Final answer:
To find the missing frequencies in the distribution, cumulative frequencies are calculated. P is confirmed as 34, and Q is determined to be 7 to maintain the total frequency at 230, while ensuring the median is within the correct class interval (40-50).
Step-by-step explanation:
The student has asked how to find the missing frequencies in a frequency distribution table when the median is known. Given the median is 46, we can conclude that half of the data values lie below this point. Therefore, we need to calculate the cumulative frequency to find out where the median lies within the data set.
To find the values of P and Q, we need to work out the cumulative frequencies. Since the total frequency is 230, and the median is 46, this implies that half of the values, which is 115 (230/2), would be at or below the median class (40 - 50). By adding the known frequencies up to the median class, we can determine P and adjust Q accordingly to ensure the total frequency remains 230.
We already know that before reaching the class where the median lies, we have a total frequency of 12 + 30 + P and P is given as 34. Therefore, the cumulative frequency before the median class is 76. This means that within the 40 - 50 class, there are 115 - 76 = 39 values that fall below the median. Since the 40 - 50 class already contains 65 values, this confirms that the median does indeed lie within this class, and our value for P is correct. The value of Q must then be adjusted to ensure the sum of all frequencies is 230. We have 76 (before the median class) + 65 (median class) + 39 (next class) + 25 + 18 = 223, which leaves us with Q = 230 - 223 = 7.