Q.The number of children in each of 40 families in a locality is distributed as follows — 0 children: 4 families, 1: 8, 2: 12, 3: 10, 4: 4, 5: 2. Calculate D4 and D7.

For a discrete series, a decile is located using the cumulative frequency column, exactly as for the median, but at position iN/10 instead of N/2.

Step 1 — Build the cumulative frequency (cf) column.

Step 2 — Locate D4. $k = \frac{4N}{10} = \frac{4(40)}{10} = 16$. The first cf ≥ 16 is 24, corresponding to X = 2.

Step 3 — Locate D7. $k = \frac{7N}{10} = \frac{7(40)}{10} = 28$. The first cf ≥ 28 is 34, corresponding to X = 3.

Cross-check via the equivalent percentiles. D4 = P40 and D7 = P70 (Section 5). Recomputing with $k=40N/100=16$ and $k=70N/100=28$ gives the identical cf-positions 16 and 28 — the same two answers, confirming the relationship D_i = P_(10i) holds exactly on this data.

Using N/10 (i.e., forgetting to multiply by i) instead of iN/10 gives the position for D1 instead of the decile actually asked for — always multiply by the specific decile number needed before dividing by 10.