Calculate quartile deviation: Marks 50 60 70 80 90 : : : : : : No. of students 2 4 8 4 1

Calculation of Quartile Deviation

Quartile deviation (QD) is a measure of dispersion that is calculated as half the difference between the third quartile (Q₃) and the first quartile (Q₁). The formula is:

$$QD=\frac{Q_3-Q_1}{2}$$

Step 1: Find the Total Number of Students

The given data is:

Total number of students ($N$) = $2+4+8+4+1=19$.

Step 2: Find the Positions of Q₁ and Q₃

The positions of quartiles are calculated as:

$$\text{Position of }{Q}_1=\frac{N+1}{4}=\frac{19+1}{4}=5$$

$$\text{Position of }{Q}_3=\frac{3(N+1)}{4}=\frac{3(19+1)}{4}=15$$

Step 3: Find Q₁ and Q₃

We use the cumulative frequency distribution to locate the quartiles.

• Q₁ (5th position) falls in the 70 marks class (since cumulative frequency reaches 6 at 60 and 14 at 70). Since the 5th position lies in the 70 marks class, $Q_1=70$.

• Q₃ (15th position) falls in the 80 marks class (since cumulative frequency reaches 14 at 70 and 18 at 80). Since the 15th position lies in the 80 marks class, $Q_3=80$.

Step 4: Calculate Quartile Deviation

$$QD=\frac{Q_3-Q_1}{2}=\frac{80-70}{2}=\frac{10}{2}=5$$

Final Answer: The quartile deviation is 5.