1. **State the problem:** We are given a frequency distribution with a missing frequency and the mean of the data is 22. We need to find the missing frequency. 2. **Given data:** Central values (x): 10, 15, 20, 25, 30 Frequencies (f): 5, 10, 15, ?, 5 Mean (\(\bar{x}\)) = 22 3. **Formula for mean:** $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ 4. **Let the missing frequency be** $f$. 5. **Calculate the sum of frequencies:** $$\sum f_i = 5 + 10 + 15 + f + 5 = 35 + f$$ 6. **Calculate the sum of products of frequencies and values:** $$\sum f_i x_i = 5 \times 10 + 10 \times 15 + 15 \times 20 + f \times 25 + 5 \times 30$$ $$= 50 + 150 + 300 + 25f + 150 = 650 + 25f$$ 7. **Use the mean formula:** $$22 = \frac{650 + 25f}{35 + f}$$ 8. **Multiply both sides by** $35 + f$: $$22(35 + f) = 650 + 25f$$ $$770 + 22f = 650 + 25f$$ 9. **Rearrange to solve for** $f$: $$770 - 650 = 25f - 22f$$ $$120 = 3f$$ $$f = \frac{120}{3} = 40$$ **Final answer:** The missing frequency is **40**.