Consider the following statements: (A) Mode can be computed from histogram (B) Median is not indepen

Q: Wait, if I multiply all data points by 2, the median also doubles. Doesn't that mean it *is* dependent on scale? Why does the statement say 'not independent'?

You have hit the nail on the head! 'Not independent' is a double negative that means 'dependent.' Since the median changes when you multiply by a constant, it is indeed dependent on the scale. You are thinking exactly like a statistician!

Q: Why is variance independent of origin? If I add 5 to every number, the values change, so shouldn't the variance change?

That is a classic trap! Variance measures the *spread* or the average squared distance from the mean. If you shift the whole dataset by 5, the distance between any two points remains exactly the same. The 'spread' is preserved, so the variance remains unchanged. Think of it as moving a rigid ruler; the length of the ruler doesn't change just because you moved it!

Q: Is the graphical method for finding the mode from a histogram always accurate?

It is a geometric construction that provides the exact mode for grouped data. It relies on linear interpolation between the modal class and its neighbors. While it is an approximation of the 'true' mode of the underlying continuous distribution, within the context of histogram analysis, it is the standard, accepted method.

Question

Consider the following statements: (A) Mode can be computed from histogram (B) Median is not independent of change of scale (C) Variance is independent of change of origin and scale. Which of these is / are correct?

Options:

(A) (A), (B) and (C)

(B) only (B)

(C) only (A) and (B)

(D) only (A)

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of Frequency: Statement (A)

Imagine you are looking at a histogram, which is a visual representation of a frequency distribution. The $x$-axis holds your class intervals, and the $y$-axis holds the frequency. The mode, by definition, is the value that appears most often.

In a histogram, this is the tallest bar, which we call the modal class. To find the exact value, we use a geometric construction. By drawing diagonal lines from the top corners of the modal bar to the corners of the adjacent bars, we create an intersection point.

Dropping a perpendicular from this intersection to the $x$-axis gives us the mode. This proves that the mode can indeed be computed from a histogram. Thus, **Statement (A) is correct**.

### The Stretching of the Number Line: Statement (B)

The median is the positional middle of your data. Imagine your data points are beads on a string; if you sort them, the median is the bead in the exact center.

Now, consider a 'change of scale,' which means multiplying every single observation by a constant $a$. If you have a set of data $x$ and you transform it to $y = ax$, you are essentially stretching the entire number line.

If the original median was $M$, the new median becomes $a 	imes M$. Because the median changes when we change the scale, it is not independent of the scale. Therefore, **Statement (B) is correct**.

### The Squared Deviation: Statement (C)

Variance measures the spread of data, calculated as the average of the squared deviations from the mean:

$$Var(X) = \frac{1}{n} \sum (x_i - \bar{x})^2$$

First, consider the change of origin: $y = x + b$. When you shift the entire dataset by $b$, the mean also shifts by $b$, meaning the distance between any two points $(x_i - \bar{x})$ remains constant. Thus, the variance is independent of the change of origin.

However, consider the change of scale: $y = ax$. When you multiply every value by $a$, the deviations also get multiplied by $a$. Because variance involves squared deviations, the new variance becomes:

$$Var(ax) = a^2 \cdot Var(x)$$

The variance is scaled by the square of the constant. Because it depends on the scale, the claim that it is independent of both origin and scale is false. Therefore, **Statement (C) is incorrect**.

### Conclusion

By dissecting these properties, we see that statistics is not just about memorizing formulas; it is about understanding how operations transform our data. We found that Statement (A) is correct, Statement (B) is correct, and Statement (C) is incorrect.

The correct choice is **only (A) and (B)**.

Animated Solution for Mathematics - Statistics: Consider the following statements: (A) Mode can be computed from histogram (B) Median is not independent of change of scale (C) Variance is independent of change of origin and scale. Which of these is / are correct?

Select Answer:

Visualized Solution

Statement (A): Mode and Histogram

Identifying the Modal Class

Graphical Computation of Mode

Statement (B): Median and Scale

Applying Change of Scale

Effect of Scale on Median

Statement (C): Variance and Origin

Variance and Change of Scale

Conclusion for Statement (C)

Final Conclusion

The Sigma Insight: Measures of Dispersion

Analyzing the Geometry of Frequency

The Stretching of the Number Line

The Squared Deviation

Conclusion

Similar Questions

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

Let the mean and variance of the frequency distribution $x : f : x_{1} = 2 4 x_{2} = 6 4 x_{3} = 8 α x_{4} = 9 β$ be 6 and 6.8 respectively. If $x_{3}$ is changed from 8 to 7, then the mean for the new data will be:

Let the mean of the data $x Frequency (f) 14324528 7 α 98$ be 5. If $m$ and $σ^{2}$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3 α}{m + σ ^{2}}$ is equal to _______.

The variance $σ^{2}$ of the data $x_{i} f_{i} 03125362106123173$ is ________.

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :

The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are $2, 3, 5, 10, 11, 13, 15, 21$ , then the mean deviation about the median of all the 10 observations is

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$ , then the variance of the first four observations in equal to

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :