If Z = 10 and M = 20 the value of X̄ will be

Question

Accepted Answer

The correct option is 2:

This question relates to the empirical relationship between the three main measures of central tendency: the Mean ($\bar{X}$), the Median (M), and the Mode (Z), which often holds true for moderately asymmetrical distributions (or skewed distributions). The formula connecting these three measures is given by: **Mode = 3 Median - 2 Mean** ($Z = 3M - 2\bar{X}$). This relationship is vital in statistics for estimating a missing value when the other two are known, especially in situations where calculating the direct mean is difficult. Given that the Mode ($Z$) is 10 and the Median ($M$) is 20, we can substitute these values into the formula: $10 = 3(20) - 2\bar{X}$. Solving for the Mean ($\bar{X}$): $10 = 60 - 2\bar{X}$, which simplifies to $2\bar{X} = 50$, thus $\bar{X} = 25$.

The incorrect option is 1 (20):
This option incorrectly assumes that the Mean is equal to the Median. This would only be true for a perfectly symmetrical distribution, but since the Mode (10) and Median (20) are different, the distribution must be skewed, meaning the Mean, Median, and Mode will typically have different values.

The incorrect option is 3 (15):
This value results from substituting the data incorrectly or attempting to find a simple average, which is not applicable here. Using the established empirical relationship is the only mathematically sound way to estimate the missing Mean from the given Mode and Median.

The incorrect option is 4 (10):
This option incorrectly sets the value of the Mean equal to the Mode. As explained, in a skewed distribution, the values for the Mean, Median, and Mode usually separate, with the Median typically lying between the Mean and the Mode. Since the Mode is 10 and the Median is 20, the Mean must be greater than the Median to maintain the proper relationship, leading to a positively skewed distribution.