Question

The mean of the data

0-10	10-20	20-30	30-40	40-50
5	2	5	x	6

is 26 , then variance of the data is?

Answer 1

Given: Intervals: \( 0-10, 10-20, 20-30, 30-40, 40-50 \)

Frequencies: \( \{5, 2, 5, x, 6\} \), and the mean \( \text{Mean} = 26 \).

Step 1: Calculate the mean:

The formula for the mean is:

\( \text{Mean} = \frac{\sum f_i x_i}{\sum f_i} \),

where \( f_i \) are the frequencies and \( x_i \) are the midpoints of the intervals.

• Midpoints: \( x_i = \{5, 15, 25, 35, 45\} \)
• Frequencies: \( f_i = \{5, 2, 5, x, 6\} \)

Substitute into the formula:

\( 26 = \frac{5(5) + 2(15) + 5(25) + x(35) + 6(45)}{5 + 2 + 5 + x + 6} \).

Simplify the numerator:

\( 26 = \frac{25 + 30 + 125 + 35x + 270}{18 + x} \).

Simplify further:

\( 26(18 + x) = 450 + 35x \).

Expand and solve for \( x \):

\( 468 + 26x = 450 + 35x \),

\( 468 - 450 = 35x - 26x \),

\( 18 = 9x \implies x = 2 \).

Step 2: Variance formula:

\( \text{Variance} = \frac{\sum f_i x_i^2}{\sum f_i} - \text{Mean}^2 \).

• Updated frequencies: \( f_i = \{5, 2, 5, 2, 6\} \)
• Compute \( \sum f_i x_i^2 \):

\( \sum f_i x_i^2 = 5(5^2) + 2(15^2) + 5(25^2) + 2(35^2) + 6(45^2) \),

\( \sum f_i x_i^2 = 125 + 450 + 3125 + 2450 + 12150 = 18300 \).

• Total frequency: \( \sum f_i = 5 + 2 + 5 + 2 + 6 = 20 \)

Substitute into the variance formula:

\( \text{Variance} = \frac{18300}{20} - 26^2 \),

\( \text{Variance} = 915 - 676 = 239 \).

Final Answer: The variance of the data is \( 239 \).