Key Concepts and Formulas

• Measures of Central Tendency: These describe the center of a dataset. Examples include Mean, Median, and Mode.
  • Mode: The value that appears most frequently in a dataset. For grouped data, it's estimated from the class with the highest frequency (modal class). The formula for grouped data is: $\text{Mode} = L + \left( \frac{f_m - f_1}{2f_m - f_1 - f_2} \right) \times h$ where $L$ is the lower limit of the modal class, $h$ is the class width, $f_m$ is the frequency of the modal class, $f_1$ is the frequency of the preceding class, and $f_2$ is the frequency of the succeeding class.
  • Median: The middle value of an ordered dataset.
• Measures of Dispersion: These describe the spread or variability of a dataset.
  • Variance ($\sigma^2$): The average of the squared differences from the mean. For a dataset $X = \{x_1, \dots, x_n\}$ with mean $\mu_X$, the variance is $\sigma_X^2 = \frac{\sum (x_i - \mu_X)^2}{n}$.
• Effect of Linear Transformations ($Y = aX + b$):
  • If $M_X$ is a measure of central tendency (Mean, Median, Mode) for dataset $X$, then for $Y = aX + b$, the new measure $M_Y = aM_X + b$.
  • If $\sigma_X^2$ is the variance for dataset $X$, then for $Y = aX + b$, the new variance $\sigma_Y^2 = a^2\sigma_X^2$. Consequently, the standard deviation $\sigma_Y = |a|\sigma_X$.
  • Change of Origin: Adding/subtracting a constant ($b \neq 0, a=1$).
  • Change of Scale: Multiplying by a constant ($a \neq 1, b=0$).

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Step-by-Step Solution

Step 1: Analyze Statement (a) - Mode can be computed from histogram.

• What we are doing: We are determining if a histogram provides sufficient information to calculate the mode.

• Why: The mode is a fundamental measure of central tendency, and a histogram is a primary tool for visualizing frequency distributions.

• Explanation:

  1. A histogram graphically represents the frequency distribution of data. The height of each bar corresponds to the frequency of the respective class interval.
  2. For ungrouped data (where each bar represents a single value), the mode is simply the value corresponding to the tallest bar. Its exact value can be read directly.
  3. For grouped continuous data, we cannot find the exact mode directly. However, the histogram clearly identifies the modal class, which is the class interval with the highest frequency (the tallest bar).
  4. The mode for grouped data is estimated using the formula: $\text{Mode} = L + \left( \frac{f_m - f_1}{2f_m - f_1 - f_2} \right) \times h$ All the components of this formula—the lower limit of the modal class ($L$), the class width ($h$), the frequency of the modal class ($f_m$), and the frequencies of the classes preceding ($f_1$) and succeeding ($f_2$) the modal class—can be directly read or derived from the histogram.
  5. Since all necessary values are available from the histogram, the mode (or its estimate for grouped data) can indeed be computed.

• Conclusion: Statement (a) is correct.

Step 2: Analyze Statement (b) - Median is not independent of change of scale.

• What we are doing: We are examining how the median responds to a change in scale (multiplication by a constant).

• Why: Understanding the effect of transformations on statistical measures is crucial.

• Explanation:

  1. The median is the middle value of an ordered dataset. It is a measure of central tendency.
  2. Consider a dataset $X$ with median $M_X$. If we apply a linear transformation $Y = aX + b$ to create a new dataset $Y$, the new median $M_Y$ is related to $M_X$ by the same transformation: $M_Y = aM_X + b$.
  3. A change of scale specifically refers to multiplying each data point by a constant $a$ (i.e., $b=0$). So, $Y = aX$.
  4. In this case, the new median becomes $M_Y = aM_X$.
  5. If $a \neq 1$, then $M_Y \neq M_X$. This means the median changes when the scale changes. For example, if $X = \{1, 2, 3\}$, $M_X = 2$. If we apply a scale change by $a=10$, $Y = \{10, 20, 30\}$, then $M_Y = 20$. Clearly, $M_Y = 10 \times M_X$.
  6. Since the median's value changes with the scale factor $a$ (if $a \neq 1$), it is not independent of a change of scale.

• Conclusion: Statement (b) is correct.

Step 3: Analyze Statement (c) - Variance is independent of change of origin and scale.

• What we are doing: We are investigating how variance is affected by both change of origin (addition/subtraction) and change of scale (multiplication).

• Why: Variance is a measure of dispersion, and its behavior under transformations is different from measures of central tendency.

• Explanation:

  1. Variance ($\sigma^2$) measures the spread of data points around the mean.
  2. Effect of Change of Origin ($Y = X + b$):
     • Let $X = \{x_1, \dots, x_n\}$ with mean $\mu_X$ and variance $\sigma_X^2$.
     • Let $Y = \{y_1, \dots, y_n\}$ where $y_i = x_i + b$.
     • The new mean will be $\mu_Y = \mu_X + b$.
     • The deviation of each new data point from the new mean is: $(y_i - \mu_Y) = (x_i + b) - (\mu_X + b) = x_i - \mu_X$
     • Since the deviations remain unchanged, their squares and sum of squares also remain unchanged.
     • Therefore, the new variance $\sigma_Y^2 = \frac{\sum (y_i - \mu_Y)^2}{n} = \frac{\sum (x_i - \mu_X)^2}{n} = \sigma_X^2$.
     • Variance is independent of a change of origin.
  3. Effect of Change of Scale ($Y = aX$):
     • Let $Y = \{y_1, \dots, y_n\}$ where $y_i = ax_i$.
     • The new mean will be $\mu_Y = a\mu_X$.
     • The deviation of each new data point from the new mean is: $(y_i - \mu_Y) = (ax_i - a\mu_X) = a(x_i - \mu_X)$
     • Squaring this deviation: $(y_i - \mu_Y)^2 = [a(x_i - \mu_X)]^2 = a^2(x_i - \mu_X)^2$
     • The new variance $\sigma_Y^2 = \frac{\sum (y_i - \mu_Y)^2}{n} = \frac{\sum a^2(x_i - \mu_X)^2}{n} = a^2 \frac{\sum (x_i - \mu_X)^2}{n} = a^2 \sigma_X^2$.
     • Variance is not independent of a change of scale; it is multiplied by the square of the scale factor ($a^2$).
  4. Since the statement claims variance is independent of both change of origin and scale, but it is demonstrably not independent of change of scale, the statement is incorrect.

• Conclusion: Statement (c) is incorrect.

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Common Mistakes & Tips

• Distinguish Central Tendency vs. Dispersion: Measures of central tendency (mean, median, mode) transform linearly ($aM_X + b$), while measures of dispersion (variance, standard deviation) are affected differently. Variance is affected by $a^2$, and standard deviation by $|a|$.
• Variance and Scale Factor: Always remember the $a^2$ factor for variance when dealing with a change of scale. This is a very common point of error.
• "Computed" for Grouped Data: For grouped data, "computed" for mode often implies estimation using a specific formula, which relies entirely on information from the histogram.

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Summary

We have systematically analyzed each statement. Statement (a) is correct because a histogram provides all necessary information to compute (or estimate) the mode using the relevant formula for grouped or ungrouped data. Statement (b) is correct because the median changes proportionally with a change of scale (multiplication by a constant), meaning it is not independent of scale. Statement (c) is incorrect because while variance is independent of change of origin (addition/subtraction), it is not independent of change of scale; it gets multiplied by the square of the scale factor. Therefore, only statements (a) and (b) are correct.

The final answer is $\boxed{\text{C}}$, which corresponds to option (C).