100 surnames were randomly picked up from a local telephone directly and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:
Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, fund the modal size of the surnames.
Consider the following table.
Here, the maximum frequency is 40 so the modal class is 7−10.
Therefore,
$l=7$
$h=3$
$f=40$
$f_{1}=30$
$f_{2}=16$
$\Rightarrow$ Mode $=l+\frac{f-f_{1}}{2 f-f_{1}-f_{2}} \times h$
$=7+\frac{10}{34} \times 3$
$=7+\frac{30}{34}$
Mode $=7.88$
Thus, the modal sizes of the surnames is 7.88.
Mean $=\frac{\sum f_{i} x_{i}}{\sum f}$
$=\frac{832}{100}$
Mean $=8.32$
Thus, the mean number of letters in the surnames is 8.32.
Median
$=l+\frac{\frac{N}{2}-F}{f} \times h$
$=7+\frac{50-36}{40} \times 3$
$=7+\frac{21}{20}$
Median $=8.05$
Thus, the median number of letters in the surnames is 8.05.
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