The formula for the mean is x = (Σ f_i x_i)/(Σ f_i). We substitute the given mean x = 14, Σ f_i = 24 + k, and Σ f_i x_i = 360 + 10k into the equation.

Example 2. The mean of the following data is $14$ . Find the value of $k$ . $\begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline f & 7 & k & 8 & 4 & 5 \\ \hline \end{array}$

Answer: $k = 6$

Step-by-step solution

Step 1: Construct the calculation table for direct method

To find the mean, we multiply each value $x_i$ by its corresponding frequency $f_i$ to get $f_i x_i$ . Adding the frequencies gives $\sum f_i = 7 + k + 8 + 4 + 5 = 24 + k$ . Adding the products gives $\sum f_i x_i = 35 + 10k + 120 + 80 + 125 = 360 + 10k$ .

Step 2: Apply the mean formula

The formula for the mean is $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ . We substitute the given mean $\bar{x} = 14$ , $\sum f_i = 24 + k$ , and $\sum f_i x_i = 360 + 10k$ into the equation.

Step 3: Solve for k

Cross-multiplying gives $14(24 + k) = 360 + 10k$ , which simplifies to $336 + 14k = 360 + 10k$ . Transposing $10k$ to the left side and $336$ to the right side yields $14k - 10k = 360 - 336$ , so $4k = 24$ . Dividing both sides by $4$ gives $k = 6$ .

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Example 2. The mean of the following data is 141414. Find the value of kkk. x510152025f7k845\begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline f & 7 & k & 8 & 4 & 5 \\ \hline \end{array}xf​57​10k​158​204​255​​

Step-by-step solution

Step 1: Construct the calculation table for direct method

Step 2: Apply the mean formula

Step 3: Solve for k

Example 2. The mean of the following data is $14$ . Find the value of $k$ . $\begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline f & 7 & k & 8 & 4 & 5 \\ \hline \end{array}$