Karl Pearson’s coefficient of skewness of a distribution is 0.32, its S.D is 6.5 and mean is 29.6. The mode and median of the distribution are

• ( A ) 27.52, 28.91
• ( B ) 26.92, 27.23
• ( C ) 25.67, 26.34
• ( D ) None of these

To find the mode and median of the distribution given Karl Pearson’s coefficient of skewness, we can use the formula for skewness, which is defined as:

Understanding Skewness

Karl Pearson’s coefficient of skewness (Sk) is calculated using the formula:

Sk = 3(Mean - Median) / Standard Deviation

In this case, we know:

• Sk = 0.32
• Mean = 29.6
• Standard Deviation (SD) = 6.5

Finding the Median

We can rearrange the skewness formula to solve for the median:

Median = Mean - (Sk * SD) / 3

Substituting the known values into the equation:

Median = 29.6 - (0.32 * 6.5) / 3

Calculating the product:

0.32 * 6.5 = 2.08

Now, dividing by 3:

2.08 / 3 = 0.6933

Finally, substituting back to find the median:

Median = 29.6 - 0.6933 = 28.9067

Rounding this, we get approximately 28.91.

Finding the Mode

Next, we can use the relationship between mean, median, and mode in a skewed distribution. The formula is:

Mode = 3 * Median - 2 * Mean

Substituting the values we have:

Mode = 3 * 28.91 - 2 * 29.6

Calculating this step-by-step:

• 3 * 28.91 = 86.73
• 2 * 29.6 = 59.2

Now, subtracting the two results:

Mode = 86.73 - 59.2 = 27.53

Rounding this gives us approximately 27.52.

Final Results

Based on our calculations, we find:

• Median ≈ 28.91
• Mode ≈ 27.52

Thus, the correct answer is option (A): 27.52 for the mode and 28.91 for the median.