The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :

Solution

Maths sir · 4 steps

Use the mean and variance to find a and b first, then the mean deviation is just a sum of distances.

1. Use the mean. The five numbers add to 5 × 6 = 30. Since 8 + 5 + 10 = 23, you get a + b = 7.
2. Use the variance. Variance = (Σ x²)/5 - 6² = 6.8, so Σ x² = 5 × 42.8 = 214. Taking away 64 + 25 + 100 = 189 leaves a² + b² = 25.
3. Find a and b. 2ab = (a+b)² - (a²+b²) = 49 - 25 = 24, so ab = 12. With a + b = 7, the two numbers are 3 and 4.

   M = (|3-6| + |4-6| + |8-6| + |5-6| + |10-6|)/5 = (3+2+2+1+4)/5 = 12/5

4. So. M = 2.4, and 25M = 25 × 2.4 = 60, which is option (a).

The idea to keep. Mean gives you a + b and variance gives you a² + b². Together they fix ab, and so the unknowns. Find every number before you compute any deviation.

Answer from the NTA final answer key. Solution by PrepWiser.