
1. If the mean of a frequency distribution is 30 and its mode is 24, what is the median?
(a) 28 (b) 27 (c) 26 (d) 25
2. If every observation x₁, x₂, x₃, …, xₙ is multiplied by p, what will happen to the mean?
(a) x̄/p (b) px̄ (c) x̄ (d) p + x̄
Multiplying every observation by p also multiplies the mean by p.
3. If the mean of a distribution is 40 and its median is 34, find the mode.
(a) 30 (b) 35 (c) 45 (d) 15
Note: The numerical options in the source do not match this calculation. Using the standard empirical relation, the result is 22.
4. Find the arithmetic mean of the first five prime numbers.
(a) 5.6 (b) 5.4 (c) 5 (d) 3.6
5. Find the mean of the first 10 odd natural numbers.
(a) 5 (b) 10 (c) 20 (d) 10.5
The first 10 odd natural numbers are 1, 3, 5, …, 19. Their mean is:
6. If y = 3x − 5 and the mode of x is 15, find the mode of y.
(a) 35 (b) 45 (c) 40 (d) None
7. The algebraic sum of deviations of all observations from their arithmetic mean is always:
(a) Equal to the number of observations (b) Positive (c) Negative (d) Zero
8. If the mean of n natural numbers is (n + 10)/4, find n.
(a) 8 (b) 9 (c) 10 (d) 11
9. If y = 2x − 5 and the median of x is 16, find the median of y.
(a) 25 (b) 27 (c) 32 (d) 28
True or False – 1 Mark
1. The mean and median of 5, 5, 6, 7, 7 are equal.
2. The graph obtained from cumulative frequency data is called an Ogive.
3. The assumed mean method is a comparatively convenient method for finding the mean of grouped data.
4. The median of 3, 4, 18, 20, 5 is 18.
Arranging the values: 3, 4, 5, 18, 20. The middle value is 5.
5. The class having the lowest frequency is called the modal class.
Fill in the Blanks – 1 Mark
1. When the mean, median and mode of a distribution coincide, the distribution is ________.
2. Mean, median and mode are measures of ________ tendency.
3. A distribution in which the frequencies decrease equally on both sides of the maximum frequency is called a ________ distribution.
4. In the step-deviation method, the class intervals generally have ________ class width.
5. Mean, median and mode are measures of ________ tendency.
Short Answer Questions – 2 Marks
Question 1: Mean Age of 40 Students
The ages of 40 friends are given below. Find their mean age using the direct method.
| Age (Years) | Number of Friends (f) | fx |
|---|---|---|
| 15 | 4 | 60 |
| 16 | 7 | 112 |
| 17 | 10 | 170 |
| 18 | 10 | 180 |
| 19 | 5 | 95 |
| 20 | 4 | 80 |
| Total | 40 | 697 |
Therefore, the mean age is approximately 17.43 years.
Question 2: Mean Age of 100 Patients
The following table gives the age distribution of 100 patients. Find their mean age.
| Age Group | Frequency | Class Mark (x) | fx |
|---|---|---|---|
| 10–20 | 12 | 15 | 180 |
| 20–30 | 8 | 25 | 200 |
| 30–40 | 22 | 35 | 770 |
| 40–50 | 20 | 45 | 900 |
| 50–60 | 18 | 55 | 990 |
| 60–70 | 20 | 65 | 1300 |
| Total | 100 | — | 4340 |
Therefore, the mean age of the patients is 43.4 years.
Question 3: Find the Median
The number of days attended by some students in June is given as:
Find the median.
Therefore, the median is 19 days.
Question 4: Find the 16th Number
The mean of 31 numbers is 50. The mean of the first 16 numbers is 46 and the mean of the last 16 numbers is 53. Find the 16th number.
The 16th number occurs in both groups.
Therefore, the 16th number is 34.
Question 5: Find the Median of Nine Observations
Find the median of:
Median = 15
Question 6: Effect of Multiplication on Mean
The mean of 10 numbers is 20. If every number is multiplied by 4, find the mean of the new numbers.
When every observation is multiplied by the same number, the mean is also multiplied by that number.
Therefore, the new mean is 80.
Question 7: Find the Median of Ten Observations
Find the median of:
Therefore, the median is 14.
Ogive – Cumulative Frequency Curve
An Ogive is a graph drawn using cumulative frequencies. It provides a graphical representation of a frequency distribution and is useful for studying cumulative data.
Types of Ogive
- Less-than Ogive: Constructed using less-than cumulative frequencies.
- More-than Ogive: Constructed using more-than cumulative frequencies.
The two curves can be used together to locate the median graphically.
Basic Steps for Drawing an Ogive
- Prepare the cumulative frequency table.
- Take the class boundaries or relevant class values on the horizontal axis.
- Take cumulative frequency on the vertical axis.
- Plot the required points.
- Join the points smoothly to obtain the Ogive.
Mean, Median and Mode – Quick Comparison
| Measure | How It Is Found | Main Use |
|---|---|---|
| Mean | Sum of observations ÷ Number of observations | Overall average |
| Median | Middle observation after arranging the data | Central position |
| Mode | Observation/class with highest frequency | Most frequent value |
| Ogive | Graph of cumulative frequency | Graphical analysis |
Important Properties to Remember
- The sum of deviations of observations from their arithmetic mean is always zero.
- If every observation is multiplied by a constant, the mean, median and mode are multiplied by the same constant.
- If a constant is added to every observation, the mean, median and mode increase by that constant.
- For an odd number of observations, the median is the middle observation after arranging the data.
- For an even number of observations, the median is the average of the two middle observations.
- The modal class is the class having the highest frequency, not the lowest frequency.
- Mean, median and mode are all measures of central tendency.
- An Ogive represents cumulative frequency graphically.
How to Solve Statistics Problems in Madhyamik Mathematics
Statistics questions become much easier when the data is organised before starting the calculation. First identify whether the question asks for mean, median, mode or an Ogive. Then select the appropriate formula or method.
- Read the data carefully and identify the required measure.
- Arrange raw observations in ascending order when finding the median.
- For frequency distributions, prepare a clear frequency table.
- Calculate the class mark when working with grouped data.
- Use Σfx/Σf for the direct mean method.
- For Ogive questions, calculate cumulative frequency accurately before plotting.
- Always write the final answer clearly with the correct unit where applicable.
Final Revision
Mean, Median, Ogive and Mode combines calculation with basic statistical interpretation. Students should be comfortable with direct mean calculations, median of raw data, transformation of mean/median/mode, mode-related relations and cumulative frequency.
For Madhyamik Mathematics preparation, practise both objective and numerical questions. Pay particular attention to arranging observations correctly, identifying the middle value and calculating frequency-based mean without arithmetic errors.