For a symmetrical frequency distribution we have :

Question

Accepted Answer

In a symmetrical frequency distribution, the mean, median, and mode are all equal. This is a fundamental property of symmetrical distributions, such as the normal distribution. Let's explore why Option 3 is correct and why the other options are incorrect: 1. **Explanation for Option 3 (Correct Answer):** - In a perfectly symmetrical distribution, the data is evenly distributed around the central point. This means that the measures of central tendency—mean, median, and mode—are all located at the center of the distribution. - The mean is the arithmetic average of all data points, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value. In a symmetrical distribution, these three measures coincide at the same point because the distribution is balanced on both sides. - Therefore, for a symmetrical frequency distribution, it is accurate to state that mean = mode = median. 2. **Explanation for why other options are incorrect:** - **Option 1 (mean > mode > median):** - This option suggests an asymmetrical distribution where the mean is greater than the mode and median. This typically occurs in a positively skewed distribution, not a symmetrical one. In a symmetrical distribution, all three measures are equal, not greater or lesser than each other. - **Option 2 (mean < mode < median):** - This option implies a negatively skewed distribution where the mean is less than the mode and median. Again, this does not apply to a symmetrical distribution, where the mean, median, and mode are equal. - **Option 4 (mode = 1/2(mean + median)):** - This option presents a formula that does not hold true for symmetrical distributions. In a symmetrical distribution, the mode is not calculated as the average of the mean and median; instead, all three values are identical. In conclusion, Option 3 is the correct answer because it accurately reflects the property of a symmetrical frequency distribution where the mean, median, and mode are equal. The other options describe scenarios that occur in skewed distributions, not symmetrical ones.