The balance point

The idea

Direct method: with class marks xi and frequencies fi, the mean of grouped data is x̄ = Σfixi/Σfi, where Σ (capital sigma) means summation over i from 1 to n.

Students on a seesaw

The 30 students are in six classes. Treat each student as one block of the same weight, sitting at the class mark of the class (17.5, 32.5, 47.5, …) on a long beam.

You may put the support (the pivot) anywhere under the beam. Where must it be so the beam does not tip?

Where do you think the beam will balance?

• At 55, the middle of the beam
• At 47.5, under the tallest stack
• Somewhere to the right of 55

Slide the pivot until it balances

Each stack is a class: its height is the number of students and it sits at the class mark. The left pull adds up (blocks × distance) on the left of the pivot, and the right pull does the same on the right. Slide the pivot until the two pulls are equal.

Balance point = mean

x̄ = Σfᵢxᵢ / Σfᵢ is the mean of grouped data, with class marks xᵢ and frequencies fᵢ. This is the direct method.

Class *i* puts fᵢ blocks at xᵢ, so its pull is fᵢ × xᵢ. Σ (capital sigma) means "add up for every class". So Σfᵢxᵢ is the total pull and Σfᵢ is the number of blocks. Dividing gives the balance point.

x̄ = 1860 / 30 = 62, exactly where your beam balanced.

• Class | fᵢ | xᵢ | fᵢxᵢ
• 10–25 | 2 | 17.5 | 35
• 25–40 | 3 | 32.5 | 97.5
• 40–55 | 7 | 47.5 | 332.5
• 55–70 | 6 | 62.5 | 375
• 70–85 | 6 | 77.5 | 465
• 85–100 | 6 | 92.5 | 555
• Total | 30 | 1860

Notes

x̄ = Σfᵢxᵢ / Σfᵢ. The mean of grouped data is the balance point of the blocks placed at the class marks.

Check yourself

Households were asked the number of family members. The class marks are 2, 4, 6, 8, 10 and the numbers of families are 7, 8, 2, 2, 1 (NCERT Example 5). Find the mean family size.

Answer: 4.2

Σfx = 14 + 32 + 12 + 16 + 10 = 84, Σf = 20, so x̄ = 84/20 = 4.2.

In Table 13.3 the class 55–70 has 6 students and class mark 62.5. Find fᵢxᵢ for this class.

Answer: 375

fᵢxᵢ = 6 × 62.5 = 375.

In Table 13.3, what is Σfᵢ?

For the percentage of female teachers in NCERT Example 2, Σfᵢxᵢ = 1390 and Σfᵢ = 35. Find the mean, correct to 2 decimal places.

Answer: 39.71

1390/35 = 39.714…, which is 39.71 to 2 decimal places.

• 30, the total number of students — correct. Yes! Σfᵢ = 2 + 3 + 7 + 6 + 6 + 6 = 30, the number of blocks on the beam.
• 1860, the total of fᵢxᵢ. That is Σfᵢxᵢ. Σfᵢ adds up only the frequencies.
• 6, the number of classes. Σfᵢ adds the frequencies of all the classes: 2 + 3 + 7 + 6 + 6 + 6.