The mean and median of a data are respectively 20 and 22. The value of mode is :

Question

Accepted Answer

To determine the value of the mode given that the mean is 20 and the median is 22, we can use the empirical relationship between the mean, median, and mode in a moderately skewed distribution. This relationship is often expressed as:

$$ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} $$

Substituting the given values into this formula:

$$ \text{Mode} \approx 3 \times 22 - 2 \times 20 $$

$$ \text{Mode} \approx 66 - 40 $$

$$ \text{Mode} \approx 26 $$

Thus, the calculated mode is 26, which corresponds to Option 2. This supports the given correct answer.

Now, let's explain why each of the other options is incorrect:

1. **Option 1 (20):** This option suggests that the mode is equal to the mean. However, given the empirical relationship and the calculated mode of 26, it is clear that the mode is not 20. The mode being equal to the mean would typically indicate a symmetric distribution, which is not the case here as the median is greater than the mean.

2. **Option 3 (22):** This option suggests that the mode is equal to the median. While in some distributions the mode can be equal to the median, the empirical relationship calculation shows that the mode is 26, not 22. Therefore, this option is incorrect.

3. **Option 4 (21):** This option suggests a mode value that is neither supported by the empirical relationship nor by the given values of mean and median. The calculated mode is 26, making 21 an incorrect choice.

In conclusion, the correct answer is Option 2 (26), as it aligns with the empirical relationship between the mean, median, and mode for a moderately skewed distribution.