Measures Of Central Tendency

27.5

Step 1: The data is already arranged in ascending order: 15, 20, 25, 30, 35, 40. Step 2: Count the observations: n = 6 (even). Step 3: For even number of observations, median = (n/2 th observation + (n/2 + 1) th observation) ÷ 2. Step 4: Find positions: 6/2 = 3rd and 4th observations. Step 5: 3rd observation = 25, 4th observation = 30. Step 6: Median = (25 + 30) ÷ 2 = 27.5.

76

Step 1: Calculate total marks: (8 × 70) + (12 × 80) = 560 + 960 = 1520. Step 2: Calculate total number of students: 8 + 12 = 20. Step 3: Apply mean formula: Mean = Total marks ÷ Total students. Step 4: Calculate: Mean = 1520 ÷ 20 = 76. Therefore, the mean score is 76 marks.

8

Step 1: Count frequency of each value. Step 2: Value 5 appears 3 times, value 6 appears 1 time, value 7 appears 2 times, value 8 appears 4 times, value 9 appears 1 time. Step 3: Compare frequencies to find the highest. Step 4: Value 8 has the highest frequency (4 occurrences). Therefore, the mode is 8.

Increases by 5

Step 1: Let original data be x₁, x₂, ..., xₙ with mean = (x₁ + x₂ + ... + xₙ)/n. Step 2: New data becomes (x₁+5), (x₂+5), ..., (xₙ+5). Step 3: New mean = [(x₁+5) + (x₂+5) + ... + (xₙ+5)]/n. Step 4: Simplify: = [(x₁ + x₂ + ... + xₙ) + 5n]/n = original mean + 5. Therefore, the mean increases by 5.