A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained: $\begin{array}{|l|c|}\hline \text{Height (in cm)} & \text{Number of girls} \\ \hline \text{Less than 140} & 4 \\ \text{Less than 145} & 11 \\ \text{Less than 150} & 29 \\ \text{Less than 155} & 40 \\ \text{Less than 160} & 46 \\ \text{Less than 165} & 51 \\ \hline \end{array}$ Find the median height.

Step 1: Convert Cumulative Frequency to Frequency Distribution

The given data is in 'less than' cumulative frequency form. To find the median, we first need to convert this into a standard frequency distribution table with class intervals and their corresponding frequencies. The frequency for each class interval is obtained by subtracting the cumulative frequency of the preceding class from the cumulative frequency of the current class.

Step 2: Calculate N/2

The total number of observations, $N$, is 51. To find the median class, we calculate $N/2$, which is $51/2 = 25.5$. This value helps us locate the median class in the cumulative frequency column.

Step 3: Identify the Median Class

We look for the cumulative frequency that is just greater than or equal to $N/2 = 25.5$. In our table, the cumulative frequency 29 corresponds to the class interval 145 - 150. Therefore, 145 - 150 is the median class.

Step 4: Identify Values for Median Formula

From the median class 145 - 150, we extract the necessary values for the median formula: $L$ (lower boundary of the median class) is 145, $N$ (total frequency) is 51, $cf$ (cumulative frequency of the class preceding the median class) is 11, $f$ (frequency of the median class) is 18, and $h$ (class size) is $150 - 145 = 5$.

Step 5: Apply the Median Formula

Now we substitute the values into the median formula. This formula is used to calculate the median for grouped data, accounting for the distribution within the median class.

Step 6: Calculate the Median

Performing the calculations: First, we subtract $cf$ from $N/2$. Then, we divide this result by $f$ and multiply by $h$. Finally, we add this value to $L$ to obtain the median height. The median height is approximately 149.03 cm.

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