6.
100 surnames were randomly picked up from a local telephone directory and the frequency
distribution of the number ofletters in the English alphabet in the surnames was obtained
as follows
Number of letters
1-4
4-7
7-10
Number of surnames
10-13
13-16
16-19
6
30
40
16
4
4.
Determine the median number of letters in the surnames. Find the mean number of letters in the
surnames? Also, find the modal size of the surnames.
1.
IWANT PROPER EXPLAIN
Answers
Answer:
Let us prepare the following table to compute the median :
Number of letters Number of surnames (Frequency) Cumulative frequency
1−4 6 6
4−7 30 36
7−10 40 76
1−13 16 92
1−16 4 96
1−19 4 10=n
We have, n=100
⇒
2
n
=50
The cumulative frequency just greater than
2
n
is 76 and the corresponding class is 7–10.
Thus, 7–10 is the median class such that
2
n
=50l=7f=40c=36 and h=3
Substitute these values in the formula
Median, M=l+
⎝
⎛
f
2
n
−cf
⎠
⎞
×h
M=7+(
40
5−36
)×3
M=7+
40
14
×3=7+1.0=8.05
Now, calculation of mean:
Number of letters Mid-Point (x
i
) Frequency (f
i
) f
i
x
i
1−4 2.5 6 15
4−7 5.5 30 165
7−10 8.5 40 340
10−13 11.5 16 184
13−16 14.5 4 58
16−19 17.5 4 70
Total 100 832
Therefore, Mean,
x
ˉ
=
∑f
i
∑f
i
x
i
=
100
832
=8.32
Calculation ofMode:
The class 7–10 has the maximum frequency therefore, this is the modal class.
Here,
l=7,h=3,f
1
=40,f
0
=30 and f
2
=16
Now, let us substitute these values in the formula
Mode =l+(
2f
1
−f
0
−f
2
f
1
−f
0
)×h
=7+
80−30−16
40−30
×3
=7+
34
10
×3=7+0.88=7.88
- Hence, median =8.05, mean =8.32 and mode =7.88