A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained:
Height (in cm)Less than 140Less than 145Less than 150Less than 155Less than 160Less than 165Number of girls41129404651
Find the median height.
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Step-by-Step Solution
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.