Understanding Mean, Median, and Mode
In statistics, the mean, median, and mode are measures of central tendency used to describe a dataset.
- Mean: The average of all values, calculated by summing all values and dividing by the number of values.
- Median: The middle value in a dataset when arranged in ascending or descending order.
- Mode: The value that appears most frequently in a dataset.
The Empirical Relationship
For moderately skewed distributions, there is an approximate empirical relationship between the mean, median, and mode. This relationship is often stated as:
Mode = 3 × Median - 2 × Mean
Alternatively, it can be expressed as:
Mean - Mode = 3 × (Mean - Median)
This formula helps us estimate one measure if we know the other two.
Applying the Formula to the Problem
We are given:
- Median = 4
- Mode = 6
We need to find the Mean.
Let's use the first version of the empirical relationship formula:
Mode = 3 × Median - 2 × Mean
Calculation Steps
Substitute the known values into the formula:
$6$ = 3 × $4$ - 2 × Mean
Now, let's solve for the Mean:
- Calculate 3 × 4:
$6$ = 12 - 2 × Mean - Rearrange the equation to isolate the term with Mean:
2 × Mean = 12 - $6$ - Perform the subtraction:
2 × Mean = $6$ - Divide by 2 to find the Mean:
Mean = $6$ / 2 - Final calculation:
Mean = $3$
Conclusion
Using the empirical relationship for skewed distributions, the calculated Mean value is 3.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
