The mean height of three groups of students consisting of 20, 16 and 14 students are 1.67 m, 1.50 m and 1.40 m, respectively. Find the mean height of all the students. Calculate the mean, correct to t

For Group 1: $N_1 = 20$ students, $\bar{X}_1 = 1.67 \text{ m}$
For Group 2: $N_2 = 16$ students, $\bar{X}_2 = 1.50 \text{ m}$
For Group 3: $N_3 = 14$ students, $\bar{X}_3 = 1.40 \text{ m}$

Sum of heights for Group 1 = $20 \times 1.67 \text{ m} = 33.4 \text{ m}$
Sum of heights for Group 2 = $16 \times 1.50 \text{ m} = 24.0 \text{ m}$
Sum of heights for Group 3 = $14 \times 1.40 \text{ m} = 19.6 \text{ m}$
Total sum of heights = $33.4 \text{ m} + 24.0 \text{ m} + 19.6 \text{ m} = 77.0 \text{ m}$

The first two significant figures are 1 and 5. The next digit is 4, which is less than 5, so we round down.

The mean height of all the students, correct to two significant figures, is $\boxed{\textbf{1.5 m}}$.