If xi ’s are the mid points of the class intervals of grouped data, f_i ’s are the corresponding frequencies and x̄ is the mean, then (f_ix_i - x̄) is equal to
a. 0
b. -1
c. 1
d. 2

Solution:

We know that

Mean (x) = Sum of all the observations/ Number of observations

Here

x = (f₁x₁ + f₂x₂ + …..+ f_nx_n) / f₁ + f₂ +……+ f_n

x = Σf_ix_i / Σf_i, Σf_i = n

x = Σf_ix_i / n

By cross multiplication

n x = Σf_ix_i --- (1)

So we get

Σ (f_ix_i - x) = (f₁x₁ - x) + (f₂x₂ - x)+ …..+ (f_nx_n - x)

Σ (f_ix_i - x) = (f₁x₁ + f₂x₂ + …..+ f_nx_n) - (x + x +….n times)

We can write it as

Σ (f_ix_i - x) = Σf_ix_i - nx

Σ(f_ix_i - x) = nx - nx (From equation 1)

Σ(f_ix_i - x) = 0

Therefore, \((f_{i}x_{i}-\overline{x})\) is equal to 0.

✦ Try This: Calculate mean \(\overline{x}\) when Σf_ix_i = 100 and Σf_i = 20.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 14

NCERT Exemplar Class 10 Maths Exercise 13.1 Problem 3

If x_i ’s are the mid points of the class intervals of grouped data, f_i ’s are the corresponding frequencies and x̄ is the mean, then (f_ix_i - x̄) is equal to a. 0, b. -1, c. 1, d. 2

Summary:

If x_i ’s are the mid points of the class intervals of grouped data, f_i ’s are the corresponding frequencies and x̄ is the mean, then (f_ix_i - x̄) is equal to 0

☛ Related Questions:

• In the formula x̄ = a + h fi ui/ fi, for finding the mean of grouped frequency distribution, ui = a. . . . .
• The abscissa of the point of intersection of the less than type and of the more than type cumulative . . . .
• For the following distribution: Class 0-5 5-10 10-15 15-20 20-25 Frequency 10 15 12 20 9 the sum of . . . .