The marks distribution of 30 students in a mathematics examination are given in the table below. Find the mode of this data.
Class interval10-2525-4040-5555-7070-8585-100TotalNumber of students (fi)23766630Class mark (xi)17.532.547.562.577.592.5
Get the complete, step-by-step math solution for: "The marks distribution of 30 students in a mathematics examination are given in the table below. Find the mode of this data. {|l|c|c|} Class interval ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the modal class
To find the mode for grouped data, we first need to identify the modal class. The modal class is the class interval with the highest frequency. In the given data, the highest frequency is 7, which corresponds to the class interval 40-55.
Step 2: Identify values for the mode formula
From the modal class 40-55, we identify the following values: l (lower limit of the modal class) = 40, h (class size) = 15 (which is 55−40), f1 (frequency of the modal class) = 7, f0 (frequency of the class preceding the modal class) = 3, and f2 (frequency of the class succeeding the modal class) = 6.
Step 3: Apply the mode formula for grouped data
Now we substitute the identified values into the formula for the mode of grouped data. This formula accounts for the distribution of frequencies around the modal class to give a more precise estimate of the mode.
Step 4: Substitute the values into the formula
We replace l, h, f1, f0, and f2 with their respective numerical values obtained in the previous step. This is the first step in calculating the mode numerically.
Step 5: Calculate the numerator of the fraction
First, we calculate the difference between the frequency of the modal class and the frequency of the preceding class, which is f1−f0. This gives us a value of 4.
Step 6: Calculate the denominator of the fraction
Next, we calculate the denominator of the fraction, which is 2f1−f0−f2. We perform the multiplication and subtractions to get a result of 5.
Step 7: Simplify the fraction
Now we substitute the calculated numerator and denominator back into the fraction. The simplified fraction is 54.
Step 8: Multiply the fraction by the class size
We multiply the simplified fraction by the class size h. This gives us 12, which is the adjustment to be added to the lower limit of the modal class.
Step 9: Add the result to the lower limit
Finally, we add the calculated adjustment to the lower limit of the modal class, l. This sum gives us the mode of the given data, which is 52.