A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household:
Family sizeNumber of families1-373-585-727-929-111
Find the mode of this data.
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Step-by-Step Solution
Step 1: Identify the modal class
The mode is the value that appears most often in a data set. For grouped data, the modal class is the class interval with the highest frequency. From the given frequency table, we need to identify which class interval has the highest 'Number of families'.
Step 2: Determine modal class parameters
The maximum frequency is 8, which corresponds to the class interval 3-5. Therefore, the modal class is 3-5. We then extract the following values: l (lower limit of the modal class) = 3, h (class size) = 5−3=2, f1 (frequency of the modal class) = 8, f0 (frequency of the class preceding the modal class) = 7, and f2 (frequency of the class succeeding the modal class) = 2.
Step 3: Apply the mode formula for grouped data
Now we use the formula for calculating the mode of grouped data. This formula helps us estimate the mode within the modal class, taking into account the frequencies of the adjacent classes. We will substitute the values identified in the previous step into this formula.
Step 4: Substitute values into the formula
Substitute the values l=3, h=2, f1=8, f0=7, and f2=2 into the mode formula. This gives us the expression we need to calculate to find the mode.
Step 5: Calculate the mode
Perform the arithmetic operations step-by-step. First, calculate the numerator and the denominator inside the parenthesis. Then, carry out the division and multiply by the class size. Finally, add the lower limit of the modal class to get the mode. We round the result to three decimal places.