Identifying the Statistical Measure That Cannot Be Negative

The question asks to identify which of the given statistical measures cannot have a negative value.

Reasoning for Standard Deviation

Standard Deviation measures the amount of variation or dispersion in a set of values. It is calculated as the square root of the variance.

• Variance is the average of the squared differences from the Mean. Since the differences are squared, the variance is always non-negative (zero or positive).
• The formula for variance ($\sigma^2$) involves summing squared terms: $\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}$.
• Standard Deviation ($\sigma$) is the square root of variance: $\sigma = \sqrt{\sigma^2}$.
• Because the variance is always non-negative ($\sigma^2 \geq 0$), its square root, the standard deviation, must also be non-negative ($\sigma \geq 0$). It can be zero only if all data points are identical.

Why Other Measures Can Be Negative

• Median: The median is the middle value of a dataset. If the dataset contains negative numbers, the median can be negative. For example, the median of {-5, -2, 1} is -2.
• Mean: The mean (average) is calculated by summing all values and dividing by the count. If a dataset has more negative values than positive values, or if the negative values are larger in magnitude, the mean can be negative. For example, the mean of {-5, -2, 1} is $\frac{(-5) + (-2) + 1}{3} = \frac{-6}{3} = -2$.
• Skewness: Skewness indicates the asymmetry of the data distribution. It can be positive (right-skewed), negative (left-skewed), or zero (symmetric). A negative skewness value indicates that the tail on the left side of the probability distribution is longer or fatter than the right side.

Therefore, Standard Deviation is the only measure among the options that inherently cannot take a negative value.

Conclusion

The statistical measure that CANNOT take a negative value is Standard Deviation.