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: $\begin{array}{|l|c|c|c|c|c|}\hline \text{Family size} & 1\text{-}3 & 3\text{-}5 & 5\text{-}7 & 7\text{-}9 & 9\text{-}11 \\ \hline \text{Number of families} & 7 & 8 & 2 & 2 & 1 \\ \hline \end{array}$ Find the mode of this data.

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) = $5 - 3 = 2$, $f_1$ (frequency of the modal class) = 8, $f_0$ (frequency of the class preceding the modal class) = 7, and $f_2$ (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 $l = 3$, $h = 2$, $f_1 = 8$, $f_0 = 7$, and $f_2 = 2$ 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.

More from this chapter

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