Arithmetic Mean Calculation: Finding the Value of x

To determine the value of 'x' when the arithmetic mean of a given frequency distribution is zero, we need to apply the formula for the arithmetic mean of grouped data. The arithmetic mean ($\bar{x}$) for a frequency distribution is calculated as the sum of the products of each number and its frequency, divided by the total sum of frequencies.

Understanding the Arithmetic Mean Formula

The formula for the arithmetic mean ($\bar{x}$) of a frequency distribution is given by:

$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

Where:

• $x_i$ represents each individual number or data point.
• $f_i$ represents the frequency of each corresponding number.
• $\sum f_i x_i$ is the sum of the products of each number and its frequency.
• $\sum f_i$ is the total sum of all frequencies.

In this problem, we are given that the arithmetic mean ($\bar{x}$) is zero.

Setting Up the Data for Calculation

Let's list the given numbers ($x_i$) and their corresponding frequencies ($f_i$):

Calculating the Total Sum of Frequencies ($\sum f_i$)

First, we sum all the frequencies:

$$\sum f_i = 3 + 7 + 11$$

$$\sum f_i = 21$$

Calculating the Sum of Products ($\sum f_i x_i$)

Next, we calculate the product of each number and its frequency, and then sum these products:

• For the first data point: $f_1 x_1 = 3(x + 3) = 3x + 9$
• For the second data point: $f_2 x_2 = 7(x - 7) = 7x - 49$
• For the third data point: $f_3 x_3 = 11(x - 4) = 11x - 44$

Now, we sum these products:

$$\sum f_i x_i = (3x + 9) + (7x - 49) + (11x - 44)$$

Combine the 'x' terms and the constant terms:

$$\sum f_i x_i = (3x + 7x + 11x) + (9 - 49 - 44)$$

$$\sum f_i x_i = 21x - 84$$

Solving for the Value of x

We are given that the arithmetic mean ($\bar{x}$) is zero. Using the formula:

$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

Substitute the calculated values into the formula and set the mean to zero:

$$0 = \frac{21x - 84}{21}$$

To solve for 'x', multiply both sides by 21:

$$0 \times 21 = 21x - 84$$

$$0 = 21x - 84$$

Add 84 to both sides of the equation:

$$84 = 21x$$

Finally, divide by 21 to find the value of 'x':

$$x = \frac{84}{21}$$

$$x = 4$$

Thus, the value of 'x' is 4 when the arithmetic mean of the given data is zero.